Laws of Exponents:
1.
aman am
2.
=
an
3. 4.
= am +n
QUADRATIC EQUATI!
a m-n
"enerall#$ an e%&ation is said to be o' "#adratic form i' it has the 'orm ax$n % &xn % c ' ( $ (here n is an integer or a 'ra)tion* s&)h as + 4 , 5+2 6 = 0 and
(a m ) n = a mn
#-3 #-32 6 = 0
( ab) n = a nb n
/&adrati) orm&la:
X
= − b ±
n
a = a n b b n
5.
n
an
=a
n
ab
=
3.
n
4.
m n
a
n
n
a
n
b
=
mn
b a
trinomial) 3. hen b2 – 4ac < 0, the roots are imaginary and unequal complex conjugates
he roots: a
X1
1.
log b MN
2.
logb
= log b M +log bN
M = logbM - logb N N logbMN =NlogbM
Important Properties :
1: a 0 2: 3: 4: 5: 6: 7:
a
-n
provided a ≠ 0
=1 1 =
a
n
m
an
=
n
a
m
=
( a) n
m
or e lnb = b a = a implies that m = n log M N implies that M = bN b log M logbN implies that M = N b log bN : log aN = log b a log b 1 !: provided b > 0, b ≠ 1 b log 1 0 = 10: provided b > 0, b ≠ 1 b a logab m
=
=
− b +
b 2
− 4ac
X2
2a
b =− −
b
2
− 4ac
2a
&m &m o' o' the the root roots$ s$ +1 +2 = - ba
Laws of Logarithms:
3.
− 4ac
2a
1. hen b2 – 4ac > 0, the roots are real and unequal . 2. hen b2 – 4ac = 0, the roots are real and equal or quadratic equation is a perfect an b
=
2
he e+pression b2 – 4ac is is )alled the discriminant
Laws of Radicals:
1. 2.
b
b
n
=
=
=
L,E-T C++! +ULTIPLE .LC+/
he lo(est )ommon m<iple
rod&)t o' the roots$ + 1 +2 = )a T)E *I!+IAL T)ERE+
89NM9;< + # 0 = 1 + B -# + # 1 = + # 2 = + # 3 = + # 4 = + # 5 = + # 6 =
>;N9N 1 +# +2 2+# #2 +3 3+2# 3+# 2 #3 +4 4+3# 6+2#2 4+#3 #4 +5 5+4# 10+3#2 10+2#3 5+#4 #5 +6 6+5# 15+4#2 20+3#3 15+2#4 6+#5 #6
;?;<@ A9;N"< 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 1
his arra# o' n&mbers is )alled the as)al@s riangle. ;n# lo(er ro( is 'ormed b# adding an# t(o adCa)ent n&mbers o' the &pper ro( and pla)e 1 at both ends so as to 'orm a triangle. as)al@s riangle is &sed to easil# re)all the n&meri)al )oe''i)ients o' the e+pansion o' the po(ers o' a binomial. 8&t 'or large po(ers o' a binomial$ as)al@s riangle be)omes in)onvenientl# to &se. or s&)h$ &se 8inomial heorem. he rth term o' + # n =
Arithmetic +ean
nn-1n-2Dn-r2 r , 1E
+n-r1#r-1
he arithmeti) mean bet(een t(o n&mbers is the n&mber (hi)h (hen pla)ed bet(een
L,E-T C++! +ULTIPLE .LC+/
he lo(est )ommon m<iple
Arithmetic +ean
he arithmeti) mean bet(een t(o n&mbers is the n&mber (hi)h (hen pla)ed bet(een the t(o n&mbers$ 'orms (ith them an arithmeti) progression.
9n general$ 'or n terms$ arithmeti) mean .A+/ = a1 a2 a3 D a n n 0eometric Progression .02P2/
-
)I0)E-T C++! 1ACTR .)C1/
he highest )ommon 'a)tor 'a)tor G? o' several nat&ral n&mbers n&mbers is the the largest nat&ral n&mber (hi)h is a 'a)tor 'a)tor o' o' all the given n&mbers. 9t ma# be 'o&nd b# taFing the prod&)t o' o' all the di''erent prime 'a)tors )ommon to the given n&mbers$ ea)h taFen the smallest n&mber o' times that it o))&rs in an# o' those n&mbers. 9' the given n&mbers have no prime 'a)tors in )ommon$ the G? is de'ined to be 1$ in this )ase the n&mbers are said to be relativel# prime.
The nth term6 a n
a se%&en)e o' terms in (hi)h ea)h term a'ter the 'irst is 'o&nd b# m<ipl#ing the pre)eding term b# a 'i+ed n&mber )alled )ommon ratio. he se%&en)e a1$ a2$ a3 are in ".. i' and onl# i': a2a1 = a3a2 = r
an = a1r n-1
-#m of the first n terms in 02P2
n = a1 1-r n 1-r (here a1 = 'irst term r = )ommon ratio n = n&mber o' terms
+ample: ind the highest )ommon 'a)tor o' 24$ 30$ 1 and 150. ol&tion: 24 = 2+2+2+3$ 30 = 2+3+5$ 1 = 2+3+3$ 150 = 2+3+5+5 G? = 2+3 = 6
Infinite 0eometric Progression
he s&m o' terms in geometri) progression )an be 'o&nd i' the )ommon ratio J r JK1$ -1 K r K 1
PR0RE--I! Arithmetic Progression .A2 P2/
a se%&en)e o' terms in (hi)h ea)h term a'ter the 'irst is obtained b# adding a 'i+ed n&mber to the pre)eding term. - a se%&en)e o' terms in (hi)h an# t(o )onse)&tive terms has a )ommon di''eren)e. hat is$ the se%&en)e a1$ a2$ a3 are in arithmeti) progression i' and onl# i':
5
-
a$ 3 a4 ' a5 3 a$
hat is$ a1$ a2$ a3Dan are in harmoni) progression 9' 1a1$ 1a2$ 1a3D1an 'orm an arithmeti) progression
=
a1
1
−
r
"eometri) Mean he term in bet(een the 'irst and last terms o' the geometri) se%&en)e.
geometri) progression
hen$ +a1 = a2+ → )ommon ratio olving 'or +: +2 = a1a2 +=
a1 a2
→
geometri) progression
)armonic Progression
e%&en)e o' terms (hose re)ipro)al 'orms an arithmeti) progression
52 7ARIATI! i. Direct Variation also dire)t proportion
hat is$ a1$ a2$ a3Dan are in harmoni) progression 9' 1a1$ 1a2$ 1a3D1an 'orm an arithmeti) progression
52 7ARIATI! i. Direct Variation also dire)t proportion
he ive tatements 8elo( Gave ame Meaning "s x increases y increase proportionately y is proportional to x y is directly proportional to x y #aries as x y #aries directly as x
)armonic +ean
9n s#mbols the above statements mean$
1a$ 1+$ 1b → in ;. .
y $ x
hen$ olving 'or +:
9n mathemati)al terms$ # = F+ (here F is )alled the )onstant o' proportionalit# proportionalit# or also )alled the )onstant variation
2+ = 1a 1b
ii. Inverse Variation also indire)t variation
1+ , 1a = 1b , 1+ → )ommon di''eren)e
o'
he 'ollo(ing tatements 8elo( Gave ame Meaning
2+ = a bab
"s x decreases y increase %and #ice #ersa) y is in#ersely proportional to x y #aries indirectly as x
+ = 2aba b )armonic +ean .)+ =
9n s#mbols the above statements mean$
L n 1a1 1a2 1a 3 D 1a n
RATI6 PRPRTI! A!D 7ARIATI! 42 RATI
he ratio o' a n&mber a to another n&mber b is the 'ra)tion ab &s&all# as a:b read a is to b. here a is )alled antecedent and and b is )alled consequent
y $ &'x
9n mathemati)al terms$ # = F+$ + not e%&al to Qero +amples 1. 8o#le@s
$2 PRPRTI!
roportion is a statement that t(o ratios are e%&al. Os&all# (ritten as a:b = ):d or ab = )d. here a and d are )alled the extremes and b and ) are )alled the means. P is )alled the fourt! proportional to to a$ b$ and ). 9' the means o' a proportion are e%&al$ as in a+ = +b$ the n&mber b is )alled the t!ird proportional to to a and +$ (hile the n&mber + is )alled the mean proportional bet(een bet(een a and b. 9t is obvio&s that the mean proportional bet(een a and b is e%&al to their geometri) mean. ; proportion ma# be altered in 'o&r di''erent (a#s s&mmariQed in the tab&lar 'orm belo(. 8asi) roportion a:b = ):d
rans'ormation b# ;lternation 9nversion ;ddition &btar)tion
rans'ormed orm a:) = b:d b:a = d:) ab:b = )d:d )d:d a-b:b = )-d:d
-the term ?+ n#n is said to be o' the degree n in terms o' +$ degree p in terms o' #$ and degree np in terms o' + and #.
•
Pol8nomial 1#nction
iii. Joint Variation y #aries jointly as x and (
9n s#mbols$
y $ x(
Mathem Mathemati ati)al )all#$ l#$
R = F+(* F+(* arnin arning: g: N Not ot # = F+( F+(
EQUATI! 1 T)E )I0)ER DE0REE • Rational integral term , a )onstant$ or a positive integral po(er o' an# variable$ or the
prod&)t o' s&)h %&alities. +. 5$ 2+4$ - 6#3$ 15+2#5
•
Degree of a term
-the term ?+ n$ (here ? is a )onstant and n is a positive integer is said to be o' degree n in terms o' +.
i' '+ is divided b# + , r$ the remainder is 'r.
•
1actor theorem
-the term ?+ n#n is said to be o' the degree n in terms o' +$ degree p in terms o' #$ and degree np in terms o' + and #.
•
Pol8nomial 1#nction
-;n algebrai) s&m o' rational integral term -; series o' a po(er o' a base (here he e+ponents are positive integer -;lso )alled rational integral '&n)tion. '+ = a0+n a1+n-1 a2+n-2 D a n pol#nomial '&n)tion in degree n (here: (here: n = non-ne non-negat gative ive intege integer r a0$ a1$ a2$D.$an are an# )onstants a0 B 0
•
• • •
0.
•
•
a he n&mber n&mber o' positive ve real roots o' '+ '+ is either e%&al e%&al to the n&mber n&mber o' variations variations in sign o' '+$ or that n&mber diminished b# a positive integer. b he n&mber negative negative (ith real Qeros o' '+ is either e%&al to the n&mber n&mber o' variations ons in sign o' '-+ or that n&mber diminished b# positive even integer.
)onsider that the roots r 1$r 2$ r 3$ D$ r n o' '+ = 0 are e%&al i' '+ '+ is e+a)tl# divisible visible b# + , r b&t not b# + , r2$ then + , r is simple root o' '+ = 0. 9' '+ = 0 is e+a)tl# divisible + ,r 2 b&t not b# + , r 13$ then r 1 is )alled do&ble root o' '+ = 9n general$ i' '+ = 0 is divisible b# + , r F b&t not b# + , r 1F1$ then r1 is a F , 'old root o' order F. +#ltiplicit8
Theorem of complex roots
Theorem on "#adratic s#rd roots
s&rd root al(a#s o))&r in pair hat is$ i' a Sbi is a root$ then a ,Sb is also a root (here a and b are rational and Sb is irrational.
•
•
Relationship *etween Coefficients and 9eros of Pol8nomial Pol8nomial
"iven an integral rational '&n)tion: '+ = a0+n a1+n-1 a2+n-2 D a n-1+ an
+#ltiple roots
)omple+ roots al(a#s o))&r in pair. hat is i' b = bi is root$ then a , bi is also a root -)onC&gate Qeros (here a$ b are real b&t bB0
•
Descartes; r#le of signs
!#m&er of roots
the n&mber o' times that r appears as roots o' '+ = 0
•
Depressed e"#ation
i' r is a root o' the e%&ation '+ = 0$ then + , r is a 'a)tor o' '+. h&s$ '+ = + , r. he e%&ation /+ = 0 is )alled depressed e%&ation o' '+ = 0$ and the roots o' /+ = 0 are the remaining roots o' '+ = 0
1#ndamental theorem
ever# e%&ation '+ = 0 o' degree n$ has n roots and no more i' root o' order F is )o&nted as F roots.
•
Conerse of factor theorem
i' + , r is a 'a)tor o' '+$ then r is a root o' '+ = 0
ever# e%&ation '+ = 0 has at least one root there e+ist at least one n&mber n&mber either real or )omple+ (hi)h )h (ill satis'# '+ =0
•
1actor theorem
i' r is a root o' the e%&ation '+$ then + , r is 'a)tor o' '+.
9ero of a f#nction
an# val&e o' the &nFno(n +$ that (ill maFe a '&n)tion '+ e%&al to Qero also )alled root o' '+ = 0
•
i' '+ is divided b# + , r$ the remainder is 'r.
Remainder theorem
. hat is the the s&m and prod&)t prod&)t o' the the roots o' the the e%&ation e%&ation 3+ 4 , 2+2 + , 6 = 0T ;ns(er: s&m = 0* prod&)t = -2 3
2
he )oe''i)ients 'or the pol#nomial '&n)tion in terms o' its Qeros )an be given as: -a1 a0= s&m o' roots a2 a0 = s&m o' prod&)t o' the roots taFen t(o at a time -a3 a0 = s&m o' prod&)t o' the roots taFen three at a time -1n an a0 = prod&)t o' all roots -#pplementar8 Pro&lems
1. Peterm Petermine ine m so so that that +3 , 2+2 m+ shall be divisible b# + 3. ;ns. m = ! 2. 9' 3+4 = F+ 3 + 2 , 16 4 is divided b# + , 2$ 'or (hat val&e o' F (ill the remainder be T ;ns. F = 2 3. or (hat (hat val&e val&e o' F (ill (ill + = 3 be a 'a)tor 'a)tor o' + 3 7+2 F+ , 12T ;ns. F = 4. ind ind the rema remaind inder er (hen (hen 14! 14!++15!2 , 375+375 10 is divided b# + 1. ;ns. 177 5. ne o' the roots roots o' o' 3+ 3+3 , m+2 23+ , 14 = 0 is 2. Petermine the val&e o' m. m. ;ns(er: m = 14 6. 2+3 ,+2 m+ n is to be e+a)tl# e+a)tl# divisib divisible le b# + 2 , 2+ , . Petermine the val&es o' m and n. ;ns. m = -2$ n = -24 7. he s&m o' the the t(o t(o roots roots o' + 3 6+ ) = 0 is 2* 'ind ). ;ns. ) = 20
7. (o prime n&mbers n&mbers (hi)h di''er di''er b# 2 are )alled )alled prime t(ins. t(ins. hi)h o' the 'ollo(ing 'ollo(ing pairs o' n&mbers are prime t(insT a. 1$3 b. 13$ 15 ). 7$! d. !$11
. hat is the the s&m and prod&)t prod&)t o' the the roots o' the the e%&ation e%&ation 3+ 4 , 2+2 + , 6 = 0T ;ns(er: s&m = 0* prod&)t = -2 !. "iven "iven the e%&ati e%&ation on +3 , 4+2 3+ , 5 = 0. 'rom the e%&ation (hose roots are : a negative negative o' the the roots roots o' the given given e%&ation e%&ation b thri)e thri)e the roots roots o' the given given e%&atio e%&ationn ) the roots roots o' the the given e%&atio e%&ationn diminished diminished b# b# 2. ;ns(er: a +3 4+2 3+ 5 = 0 b +3 12+2 27+ , 135 = 0 ) +3 2+2 , + , 7 = 0 10. he s&m o' the the roots roots o' 2+ 3 m+2 , 5+ - 3 = 0 e%&als t(i)e the prod&)t o' the roots. Petermine m. ;ns. m = -6 11. he prod&)t prod&)t o' the t(o roots o' the e%&ation +3 5+ 12 = 0 is 4. ind the third root. ;ns. -3 12. he s&m o' o' the roots roots 3+2 , 2m+2 4 = 0 is 6. ind m. ;ns. m = !
+I-CELLA!EU- QUE-TI!-
1. ; set o' elements that is taFen (itho&t regard to the order in (hi)h the elements arranged is )alled: a. )ombination b. se%&en)e ). perm&tation d. series
.
; relat relatio ionn in (hi)h (hi)h ever# ever# ordere orderedd pair pair +$# +$# has one and and onl# onl# one val&e val&e o' # )orresponds to the val&es o' + is )alled a. term b. '&n)tion ). )oordinated d. abs)issa
!.
oss ossin ingg a )oi )oinn is gene genera rallll## )all )alled ed a. an e+periment b. an event
10.
; pol#nom pol#nomial ial (hi)h (hi)h is is e+a)tl# e+a)tl# divisi divisible ble b# t(o or or more more pol#nomia pol#nomials ls is )alled )alled as: a. least )ommon denominator b. )ommon m<iple ). 'a)tors d. binomial
11.
he roots roots o' the the e%&a e%&atio tionn 2+2 , 13+ 20 = 0 are a. real and e%&al b. real and &ne%&al ). )omple+ and e%&al d. )omple+ and &ne%&al
12.
a)h o' o' t(o or or more n&mbers n&mbers (hi)h (hi)h is m<ip m<iplied lied toget together her to 'orm 'orm a prod&)t prod&)t is )alled ed a. term b. m<iplier ). Filogram d. Filo(att
13.
9n the 9 &nit$ &nit$ the the small letter letter F means Filo lo (hile the the )apital )apital letter letter U means a. Uilometer b. Uelvin ). Uilogram d. Uilo(att
are
2.
hen a logarith logarithm m is e+pres e+pressed sed as an in integer teger pl&s a de)imal mal bet(ee bet(eenn 0 and 1$ the integer is )alled the a. 8riggs
3.
;n# positi positive ve integ integers ers as as 1$ 2$ 2$ 3$ et). et). is also also )alle )alledd a. Aeal N&mber ). Nat&ral N&mber b. Aational N&mber d. 9rrational N&mber
4.
he set o' intege integers rs that that does does not not sa satis' tis'## the )los&re )los&re propert propert## &nder &nder the the operation operation o' a. addition b. s&btra)tion ). m<ipli)ation d. division
). an o&t)ome
d. a trial
14. ;n# n&mber n&mber that that )an )an be e+presse e+pressedd as a %&otien %&otientt o' t(o t(o integers integers divisi division on o' Qero e+)l&ded is )alled a. irrational n&mber ). rational n&mber b. ima imagi gina nar# r# n&m n&mbe berr d. odd odd n&m n&mbe ber r
5. ;n e%&ation e%&ation (hi)h is satis'ied satis'ied b# some$ b&t not all$ o' the val&es val&es o' the variables variables (hi)h the members o' the e%&ation are not de'ined is )alled a a. linear e%&ation ). rational e%&ation b. )onditional )onditional e%&ation e%&ation d. irrational irrational )ondition tion
'or
6. ;n e%&ation e%&ation (hi)h is satis' satis'ied ied b# all all o' o' the the val&es val&es o' the variables variables 'or (hi)h the members o' the e%&ation are de'ined is a. linear e%&ation ). rational e%&ation b. )onditiona )onditionall e%&ation e%&ation d. irrational irrational e%&atio e%&ationn
1!. ; s&))ession s&))ession o' n&mbers n&mbers in (hi)h (hi)h one n&mber n&mber is designate designatedd as 'irst$ 'irst$ se)ond$ another as third and so on is )alled a a. series ). order o' n&mbers
7. (o prime n&mbers n&mbers (hi)h di''er di''er b# 2 are )alled )alled prime t(ins. t(ins. hi)h o' the 'ollo(ing 'ollo(ing pairs o' n&mbers are prime t(insT a. 1$3 b. 13$ 15 ). 7$! d. !$11
another anot her
as
15.
; re)tang&l re)tang&lar ar arra# o' o' n&mbers n&mbers 'orming 'orming m ro(s ro(s and n )ol&mns )ol&mns are are )alled )alled as a. determinants ). elements b. as)al@s triangle d. none o' the above
16. 16.
9n the the e+p e+pre ress ssio ionn nSa$ the letter n represents the a. po(er b. order ). e+ponent
d. radi)and
17. ; n&mber n&mber o' o' the the 'orm a bi (ith a and and b real real )onstants )onstants and i is s%&are s%&are root o' -1 is )alled a. imagin imaginar# ar# n&m n&mber ber ). )omple )omple++ n&mb n&mber er b. radi)al d. )ompo&nd n&mber 1.
hi)h o' o' the 'ollo( 'ollo(ing ing nontermi nonterminating nating de)imals de)imals is rational rational a. 3.1415!265D ). 2.470470D b. 2.71212D d.1.141421356
35.
9' 3+ = !# and 27# = 1Q$ then is e%&al to a. 37 b. 35
). 34
d. 3
1!. ; s&))ession s&))ession o' n&mbers n&mbers in (hi)h (hi)h one n&mber n&mber is designate designatedd as 'irst$ 'irst$ se)ond$ another as third and so on is )alled a a. series ). order o' n&mbers b. arrangement d. se%&en)e 20.
another anot her
as
;n e%&ation e%&ation in (hi)h a variabl variablee appears appears &nder &nder a radi)al radi)al sign is )alled )alled a. irradi)al e%&ation ). irrational e%&ation b. %&adrati) e%&ation d. linear e%&ation the the
22. 9' 14 and , 72 are the roots o' the %&adrati) e%&ation ;+2 8+ ? = 0$ (hat (hat is val&e o' ?T a. -2 b. - 7 ). 4 d . 26
the the
Aadi)als Aadi)als )an )an be added added i' the# the# have have the same same radi)and radi)and and a. e+ponent b. po(er ). order
d. )oe''i)ients
2
24. 24.
9' the the roo roots ts o' a+ b+ ) = 0$ are real and e%&al$ then a. b2 , 4a) V 0 ). b2 , 4a) K 0 b. b2 , 4a) = 0 d. b2 , 4a) K 0
25.
he s&m s&m o' the integer integerss bet(een bet(een 2 and and 7 that that are e+a)tl# e+a)tl# divisib divisible le b# 15 is a. 23$700 b. 22$15 ). 21$00 d. 24$150
26.
he s&m s&m o' the pprime rime n&mbers n&mbers bet(ee bet(eenn 1 an andd 15 a . 42 b . 41 ). 3!
d . 3
27.
hat is the s&m s&m to in'ini n'init# t# o' the the se%&en)e se%&en)e 1 13 1! D a. 25 b. 56 ). 23
d. 32
31.
he term term 'ree 'ree o' o' # in the the e+pans e+pansion ion o' o' is a . 46 b . 4
d . 4!
32.
9' '+ '+ = + 2 and g# g# = # 2$ then 'Hg2I 'Hg2I e%&als e%&als +,2 a. 6 b. 5 ). 4 3
). 47
d. 3
1.
37. Go( man# terms in the progression 3$ 5$ 7$ !$ D m&st be taFen in order that s&m is 2$600T a. 53 b. 52 ) . 51 d . 50 3. 3. he he oth other er 'orm 'orm o' log logaN = b is a. N = ab b. N = b a ). N = ab d. N = ab 3!.
he val&e val&e o' o' U (hi) (hi)hh (ill (ill maFe maFe 4+2 , 4F+ 5F a per'e)t s%&are trinomial is a. 6 b. 5 ). 4 d.3
olv olvee the the 'oll 'ollo( o(in ingg e%&a e%&atition ons: s: ; ind the val&e val&e o' +: a b+ = a2 2ab b 2+-1 4
the
9n the %&adra %&adrati) ti) e%&ati e%&ation on ;+ ;+2 8+ ? = 0$ the prod&)t o' the root is: a. ?; b. ,8; ). ,?; d. 8;
40. (o st&dents st&dents (ere (ere solving solving a problem em that (o&ld (o&ld red&)e red&)e it to a %&adrati %&adrati)) e%&ation. e%&ation. he he 'irst st&dent )ommitted )ommitted an error in the )onstant term and 'o&nd the roots to be 5 and 7 (hile the se)ond st&dent made an error in the 'irst degree term and gave the roots as 2 and 16. i' #o& (ere to )he)F their sol&tions$ the right e%&ation is: a. +2 12+ 35 = 0 b. + 2 1+ 32 = 0 ). +2 7+ , 14 = 0 d. + 2 , 12+ 32 = 0 41. Petermine Petermine the vval&e al&e o' o' F so that the ee%&ati %&ation on +2 F-5+ F , 2 = 0 is a per'e)t trinomial s%&are. "ns or &&
42. he e+pression +4 a+3 5+2 b+ 6 (hen divided b# + , 2 leaves the remainder 16$ and (hen divided b# + , 1 leaves the remainder remainder 10. ind the val&es o' a and b. "ns a = * &&', &&', b = +' +'
43.
"iven the e%&ation +4 +2 1 = 0. hi)h o' the 'ollo(ing is not a rootT a. 1 120W b. 1 135W ). 1 240W d. 1 300W
44. he area area o' a s%&are s%&are 'ield e+)eeds e+)eeds anothe anotherr s%&are b# 56 56 s%&are meters meters.. he perimeter meter o' the larger 'ield e+)eeds one hal' o' the smaller b# 26 meters. hat are the sides o' ea)h 'ieldT "ns larger field, m or 2+'m- smaller field, +m or &&'m
2
33. 9' + 3+ U 5+ 2 , U is divided b# + 1 and the remainder is 3$ then the val&e o' U is a. - 2 b. - 4 ). -3 d. - 5 34.
9' 3+ = !# and 27# = 1Q$ then is e%&al to a. 37 b. 35
). 34 d. 3 . 36. ; )ertai )ertainn (orF (orF )an be done done in in as as man# man# da#s da#s as as ther theree are are men in the gro&p. gro&p. 9' the n&mber o' men in the gro&p is red&)ed b# 3$ the Cob (ill be dela#ed b# 4 da#s. he n&mber o' men originall# in the gro&p is a. men b. 10 men ). 12 men d. 14 men
21. 9' 14 and , 72 are the roots o' the %&adrati) e%&ation ;+2 8+ ? = 0$ (hat (hat is val&e o' 8T a. -2 b. - 7 ). 4 d . 26
23.
35.
45. he s&m o' the areas o' t(o &ne%&al &ne%&al s%&are s%&are lots ots is 5$20 5$2000 s%&are meters. meters. 9' the lots (ere adCa)ent to ea)h other$ the# (o&ld re%&ire 320 meters o' 'en)e to en)lose the )ombined 40m or ./m and 24m 24m area 'ormed b# them. ind the dimensions o' ea)h lot. "ns .0m and 40m *I!+IAL E
1.
; 'ather 'ather is is t(i)e t(i)e older older than than his son an andd the s&m o' their ages is 4. Go( old is ea)hT ea)hT a. $ 40 b. 12$ 36 ) . 16$ 32 d. none o' these
1.
2. 3.
1.
olv olvee the the 'oll 'ollo( o(in ingg e%&a e%&atition ons: s: ; ind the val&e val&e o' +: a b+ = a2 2ab b 2+-1 8 ind the roots o' the e%&ation 4+ 4 1 = 0 itho&t itho&t e+panding$ e+panding$ 'ind the term term involving involving +4 o' 3+2 , 2+-1. ;ns. !0$720+4 ; +pa +pand nd to 4 term termss + +23 , X+13 th 8 ind the ! term in the e+pression o' + 2 X13 ? rite the 'irst 'o&r terms and the last term o' the e+pansion o' 3+ , +365 P ind the term independent o' # in the e+pansion o' # 2 , #-1!. hi)h hi)h o' o' the the 'oll 'ollo(i o(ing n g has has no no midd middle le term termTT a. + # 3 b. a , b4
5.
ind ind the the mid middl dlee ter term m o' o' + + 2 , 2#10
). & v 6
d. + , #
9' the the midd middle le term term in the the e+pa e+pansi nsion on o' o' + 2# 2# n is F+4#4$ 'ind F and n. 9' the rth term o' + 2 , 2#3n is ?+#12$ 'ind the val&e &e o' ?. ;ns. 1120
.
ith itho& o&tt e+p e+pan andi ding ng$$ 'in 'indd the the 10th term o' the e+pansion o' , 2t 214
!.
9n the the e+pa e+pans nsio ionn o' o' + +2 1+ 12 'ind: a the 6th term b the middle term ) the term involving +6 d the term 'ree o' +
10.
6. edro edro is as old as Z&an (as (hen (hen Z&an is t(i)e t(i)e as as old as edro edro (as. hen edro (ill be as old as Z&an is no($ the di''eren)e bet(een their ages is 6 #ears. ind the age o' ea)h no(. ;ns. Z&an$ 24 #ears old and edro$ 1 #ears old
. he s&m o' o' the the age agess o' t(o bo#s bo#s is is 'o&r 'o&r times times the s&m o' o' the the age agess o' a )ertain )ertain n&mber o' girls. o&r #ears ago$ the s&m o' the ages o' the girls (as one eleventh o' the s&m o' the ages o' the bo#s and eight #ears hen)e$ the s&m o' the ages o' the girls (ill be one hal' that o' the bo#s. Go( man# girls are thereT ;ns. 4 girls !. 9n a 'amil#$ 'amil#$ there there are )hildren$ )hildren$ t(o t(o o' them them are are t(ins. t(ins. he #o&nge #o&ngest st is 3 #ears #ears old and and the eldest is 21 #ears old. heir ages are in arithmeti) progression. here are three )hildren #o&nger than the t(ins. Go( old are the t(insT ;ns. 12 #ears
ind the ssi+th i+th term in the the e+pansion e+pansion o' +2 +2 # !
10. he s&m s&m o' the aages ges o' t(o men men e% e%&als &als !!. !!. 9' the invert inverted ed age o' the the elder elder is added added to the age o' the #o&nger$ the s&m is 10. Go(ever$ i' the age o' the #o&nger is inverted and s&btra)ted 'rom the age o' the older$ the di''eren)e is 44. ind the age o' the older man. a. 67 b. 32 ) . 53 d. 46
"ns .'/x 4y +
11.
; 'ather 'ather is t(i)e t(i)e as old as his son son and and the the s&m o' their their ages is 4. 4. G Go( o( old old is ea)hT ea)hT a. $ 40 b. 12$ 36 ) . 16$ 32 d. nota
7. 9 am am three three times times as old as #o& #o& (ere (ere (hen (hen 9 ( (as as as as old old as #o& are no(. hen #o& got to be m# age together o&r ages (ill be 4. Go( old are (e no(T a. 24 Y 36 b. 24 Y ) . 16 Y d . 1 Y 27
"ns n = /, = &,&20
7.
3. he s&m s&m o' the the ages ages o' the the 'athe 'atherr and his his son is is !!. 9'9' the age age o' the the son son is added added to the inverted age o' the 'ather$ the s&m is 72. 9' the inverted age o' the s on is s&btra)ted 'rom the age o' the 'ather$ the di''eren)e is 22. hat are their agesT ;ns. 74 Y 25
5.
"ns */,0.4x &0 y +
6.
2. Maria Maria is 36 #ears #ears old$ Maria (as t(i)e t(i)e as old as ;nna ;nna (as (hen Maria Maria (as as old as ;nna is no(. Go( old is ;nna no(T ;ns. 24 #ears old
4. Maria Maria is 24 24 #ears old no(. Maria Maria (as (as t(i)e as old as ;na (hen Maria Maria (as as old as ;nna is no(. Go( old is ;nna no(T ;ns. 16 #ears old
"ns /4
4.
; 'ather 'ather is is t(i)e t(i)e older older than than his son an andd the s&m o' their ages is 4. Go( old is ea)hT ea)hT a. $ 40 b. 12$ 36 ) . 16$ 32 d. none o' these
ind ind the the term term )ont )ontain aining ing +26 in the e+pansion o' + -2 +312 "ns ..x 2.
12. 12.
he he ter term m )ont )ontai aini ning ng +! in the e+pansion o' + 3 1+ 15
13.
ind the )oe''i)ie )oe''i)ient nt o' the e+pansion e+pansion o' + +2 #10 )ontaining +10#5 a. 14! b. 252 ). 105
I!TE0ER A!D DI0IT PR*LE+-
d. 10$1
1. eparate eparate 132 132 into into 2 parts parts s&)h that that the the larger larger divided divided b# the the smaller smaller the the %&otient %&otient is 6 and the remainder is 13. hat are are the partsT ;ns. 17 and 115
A0E PR*LE+-
2. ; n&mber n&mber o' t(o t(o digit digitss divided divided b# th thee s&m o' the the digits digits the %&ot %&otient ient is 7 and the remainder is 6. 9' the digits o' the n&mber are inter)hanged$ the res<ing n&mber e+)eeds three times the s&m o' the digits b# 5. hat is the n&mberT ;ns. 3
2. Go( m&)h m&)h silve silverr and ho( m&)h m&)h )oppe )opperr m&st be be added added to 20Fg 20Fg o' an allo# allo# )ontai )ontainin ningg 10[ silver and 25[ )opper to obtain an allo# )ontaining 36[ silver and 3[
2. ; n&mber n&mber o' t(o t(o digit digitss divided divided b# th thee s&m o' the the digits digits the %&ot %&otient ient is 7 and the remainder is 6. 9' the digits o' the n&mber are inter)hanged$ the res<ing n&mber e+)eeds three times the s&m o' the digits b# 5. hat is the n&mberT ;ns. 3 3. i+ times times the the middle middle o' a three three digit n&mber n&mber is the the s&m o' o' the other other t(o. t(o. 9'9' the n&mbe n&mberr is divided b# the s&m o' the digits$ the ans(er is 51 and the remainder is 11. 9' the digits are reversed$ the n&mber be)omes smaller smaller b# 1!. ind the n&mber. ;ns. 725 4. ind the n&mber n&mber s&)h s&)h that that their their s&m s&m m<ipl m<iplied ied b# the s&m s&m o' ttheir heir s%&ares s%&ares is 65$ aand nd their di''eren)e m<iplied b# the di''eren)e o' their s%&ares is 5. ;ns. 2 and 3 5.
hree hree n&mbers n&mbers are in the the ratio ratio 2:5:. 2:5:. 9' their their s&m is is 60$ 'ind 'ind the n&mbers n&mbers.. ;ns. $ $ 20$ 32
2. Go( m&)h m&)h silve silverr and ho( m&)h m&)h )oppe )opperr m&st be be added added to 20Fg 20Fg o' an allo# allo# )ontai )ontainin ningg 10[ silver and 25[ )opper to obtain an allo# )ontaining 36[ silver and 3[ )opperT a. 14Fg$ 12Fg b. 16Fg$14Fg ). 12Fg$10Fg d. 16Fg$ 1Fg 3. ; tanF '&ll o' o' al)ohol )ohol is emptied o' one one third rd o' its )ontent and then then 'illed lled &p (ith (ater and mi+ed. 9' this is done si+ times$ (hat 'ra)tion o' the vol&me original o' al)ohol remainsT ;ns.6472! 4. Go( m&)h m&)h ttin in and and ho( m&)h iron m&st m&st bbee add added ed to 50 Filogra Filograms ms o' an allo# o# )ontaining )ontaining 10 per)ent tin and 25 per)ent iron to obtain an allo# )ontaining 25 per)ent tin and 50 per)ent ironT ;ns. 27.5 Fgtin$ 52.5 Fg iron
6. he s&m o' the the digit digitss o' a three-d three-digit igit n&mber n&mber is is t(elve. t(elve. he s&m o' the s%&ares s%&ares o' the h&ndreds@ digit and the tens@ digit is e%&al to the s%&are o' the &nits@ digits. 9' the h&ndreds@ digit is in)reased b# t(o$ the digits (ill be reversed. ind the n&mber. ;ns. 345
5. Go( man# man# liters liters o' o' (ater (ater m&st m&st be added added to to 45 liters ters o' sol&tion sol&tion (hi)h (hi)h is !0[ al)oho al)oholl in order to maFe the res<ing sol&tion 0[ al)oholT ;ns. 5.63<
7. ;pril ;pril 1!7. 1!7. he s%&are s%&are o' a n&m n&mber ber in)rea in)reased sed b# 16 is the same same as 10 times times n&mber. ind the n&mber. ;ns. $ 2
the
6. ; 40-gram 40-gram sol&tion sol&tion o' o' a)id a)id and (ater (ater is 20[ 20[ a)id a)id b# (eight (eight.. Go( m&)h p&re p&re a)id m&st be added to this sol&tion to maFe it 30[ a)idT ;ns. 5.71 grams
. he s&m s&m o' the ddigit igitss o' a three-digi three-digit n&mber n&mber is 12. he he middle ddle digit digit is e%&al e%&al to the the s&m o' the other t(o digits and the n&mber shall be in)reased b# 1! i' its digits are reversed. ind the n&mber. ;ns. 264
7. Go( m&)h (ater (ater m&st m&st be evaporated evaporated 'orm 0 liters liters o' o' 12[ 12[ sol&tion sol&tion o' salt in order order to obtain a 20[ sol&tion o' saltT ;ns. 32 < 3!. Go( man# liters o' (ater m&st be added to 100 liters o' 5[ s&l'&ri) a)id sol&tion to prod&)e 60[ s&l'&ri) a)id sol&tionT ;ns. 41.67 <
!. ind three three )on )onse)&t se)&tive ive odd odd intege integers rs s&)h s&)h that that t(i)e t(i)e the s&m s&m oo'' the the 'irst 'irst and the se)ond integers pl&s 'o&r times the third is e%&al to 60. ;ns. 5$ 7$ !
. ; )ertain )ertain sol&tion sol&tion sho&ld )ontain [ [ al)ohol. al)ohol. 9' it (as (as mistaFenl# mi+ed to )ontain )ontain 6[ al)ohol$ ho( m&)h m&st be dra(n 'rom a 5-liter tanF and repla)ed b# 10 per)ent al)ohol sol&tion to provide the proper )on)entrationT ;ns. 2.5 <
10. he s&m o' the digits o' a three-digi three-digit n&mber is 12. he s&m o' the s%&ares o' the h&ndreds digit and the tens digit is e%&al to the s%&are o' the &nits digit. 9' the h&ndreds digit is in)reased b# 2 and the &nits digit is de)reases b# 2$ the digits o' the original n&mber (ill be reversed. ind the n&mber. ;ns. 345 11. he e+)ess o' the s&m o' the 'i'th and the seventh parts over the di''eren)e o' the hal' and third parts o' n&mber is 25!. hat is the n&mberT ;ns. 1470 12. he s&m o' the digits o' the three-digit n&mber is 6. he middle digit is e%&al to the s&m o' the t(o other digits and the n&mber shall be in)reased b# !! i' the digits are reversed. ind the n&mber. ;ns. 132
!. ; )ertain )ertain amo&n amo&ntt o' 0[ s&gar s&gar sol&ti sol&tion on added added to anothe anotherr amo&nt amo&nt o' 40[ s&gar s&gar sol&tion #ields a sol&tion that )ontains 14 Fg o' s&gar. Gad the amo&nt been reversed$ the sol&tion (o&ld have )ontained 16 Fg s&gar. Go( m&)h o' the 0[ sol&tion (as thereT ;ns. 10 Fg 10. en liters liters o' 25[ salt salt sol&tion sol&tion and 15 liter liter o' 35[ salt salt sol&tion sol&tion are po&red po&red into a dr&m originall# )ontaining 30 liters o' 10 [ salt sol&tion. hat is the per)ent ) on)entration o' salt in the mi+t&reT a.1!.55[ b. 22.15[ ). 27.05[ d. 25.72[ RATE A!D +TI! PR*LE+-
+I
1. he tanF o' a )ar )ontains )ontains 50 liters liters o' al)ogas 25[ o' (hi)h (hi)h is p&re al)ohol. Go( m&)h o' the mi+t&re m&st be dra(n o'' (hi)h (hen repla)ed b# p&re al)ohol (ill #ield a 5050[ al)ogasT a. 16 23 b. 15 13 ). 14 d. 2 0
2. ;t the re)ent re)ent l#mpi l#mpi)) gam games es in Mon Montre treal$ al$ ?anada ?anada$$ a team team (hi)h (hi)h parti) parti)ipa ipated ted in 1600 meters rela# event had the 'ollo(ing individ&al speed. irst r&nner$ 24 Fph$ se)ond r&nner$ 20 Fph$ third r&nner$ 22 Fph and 'o&rth r&nner 23 Fph. hat (as the team@s speed.
1. ; motorist motorist is traveling traveling 'rom 'rom to to(n (n ; to to(n to(n 8 at 60 Fph and and ret&rns ret&rns 'rom 'rom to(n to(n 8 to to(n to(n ; at 30 Fph. Gis average velo)it# 'or 'or the ro&ndtrip is ;. 45 Fph 8. 40 Fph ?. 35 Fph P. N;
per ho&r. he airplane )an 'l# 20 Filometers per ho&r in still air. 9' the pa)Fage )arrier taFes 3 23 ho&rs in going 'rom ; to ? and 3 16 ho&rs 'or the ret&rn trip$ (hat is the total distan)e o' travel )overed b# the manT P = 605 Fm. t1 = 1.5 hr.$ t 2 = 53 hr.
2. ;t the re)ent re)ent l#mpi l#mpi)) gam games es in Mon Montre treal$ al$ ?anada ?anada$$ a team team (hi)h (hi)h parti) parti)ipa ipated ted in 1600 meters rela# event had the 'ollo(ing individ&al speed. irst r&nner$ 24 Fph$ se)ond r&nner$ 20 Fph$ third r&nner$ 22 Fph and 'o&rth r&nner 23 Fph. hat (as the team@s speed. ;ns. 22.14! Fph 3. ; troop troop o' soldiers e rs mar)h mar)hed ed 15 Fm$ going going to the the )on)ent )on)entrat ration ion )amp )amp a't a'ter er the# the# (ere 'or)ed to s&rrender$ at the same time that the vi)torio&s general (ho is s&pervising the mar)h rode 'rom the rear o' the troop to the 'ront and ba)F at on)e to the rear. 9' the distan)e )overed b# the vi)torio&s general is 25 Fm. and both the troop and general traveled at &ni'orm rate$ ho( long is the troopT ;ns. Fm. 4. (o )#)lists sts are pra)ti)ing on on a )ir)&lar tra)t o' )ir)&m'eren)e )ir)&m'eren)e 276 meter. tarting tarting at the same instant and 'rom the same pla)e$ (hen the# r&n in opposite dire)tions the# pass ea)h other ever# 6 se)onds and (hen the# r&n the same dire)tion the 'aster passes the slo(er at ever# 23 se)onds. Petermine their rates. ;ns. = 2! ms$ = 17ms 5. (o )ars )ars ; and 8 start at the the same same point point and and at the the same same time time and travel travel in opposite opposite dire)tions$ dire)tions$ )ar 8 traveling traveling 20 Fmhr slo(er than ;. 9' the# are 420 Filometers Filometers apart a'ter 3 ho&rs$ 'ind the rate o' ea)h. ;ns. 60 Fph$ 0 Fph 6. (o )ars )ars ; and and 8 are to ra)e ra)e aro&nd aro&nd a 1$500 1$500-me -meter ter )ir)&l )ir)&lar ar tra) tra)F. F. 9'9' the# the# (ill (ill start start at the same point and travel opposite dire)tions$ the# (ill meet 'or the 'irst time in 3 min&tes. 8&t i' the# (ill travel in the same dire)tion$ (ith the same starting point$ )ar ; (ill rea)h the starting point (ith )ar 8 trailing behind b# 500 meters. hat sho&ld be the rates o' ea)hT ;ns. 300 mmin$ mmin$ 200 mmin mmin 7. ; one Filometer long )aravan o' men is (alFing at a )onstant rate. ; man 'rom the rear ends (alF to(ards the head and ba)F to the rear at the instant (hen the )aravan has )overed a distan)e o' one Filometer. ind the total distan)e traveled b# the man. ;ns. 2.414 Fm . ; bo# starte startedd one ho&r ho&r and t(ent# t(ent# min&te min&tess earlie earlierr than than a man man.. 9' the man ran at 6 Fph 'aster than the bo# and overtooF the bo# in 40 min&tes$ 'ind the rate o' ea)h. ;ns. 3 Fph 'or the bo# and ! Fph 'or the man !. ; man man (alFed (alFed 24 Fm in time . P&ring P&ring the ''irst irst part o' this this time$ time$ hhee (alFed (alFed at 6 Fph and the last part at 4 Fph. Gad he reversed his rates$ he (o&ld have (alFed t(o Fm more. ind the time. ;ns. 5 hrs. 10. ; tra''i) tra''i) )he)F )he)F )o&nte )o&ntedd 3!0 )ars )ars passing passing a )ertai )ertainn spot on on one da# da# and 430 430 )ars at the the same spot on the se)ond da#. n the 'irst da#$ there (ere three times as man# )ars going east and hal' as man# going (est on the se)ond da#. hat (as the total total n&mber o' (est bo&nd )ars 'or the t(o da#sT ;ns. 20 east$ 540 (est 11. ; man is sent to deliver deliver an important important pa)Fage pa)Fage ant travels travels b# )ar 75 Filometers Filometers per ho&r 'rom point ; to 8 and then b# airplane to point ? against a (ind blo(ing 40 Filometers
4. he he s&m s&m o' three three n&mbe n&mbers rs in arit arithm hmet eti) i) progr progres essi sion onss is 60. 9' the the n&mb n&mber erss are are in)reased b# 2$ 1$ and 2$ respe)tivel#$ the ne( n&mbers (ill be in geometri) progression. ind the arithmeti) progression.
per ho&r. he airplane )an 'l# 20 Filometers per ho&r in still air. 9' the pa)Fage )arrier taFes 3 23 ho&rs in going 'rom ; to ? and 3 16 ho&rs 'or the ret&rn trip$ (hat is the total distan)e o' travel )overed b# the manT P = 605 Fm. t1 = 1.5 hr.$ t 2 = 53 hr. 12. ; motor)# motor)#)le )le messeng messenger er le't le't the the rear rear o' a motoriQed motoriQed troop Filomete Filometers rs long long and rode to the 'ront o' the troop$ ret&rning at on)e to the rear. Go( 'ar did he ride$ i' the troop traveled 15 Filometers d&ring this time and ea)h traveled at a &ni'orm rateT ;ns. 25Fms. 13. ;n arm# o''i)er o''i)er made the 'irst part o' the trip on a plane (hi)h 'le( at the rate o' 210 Filometers per ho&r. ;t the landing 'ield$ he (as met b# a Ceep (hi)h tooF him the rest o' the (a# to his destination at a rate o' 40 Filometers per ho&r. he trip re%&ired 3 ho&rs and 15 min&tes. n his ret&rn trip$ the Ceep traveled at the rate o' 50 Filometers per ho&r and the plane (hi)h he tooF 'le( at the rate o' 200 Filometers Filometers per ho&r. he ret&rn Co&rne# re%&ired the same amo&nt o' time$ b&t this in)l&ded a min&te (hi)h he spent (aiting 'or the plane to taFe o''. ind the total distan)e that he 'le( and the total distan)e that he traveled b# Ceep. ;ns. 532Fm b# plane and 2 23Fm b# Ceep 14. ; 8M )ar )ar drives 'rom 'rom ; to(ard to(ard ? at 30 mileshr. eshr. ;nother ;nother )ar starting starting 'rom 'rom 8 at the the same time$ drives to(ards ; at 20 mihr. 9' ;8 = 20 miles$ 'ind (hen the )ars (ill be nearest together. ;ns. 24 min. 15. (o boat boatss starte startedd their their vo#ages vo#ages is is in a straigh straightt line line to(ards to(ards ea)h ea)h other. other. ne ne has an average navigational speed o' 30 Fmh and the other one has an average o' 20Fph. ;ss&ming that the# )an not avoid a )ollision$ ho( long (ill it taFe be'ore the )ollision ision o))&rsT Go( 'ar (o&ld ea)h boat have traveled be'ore the )ollisionT ;ns. 4 hrs$ 120 Fm$ 0 Fm 16. ; man traveling 40 Fm 'inds that b# traveling one more Fm per ho&r$ he (o&ld the Co&rne# in 2 ho&rs less time. Go( man# Filometer per ho&r did he a)t&all# travelT a. 4 b. ) . 1 d. 6
made
PR0RE--I!
1. (o n&mbers n&mbers di''er di''er b# 40 and their their arithm arithmeti) eti) mean e+)eeds e+)eeds their their positive positive geometri) geometri) mean b# 2. he n&mbers are a. 45$ 5 b. 64$ 104 ) . 1 $ 12 1 d.100$ 140 2. ; sets sets o&t to (alF (alF at the rate o' 'o&r Fm per ho&r. ;'ter ;'ter he he had had been been (alFin (alFingg 'or 2-34 ho&rs$ 8 sets o&t to overtaFe and (ent 4-12 Fm the 'irst ho&r$ 4-34 Fm the se)ond hr.$ 5 Fm the third hr and so on gaining 1 %&arter o' a Fm. ever# ho&r. 9n ho( man# ho&rs (o&ld 8 overtaFe ;T ;ns. ho&rs 3. ; besieg besieged ed 'ortre 'ortress ss is held held b# 5700 5700 men (ho have have provis provision ionss 'or 66 da#s. da#s. 9' the garrison loses 20 men ea)h da#$ 'or ho( man# da#s )an the provisions hold o&tT ;ns. 76 da#s
17.
9n the the series series 1.01$ 1.01$ 11.0$ .0$ .0!!$ .0!!$ .0!D ind the 0th term.
4. he he s&m s&m o' three three n&mbe n&mbers rs in arit arithm hmet eti) i) progr progres essi sion onss is 60. 9' the the n&mb n&mber erss are are in)reased b# 2$ 1$ and 2$ respe)tivel#$ the ne( n&mbers (ill be in geometri) progression. ind the arithmeti) progression. 5. hree hree n&m n&mber berss are are in arithm arithmeti eti) prog progres ressio sion. n. heir heir s&m is 15$ and the s&m o' s%&ares is 3. ind the n&mbers.
their their
6. ; 20-liter 20-liter )ontainer )ontainer is 'illed 'illed (ith p&re a)id. ive liters liters are are dra(n o'' and repla)ed repla)ed (ith (ater* then 5 liters o' the mi+t&re dra(n o'' and repla)ed (ith (ater$ and so on &ntil 5 dra(ings and 5 repla)ements have been made. ind the amo&nt o' a)id in the 'inal mi+t&re. 7. ; r&bber r&bber ball ball is dropped dropped 'rom a height height o' 27 27 met meters. ers. a)h a)h time time th that at it hits hits the the gro&nd gro&nd it bo&n)es to a height 23 o' that 'rom (hi)h it 'ell. ind the distan)e that it travels &p to the time that it hits the gro&nd 'or the 5 th time. he total distan)e traveled b# the ball &ntil it )omes to rest. . ; man (ishes (ishes to to b&# b&# a pie)e pie)e o' land (orth (orth 150$000 150$000 pesos. 9' it (ere possible possible 'or him to save one )entavo on the 'irst da#$ t(o ) entavos on the se)ond da#$ 4 )entavos on the third da# and so on$ in ho( man# da#s (o&ld he save to be able to b&# the landT ;ns. abo&t 24 da#s !. he s&m s&m o' the t(o t(o n&mb n&mbers ers is 20 and and thei theirr positiv positivee geom geometr etri) i) mean mean is one greate greaterr than one hal' o' their arithmeti) mean. ind their di''eren)e. 10. ; man )&ts )&ts a pie)e o' paper paper 0.03 0.03 mm thi)F thi)F into three three e%&al e%&al parts. parts. hen he )&ts )&ts ea)h o' these parts into three e%&al parts again and the pro)ess is repeated 10 times a'ter (hi)h he piles together the pie)es o' paper. Go( thi)F is the pileT 11. ; ri)h man man )alled )alled his seven seven sons. sons. Ge had (ith (ith him him a n&mber n&mber o' pebbles$ pebbles$ ea)h ea)h pebble pebble representing a gold bar. o his 'irst son$ he gave hal' the pebbles that he had and one pebble more. o his se)ond son$ he gave hal' the remaining pebbles and one pebble more. Ge did the same to ea)h to his 'ive other sons and then 'o&nd o&t that he had one pebble le't. Go( man# pebbles (ere there initiall#T ;ns. 32 12. ; man man re)eives re)eives a salar# salar# o' 36$00 36$0000 per ann&m 'or the the 'irst 'irst #ear #ear and and a 10 10[ [ raise ever# #ear 'or ten #ears. hat is his salar# d&ring the 'i'th #earT ;ns. 52$707.60 13. ; )ar r&nning r&nning at 25 Filome Filometers ters per ho&r ho&r )an )over )over a )ertain distan)e distan)e in ho&rs. ho&rs. 8# ho( man# Filometers per ho&r m&st its rate be in)reased in order to )over the same distan)e in three ho&rs lessT ;ns. 15Fmhr 14.
ind the harmoni) harmoni) mean 7$ 1$ 1$ 5$ 2$ 6 and 3 a. 2.36 b. 2.46
15 .
he term o' an ; is 3 (hile its 4 term is 273. ind the 35 term.
16.
ind the the s&m to in'init in'init## o' 13$ 13$ 127$1 127$1243D 243D ;ns. ;ns. 3 3
th
). 2.56
th
d.2.66 th
1. 9' 4 men )an plo( 12 he)tares in ho&rs$ ho( man# men are needed to plo( 24 he)tares in 24 ho&rsT ;ns. 6 men
17.
9n the the series series 1.01$ 1.01$ 11.0$ .0$ .0!!$ .0!!$ .0!D ind the 0th term.
1.
ind the the s&m o' 30.4 30.40.0 0.050. 50.0040 0040.000 .0005D 5D ;ns. ;ns. 311 311
1!. ;n arithmeti) thmeti) progressio progressionn starts starts (ith (ith 1$ has ! terms$ terms$ and the m middle iddle term is is 21. Petermine the s&m o' the 'irst ! terms. 20. ; pend&l&m pend&l&m s(ings s(ings 24 in)hes in)hes 'or the 'irst 'irst time. time. 9t is s(ingi s(inging ng 1112 1112 o' its previo previo&s &s s(ing. hat (o&ld be the total distan)e traveled (hen the pend&l&m stoppedT a. 246 b.264 ).2 d. 312 21. ; small line ne tr&)F ha&ls ha&ls poles 'rom 'rom s&bstation s&bstation sto)F#ar sto)F#ardd to pole sites sites along a proposed proposed distrib&tion line. he tr&)F )an handle onl# one pole at a time. he 'irst pole site is 150 meters 'rom the s&bstation s&bstation and the poles are to be 50 meters meters apart. apart. Petermine Petermine the total total distan)e traveled b# the line tr&)F$ ba)F and 'orth$ a'ter ret&rning 'rom delivering th the 30 pole. a. 35.0Fm b. 30.0Fm ). 37.5Fm d. 40.0Fm 22. (o positive positive n&mbers n&mbers ma# be inserted inserted bet(een bet(een 3 and and ! s&)h s&)h that the 'irst three are in geometri) progression$ (hile the last three are in arithmeti) progression. hat is the s&m o' these t(o positive integersT a.1.25 b. 12.25 ). 11.25 d.6.25 23. ; man piles 150 logs in la#ers so that the top la#er )ontains 3 logs ogs and ea)h lo(er la#er has one more log than the la#er above. Go( man# logs are at the bottomT ;ns. 17 logs 24. ; bod# 'alls 16.1 meters d&ring the 'irst se)ond$ 4.3 meters d&ring the se)ond$ 0.5 meters d&ring the third se)ond and so on. Go( long (ill it taFe the bod# to rea)h the gro&nd i' it (as released at an altit&de o' 15$000 metersT ;ns. 30.5 se)onds 25. he 1th and the 52 nd terms terms o' an arithmeti) thmeti) progression are 3 and 173$ respe)tivel#. he 25th term is a. 3 b. 35 ) . 2 d . 25 25. ind the n&mber o' terms o' a geometri) progression in (hi)h the 'irst term last term is 34 and the s&m o' the terms is 720. ;ns. 4 terms
is 4$ the
26. he s&m o' three three n&mbers in ;.. is 27. 9'9' the 'irst rst n&mber is in)reased b# 2$ the se)ond b# 7$ and the third b# 20$ the res<ing n&mbers (ill be in ".. ind the original n&mbers. a. 3$ !$ 15 b. 4$ !$ 14 ). 5$ !$ 13 d. 6$ !$ 12 ,R= A!D DI-C)AR0E PR*LE+-
12. ; steel )ompan# )ompan# has three blast '&rna)es '&rna)es o' var#ing var#ing siQes. siQes. 9' '&rna)es '&rna)es ;$ 8$ ? are &sed '&ll time$ 00 metri) tons o' steel are prod&)ed per da#. 9' ; and 8 are &sed hal' time and ? '&ll time$ 545 metri) tons are prod&)ed. 9' ; is not &sed$ 8 is &sed '&ll time$ and ? hal'
1. 9' 4 men )an plo( 12 he)tares in ho&rs$ ho( man# men are needed to plo( 24 he)tares in 24 ho&rsT ;ns. 6 men 2. ; garden garden )an be be )<ivated tivated b# bo#s bo#s in 5 da#s. da#s. he he same same Cob Cob )an )an be done 5 me menn in 6 da#s. Go( long (ill it taFe to 'inish the Cob i' ; the bo#s and 5 men (ill (orF togetherT 8 onl# 6 bo#s ands 3 men (ill (orF togetherT ? t(o da#s a'ter the 5 men (ere (orFing the bo#s arrived to helpT 3. ;$ 8 and ? )an do a pie)e pie)e o' (orF in in 10 dda#s$ a#s$ ; and 8 )an ddoo it in 12 dda#s$ a#s$ ; and ? in 20 da#s. Go( man# da#s (o&ld it taFe ea)h to do the (orF aloneT ;ns. 30$ 20 $60 4. ; boat@s boat@s )re( ro(ing ro(ing at hal' their their &s&al &s&al rate rate )a )ann neg negotiat otiatee 2Fm. 2Fm. do(n do(n a river and ba)F in one ho&r and 40 min&tes. ;t their &s&al rate in still (ater$ the# (o&ld have gone over the same )o&rse in 40 min. ind their rate o' ro(ing in still (ater. ;ns. 325 Fmhr 5. (o pipes pipes r&nnin r&nningg sim& sim<a ltaneo neo&sl &sl## )an )an 'ill a tanF tanF in in 3 ho&rs ho&rs and 20 min&te min&tes. s. 9' both both pipes r&n 'or 2 ho&rs and the 'irst is then sh&t o''$ it re%&ires 2 ho&rs more 'or the se)ond to 'ill the tanF. Go( long does it taFe ea)h pipe to 'ill it aloneT 6. ; and and 8 )an do do a Cob in in 12 da#s. da#s. ; and ? )an )an do the the same same Co Cobb in 1 da#s da#s (hil (hilee 8 and ? )an do it in 24 da#s. Go( (ill it taFe ;$ 8 and ? to do the Cob togetherT 7. ; man )an 'inished 'inished a )ertain )ertain Cob in three-'o&r three-'o&rths ths the time that the bo# )an* the bo# )an 'inish the same Cob in t(o-thirds the time that a girl )an* and the man and the girl (orFing onl# Cointl# )an 'inish the Cob in 4 ho&rs. Go( long (ill it taFe to 'inish the Cob i' the# all (orF togetherT ;ns. 3 hr. . he intaFe intaFe pipe to a reservoir reservoir is )ontrolled )ontrolled b# a valve (hi)h a&tomati)a a&tomati)all# ll# )loses )loses (hen the reservoir is '&ll and opens again (hen 'o&r-'i'ths o' the (ater had been drained o''. he intaFe pipe )an 'ill the reservoir in 4 ho&rs and the o&tlet pipe )an drain it in 10 ho&rs. 9' the o&tlet pipe remains open$ ho( m&)h time elapses bet(een the t(o instants that the reservoir is '&llT ;ns. 13.3 hr.
12. ; steel )ompan# )ompan# has three blast '&rna)es '&rna)es o' var#ing var#ing siQes. siQes. 9' '&rna)es '&rna)es ;$ 8$ ? are &sed '&ll time$ 00 metri) tons o' steel are prod&)ed per da#. 9' ; and 8 are &sed hal' time and ? '&ll time$ 545 metri) tons are prod&)ed. 9' ; is not &sed$ 8 is &sed '&ll time$ and ? hal' dime$ 410 metri) tons are prod&)ed. Go( man# metri) tons per da# does ea)h '&rna)e prod&)eT 13. hree observat observation ion planes planes ;$ 8$ and ?$ (orFing (orFing togeth together$ er$ )an map the the region region in 4 ho&rs. lanes ; and 8 )an map the region in 6 ho&rs$ planes 8 and ? )an map it in 6 ho&rs and 40 min&tes. Go( long (o&ld it taFe ea)h o' the planes (orFing alone to map this regionT 14. ; p&mp dis)harg dis)harging ing ! gpm re%&ires re%&ires 36 ho&rs ho&rs to 'ill 'ill a tanF. 9' the the p&mp is repla)ed repla)ed b# one that (ill dis)harge 16 gpm$ ho( long (ill it taFe to 'ill the tanFT a. 64 hr b. 16 hr. ). 20.25 hr d. 40.5 hr 15. (o pipes r&nning ng sim<ane sim<aneo&sl# o&sl# )an 'ill a s(imming s(imming pool in 6 ho&rs. ho&rs. 9' both pipes r&n 'or 3 ho&rs and the 'irst pipe is then sh&t o''$ it re%&ires 4 ho&rs more 'or the se)ond to 'ill the pool. Go( long does it taFe ea)h pipe r&nning separatel# to 'ill the poolT ;ns. Y 24 16. ; man man and and a bo# )an dig a tren)h tren)h in 20 da#s. da#s. 9t (o&ld taFe the bo# ! da#s da#s longer longer to dig it alone than it (o&ld taFe the man. Go( long (o&ld it taFe the bo# to dig aloneT a. 45 da#s b. 16 da#s ).25 da#s d. 4 da#s 17. ; Cob )an be done in as man# man# da#s as there are men in the gro&p. gro&p. 9' the n&mber o' men is red&)ed b# 3$ the Cob (ill be dela#ed b# 4 da#s. Go( man# men are there originall# in the gro&pT a. 6 b. 12 ) . 20 d . 30 7E!! DIA0RA+
10. ; s(imming s(imming pool holds 54 )&bi) meters meters o' (ater. (ater. 9t )an be drained drained at a rate o' one )&bi) meter per min&te 'aster than it )an be 'illed. 9' it taFes ! min&tes longer to 'ill it than to drain it$ 'ind the drainage rate. ;ns. 3 )&.mmin
1. ; )ertain )ertain part )an be de'e)ti de'e)tive ve be be)a&se )a&se it has has on onee or more o&t o' o' three three possible possible de'e)ts* ins&''i)ient tensile strength a b&rr or diameter o&tside o' toleran)e limits. 9n a lot o' 500 p)s: 1! have a tensile strength de'e)t 17 have a b&rr 11 have an &na))eptable diameter 12 have tensile strength and b&rr de'e)ts 7 have tensile strength and diameter de'e)ts 5 have b&rr and diameter de'e)ts 2 have all three de'e)ts a. ho( man# have 'o&r de'e)ts b. ho( man# p)s have onl# a b&rr de'e)t ). ho( man# p)s have e+a)tl# t(o de'e)ts. ;ns. 475$ 2$ 1
11. ne inp&t pipe )an 'ill a tanF tanF alone in hrs. another another inp&t pipe )an 'ill it alone in 6 ho&rs and a drain pipe )an empt# the '&ll tanF in 10 ho&rs. 9' the tanF is empt# and all the pipes are (ide open$ ho( long (ill it taFe to 'ill the tanFT ;ns.5.22 ho&rs
2. P&ring the ele)tion$ the total n&mber o' votes re)orded in a )ertain m&ni)ipalit# (as 12$400 had 25 o' the s&pporters o' <;8;N )andidate stampede a(a# 'rom the pools and X o' the s&pporters o' ";P )andidate behaved liFe(ise$ the <;8;N )andidates maCorit# (o&ld
have been red&)ed b# 100. Go( man# votes did the <;8;N Y ";P )andidates a)t&all# re)eivedT ;ns. 7$000$ 5$400
. ; st&den st&dentt le't le't his home to attend attend a part# one morning ng at at past past 6 o@)lo)F )lo)F and ret&rned ret&rned at past 3 o@)lo)F. Ge noti)ed the hands o' the (all )lo)F have e+)hanged position. hat e+a)t time did he arriveT
!. (o brothers brothers (ashed (ashed the 'amil# 'amil# )ar in 24 min&tes. min&tes. revio&sl#$ o&sl#$ (hen ea)h had (ashed the )ar alone$ the #o&nger bo# tooF 20 min. longer to do the Cob than the older bo#. Go( long did it taFe the older bo# to (ash the )ar aloneT ;ns. 40 min&tes
have been red&)ed b# 100. Go( man# votes did the <;8;N Y ";P )andidates a)t&all# re)eivedT ;ns. 7$000$ 5$400 3. he resident resident C&st re)entl# re)entl# appointed nted 25 "enerals "enerals o' the hil. ;rm# o' these 14 have alread# served in the (ar o' Uorea$ 12 in the (ar o' \ietnam and 10 in the (ar o' Zapan. here'ore 4 (ho have served both in Uorea and Zapan$ 6 have served both in \ietnam and Uorea and 3 have served in Zapan$ Uorea$ and \ietnam. ;ns. 2 generals 4.
; s&rve# s&rve# o' 500 500 .\. .\. vie(e vie(ers rs pro)ee pro)eedd the 'ollo 'ollo(in (ingg res<: res<: 25 (at)h 'ootball games 1!5 (at)h ho)Fe# games 115 (at)h basFetball games 45 (at)h 'ootball and basFetball 70 (at)h 'ootball and ho)Fe# 50 (at)h ho)Fe# and basFetball 50 do not (at)h an# o' the 3 games$ Go( man# (at)h the basFetball games onl#T a.50 b.40 ).30
!. Go( man# times times in one )omplete )omplete da# (ill the ho&r and the min&te min&te hands (ith ea)h otherT a. 24 b. 23 ) . 22 d . 25
)oin)ide de
RATI6 PRPRTI! A!D 7ARIATI!
1. he Filo(att Filo(att that )an be transmitt transmitted ed sa'el# sa'el# b# b# a sha't varies varies dire)tl# re)tl# as as the the n&mber n&mber o' revol&tion it maFes per min&te and the )&e o' its diameter. 9' a sha't 3 )entimeters in diameter maFing 200 revol&tions per min&te )an sa'el# transmit 60 Filo(att$ (hat Filo(att )an be sa'el# transmitted b# a 2-)entimeter sha't maFing 300 revol&tions per min&teT d.60
CLC= PR*LE+-
1.
. ; st&den st&dentt le't le't his home to attend attend a part# one morning ng at at past past 6 o@)lo)F )lo)F and ret&rned ret&rned at past 3 o@)lo)F. Ge noti)ed the hands o' the (all )lo)F have e+)hanged position. hat e+a)t time did he arriveT a. 3:26.07 b. 3:34.62 ). 3:31.47 d. 3:32.1!
;t (hat (hat time me be bet(een t(een 7 and o@)lo)F o)F are the hhands ands o' o' the )lo)F are ; at right angles 8 straight line ? )oin)iding
2. he time is 3:00 o@)lo)F o@)lo)F and the hands o' the )lo)F are at right angles angles to ea)h other. hat is the nearest time o' the )lo)F s&)h that the hands o' it (ill be at right angles againT ;ns. 32.72 min&tes 3. Go( long long (ill (ill it be 'rom 'rom the time time the the ho&r hand hand and the the min&t min&tee hand o' a )lo)F )lo)F are are together &ntil the# (ill be together againT ;ns. 1 hr. and 5.45 min 4. ;t (hat time bet(een 4 and 5 o@)lo)F do the hands o' the )lo)F )oin)ideT ;ns. 4:21.2 o@)lo)F 5. 9t is e+a)tl# e+a)tl# 3 o@)lo)F o@)lo)F.. 9n ho( ho( man# man# se)onds se)onds (ill (ill the the ang angle le 'ormed 'ormed b# the hho&r o&r hand hand and the min&te hand be t(i)e the angle 'ormed b# the ho&r and the se)ond handT ;ns. 22.4 se)onds 6. 9t is no( bet(een bet(een ! and 10 o@)lo)F o@)lo)F.. 9n 4 min&tes min&tes$$ the the ho& ho&rr hand hand (ill (ill be e+a)tl# e+a)tl# opposite the position o))&pied b# the min&te hand 3 min&tes ago. hat is the time no(T ;ns. !:20 7. Go( man# min&tes min&tes a'ter 2:00 o@)lo)F o@)lo)F (ill the hands o' the )lo)F e+tend e+tend in the opposite dire)tions 'or the 'irst timeT a. 40.636 b. 41.636 ). 43.636 d. 42.636
!. 9' the s%&are s%&are root root o' + varies varies dire)tl# dire)tl# as # and and inversel# inversel# as the the s%&are s%&are o' Q and i'i' + = 16 (hen # = 24 and Q = 2$ 'ind Q (hen + = ! and # =2 2 ;ns. 23
2. he time time re%& re%&ire iredd 'or 'or an elevat elevator or to li't li't a (eight (eight varies varies dire)t dire)tl# l# (ith (ith the the (eight (eight and distan)e thro&gh (hi)h it is li'ted and inversel# as the po(er o' the motor. 9' it taFes 30 se)onds 'or a 10 G motor to li't 100 lbs thro&gh a height o' 50 't$ (hat siQe o' motor is re%&ired to li't 00 lbs in 40 se) thro&gh the height o' 40 'tT ;ns. 4 G 3. ight men )an e+)avate 15m3 o' drainage open )anal )anal in 7 hrs. hree men )an ba)F'ill 10m3 in 4 hrs. Go( long (ill it taFe 10 men to e+)avate and ba)F 'ill 20 m 3 in the proCe)tT ;ns. !.7 hrs. 4. ; man is sent sent to to deli deliver ver an import important ant pa)Fag pa)Fagee and and travel travelss b# b# )ar )ar 75 Filom Filomete eters rs per ho&r 'rom point ; to 8 and then b# airplane to point ? against a (ind blo(ing 40 Filometers per ho&r in still air. 9' the pa)Fage )arrier taFes 3 23 ho&rs in going 'rom ; to ? and 3 16 ho&rs 'or the ret&rn trip$ (hat is the total distan)e o' ravel )overed b# the manT 5. ; sphere sphere 30 )m in diamete diameterr is divided divided into into t(o segments. segments. ne ne o' (hi)h (hi)h is t(o t(o times times as high as the other. ind the vol&me o' the bigger segment. 6. (o 'l#(h 'l#(heels eels are are )onne)te )onne)tedd b# a belt. belt. he radi&s radi&s o' the the 'l#(heel 'l#(heelss are 30 in and 50 in. he small 'l#(heel has a speed o' 350 rpm. Petermine the velo)it# o' the belt in 'tse). hat (o&ld be the ang&lar velo)it# o' the larger 'l#(heelT 7. ; )#lindri )#lindri)al )al tin )an has its its height height e%&al e%&al to to the diameter diameter o' its its base. base. ;nothe ;notherr )#lindri)a )#lindri)all tin )an (ith the same )apa)it# has its height e%&al to t(i)e the diameter o' its base. ind the ratio o' the amo&nt o' tin re%&ired 'or maFing the t(o )ans (ith )overs. ;ns. 0.!524 . he diamet diameters ers o' t(o sphere spheress are in the ratio ratio 2:3 and the s&m o' their their 1$260 )&bi) meters. ind the vol&me o' the larger sphere. ;ns. !72 )&. M
+0 = X + 1 +2
*
#0 = X # 1 #0
Angle *etween Two Conc#rrent Lines
vol&me vol&mess
is
!. 9' the s%&are s%&are root root o' + varies varies dire)tl# dire)tl# as # and and inversel# inversel# as the the s%&are s%&are o' Q and i'i' + = 16 (hen # = 24 and Q = 2$ 'ind Q (hen + = ! and # =2 2 ;ns. 23 10.
9' a:b a:b = 2:3$ 2:3$ b:)= b:)= 4:5$ 4:5$ (hat (hat is a:b: a:b:)T )T a. 2:34:5 b. :12:16
). :12:16
d. 6:!:12
Distance *etween Two Points P 4 .x4684/ and P $ .x$68$
SULEMENTARY RO!LEMS
+
d ' .x$ 3 x4/$ % .8$ 3 84/$ Area of Pol8gon .!on>oerlapping/ .!on>oerlapping/ of n>sides 0ien 7ertices
"iven verti)es +1$ #1$ +2$ #2$ DDD +n$ #n oriented )o&nter)lo)F(ise
A'
; = X H +1#2 +2#3 +3#4 DD. + n#1 , #1+2 #2+3 #3+4 DD. # n+1I Diision of Line -egment
+$#
1+1$#1 o
3. ind the length length o' the segment segment Coining Coining the t(o midpoin midpoints ts o' the sides o' the triangl trianglee i' the length o' the third side opposite to it is 30 )m. ;ns. 15 )m.
5. (o verti)es o' a triangle are are 0$ - and and 6$ 0. 9' the medians interse)t at !$ -3$ 'ind the third verte+ o' the triangle. Ans. *+%, +-
85 ????84
#
2. ind the lo)&s o' points +$ # s&)h that the distan)e distan)e 'rom to to 3$ 0 is t(i)e )e its distan)e to 1$ 0. ;ns. 3 " # $ %& # $ #" $ ' ( )
4. ; line 'rom 'rom 1$ 1$ 4 to /4$ -1 -1 is e+tended e+tended to a point point A so that that A = 4/. ind n d the )oordinate o' A. ;ns. A13$ -16
x5 ????x4 8$
' m$ 3 m4 4 % m4m$
1. ; point +$ 3 3 is e%&idistant 'rom 'rom points ;1$ 5 and 8-1$ 2. ind +. ;ns. ]
o
84
1+1$#1
x$
#0 = X # 1 #0
2+2$#2
d
x4
*
Angle *etween Two Conc#rrent Lines
ANALYTIC GEOMETRY GEOMETRY
#
+0 = X + 1 +2
x ' x4 % @ .x$ 3 x4/ 8 ' 84 % @ .8$ 3 84/
+
9' F = X$ then 'orm&la above be)omes a midpoint formula
, the lo)&s o' an an e%&ation is a )&rve )ontaining onl# those those points (hose )oordinates )oordinates satis'# the e%&ation.
6. he area o' a triangle angle (ith verti)es 6$ 2$ +$ 4 and and 0$ -4 is 26. ind ind +.;ns. , 23 and 503 7. ind the length o' o' the median 'rom 'rom ; o' a triangle triangle ;8? given verti)es verti)es ;1$ 6$ 8-1$ 3 and ?3$ -3. ;ns. 6 . 9' the midpoint dpoint o' a segment is 5$ 5$ 2 and one endpoint is 7$ -3$ (hat are the )oordinates )oordinates o' the other endT Ans. *%, / !. "iven verti)es verti)es o' o' a triang triangle le ;8? ;8? : ;1$ 5$8-1$ 1 and ?6$ 3. ind nd the interse)tion o' the median. median. Ans.*#, % 10. ind the in)lination o' the line 2+ 5# = 10. ;ns. 15.2 ° Loc#s
point.
, the )&rve tra)ed b# an arbitrar# point as it moves in a plane is )alled lo)&s o' a
#
, the lo)&s o' an an e%&ation is a )&rve )ontaining onl# those those points (hose )oordinates )oordinates satis'# the e%&ation.
# 4 (o* (o* 6nte 6nterce rcept pt orm orm x % 8 '4
EQUATI! 1 A -TRAI0)T LI!E Line , is a lo)&s o' points (hi)h has )onstant slope.
a
!eorems 3
(here a = + inter)ept
• •
20$ b +$ #
&
b = # inter)ept
ver# straight line )an be represented b# a 'irst-degree e%&ation. he lo)&s o' an e%&ation o' the 'irst degree is al(a#s a straight line.
1a$ 0
+
0eneral E"#ation of a Line + 7ormal 7ormal 8quatio 8quation n of a 5traig!t 5traig!t 9ine
* ;$ ;$ 8$ 8$ ? are )onstants )onstants * ; and 8$ not Qero at the same time
Ax % *8 % C ' (
#
-tandard E"#ation of a Line & (o (o 1oi 1oint nt orm orm
R
2+2$#2
+$#
8# similarit# o' triangles
1+1$#1
x
+
1
8 3 84 ' 8$ 3 84 .x 3 x4/
"iven ρ = normal inter)ept
x$ 3 x4
Nρ)osθ$ρsinθ
ρ
= segment 'rom the origin #
θ
perpendi)&lar to the re%&ired line
θ = normal angle 2. 1oint*5lope orm 9n 1 1 repla)ing #2 , #1 b# m$
+$ #
= in)lination o' the normal inter)ept
+2 , +1
rom the point slope 'orm :
θ
8 3 84 ' m .x 3 x 4/
# , #1 = m + , + 1
+
# - ρsin θ = -1 tan θ +,ρ)os θ impli'#ing$ +)osθ #sinθ = ρ
0° ≤ θ ≤ 10°
#
Red#ction to !ormal 1orm :
"iven the line ;+ 8# ? = 0
0$ b
(here m = slope
(here +1 = ρ)os θ$ #1 = ρsin θ m< = -1 tan θ
m = tan θ
5lope* 5lope*6nt 6nterc ercept ept orm orm 8 ' mx % &
ρ)osθ
he normal 'orm is :
+$#
;
b = # inter)ept
+
8
#
+
±√ ;2 82
±√ ;2 82
±√ ;2 82
!ote : he sign o' the radi)and m&st be )hosen s&)h that the last term (ill be)ome negative sin)e ρ V 0.
a 'ind tthe he e%&ation e%&ation o' the median median 'rom 'rom ; b 'ind tthe he e%&ation e%&ation o' the altit&de altit&de ''rom rom 8 ) the interse interse)tion )tion o' medians medians 'rom 'rom 8 to to ?
?
=0
ρsinθ
±√ ;2 82
±√ ;2 82
±√ ;2 82
!ote : he sign o' the radi)and m&st be )hosen s&)h that the last term (ill be)ome negative sin)e ρ V 0.
a 'ind tthe he e%&ation e%&ation o' the median median 'rom 'rom ; b 'ind tthe he e%&ation e%&ation o' the altit&de altit&de ''rom rom 8 ) the interse interse)tion )tion o' medians medians 'rom 'rom 8 to to ? Ans. a " $ #& $ # ( )
4 " 0 & $ - ( )
c *1, -
12. ind the normal inter)ept and the the normal angle o' line 5+12#,3! 5+12#,3! = 0 Ans.
-pecial Cases of a -traight Line
(%,
( 2/.%6
#
Α ;. %&ation o' the + , a+is:
#=0 %&ation o' a horiQontal line : # = b
8. %&ation o' the R-a+is : %&ation o' a verti)al line :
(here b is a )onstant
+=0 +=a
Dist Distan ance ce *et *etwe ween en Para Parall llel el Line Lines s
L4
(here a is a )onstant
<1 : ;+ 8# ? 1 = 0
-UPPLE+E!TAR PR*LE+-:
L$
<2 : ;+ 8# ? 2 = 0
ind the e%&ations o' the lines satis'#ing the given )onditions.
he distan)e bet(een the t(o
1. assing thro&gh 1$ -2 and perpendi)&lar to the line thro&gh 2$ -1 and -3$ 2
d
lines is given b# the 'orm&la
Ans. '" $ %& $ -- ( )
d = ?2 , ?1 L L
2. ith + inter)ep inter)eptt o' 5 and passing passing thro&gh thro&gh 3$ 3$ 4 ;ns. #" 0 & $ -) ( ) 3. assing assing thro&gh thro&gh -3$ -3$ 4 and (ith th e%&al e%&al inter)epts inter)epts Ans. " $ & 0 / ( ) and " 0 & $ - ( )
√ ;2 82 -ample Pro&lems:
4.
MaFing an angle o' 45° (ith the +-a+is and passing thro&gh 2$ 3 Ans. " $ & $ - ( )
5. it ith slop slopee - 125 )rosses the 'irst %&adrant and 'orms (ith the a+es a triangle (ith perimeter o' 15. Ans. '" 0 -#& $ % ( )
1. ind the the distan)e distan)e 'rom 'rom point 3$ 3$ -1 to the the line 3+ 3+ , 4# , 3 = 0 ol&tion : Gere$ ; = 3$
8 = - 4$
Osing the 'orm&la d = ;+0 8#0 ?
√ ;2 82
6. assing assing thro&gh thro&gh 7$ -4 and and at a distan)e distan)e o' 1 &nit 'rom the the point point 2$ 1 Ans. 1" 0 %& $ -2 ( )
3
%" 0 1& $ ' ( )
7. assing assing thro&gh thro&gh the midpoint nt o' the segment segment Coining Coining the points points 1$ 3 and 5$ 1 and parallel to the line 2+ , # 5 = 0 Ans. #" $ %& $ '
9.
c 1 5 ' '
d ' $ #nits the point 3$ -1 and the origin are on the opposite side o' the line
P = ?2 , ?1 = 1 , 1
√ ; 8 2
ind the e%&ations o' the lines parallel to the line + 2# , 5 = 0 and passing at a distan)e 2 'rom the origin Ans. " 0 #& 0 # ' ( ) and " 0 #& + # ' ( ) Ans. " $ & 0 - ( )
= 33 -4-1 , 3 √ 32 42
; = $ 8 = 15$ ?2 = 1$ ? 1 = 1
d - 5 %
10. ind the e%&ation o' the perpendi)&lar perpendi)&lar bise)tor o' the segment Coining Coining 2$ 5 and
0+0$ #0 ↔ 3$ -1
2. ind the distan)e bet(een parallel lines nes + 15# 1 = 0 and + 15# 1 = 0. ol&tion :
. ind the the val&e val&e o' paramete parameterr F so that the the line 3+ 3+ , 5F# 5 = 0 a (ill (ill pass pass thro& thro&gh gh 0$ 0$ 1 b (ill (ill be para paralle llell to + 2# = 5 ) (ill be perpendi perpendi)&lar )&lar to 4+ 3# = 2 d has the the #-inter #-inter)ept )ept e%&al to 3 Ans. a - 4 $2 5 ' '
? = -2
4$ 3.
2
= 4 #nit
Answer
√ 15 2
2
Distance from a Point to a Line
he dire)ted distan)e 'rom a point + 0$#0 to a line ;+ 8# ? = 0 is given b# the 'orm&la: d = ;+0 8#0 ?
± √ ;2 82
11. "iven verti)es o' a triangle ;8?$ ;8?$ ;2$ 0* 83$ -2 and ?7$ 5
(here the sign o' the radi)al is )hosen to be the opposite that o' ?. Remar@s:
A-+PTTE - a straight line (hi)h the the )&rve '+$ # = 0 approa)hes inde'initel# inde'initel# near as its
tra)ing point approa)hes to in'init#. • o 'ind the verti)al as#mptote$ solve the e%&ation 'or # in terms o' + and set the
(here the sign o' the radi)al is )hosen to be the opposite that o' ?.
A-+PTTE - a straight line (hi)h the the )&rve '+$ # = 0 approa)hes inde'initel# inde'initel# near as its
!otes:
tra)ing point approa)hes to in'init#. • o 'ind the verti)al as#mptote$ solve the e%&ation 'or # in terms o' + and set the linear 'a)tors o' the denominator e%&al to Qero. • o 'ind the horiQontal as#mptote$ solve the e%&ation 'or + in terms o' # and set the linear 'a)tors o' the denominator e%&al to Qero.
Aegardless o' the lo)ation o' the point 0+0$ #0$ the distan)e being al(a#s positive the 'orm&la )an be e+pressed &sing the absol&te val&e as:
CIRCLE :ircle is the lo)&s o' a point (hi)h moves so that it is al(a#s e%&idistant 'rom a
Remar@s:
1. 9' d V 0$ the origin origin and lie on opposite opposite sides sides o' the given given line. line. 2. 9' d K 0$ the the origin origin and lie on on the same same side o' the the line. line.
d = ;+0 8#0 ?
√ ;2 82
'i+ed point.
!ote: 'i+ed point is )alled the center i+ed distan)e is )alled the radius
Line Thro#gh the Intersection of Two Lines
E78ation o9 a Circ:e
9n normal 'orm ?onsider a )ir)le o' radi&s r (ith )enter at ?&$ F
(here F is an arbitrar# )onstant.
+$ +$# # r ?h$F
I!TERCEPT 1 A CUR7E • x*intercept , , dire)ted distan)e 'rom the origin to the point (here the )&rve )rosses the +-
a+is
•
the #-a+is to 'ind the #- inter)ept o' a )&rve$ set + = 0$ then solve 'or #. -++ETR
•
9' the e%&ation o' a )&rve does not )hange &pon repla)ement o' # b# ,#$ then the lo)&s is s#mmetri) (ith respe)t to the +-a+is. '+$ -# = '+$# =0 • 9' an e%&ation o' a )&rve does not )hange &pon repla)ement o' + b# ,+$ then the lo)&s is s#mmetri) (ith respe)t to the #-a+is '-+$# = '+$# = 0 • 9' an e%&ation o' a )&rve does not )hange &pon repla)ement o' + b# ,+ and # b# ,#$ then the lo)&s is s#mmetri) (ith (ith respe)t to the the origin. '-+$ -# = '+$ # = 0
Radical Axis of Two Circles
?onsider the t(o non-)on)entri) )ir)les +2 #2 P1+ 1# 1 = 0 2
8# #th #thag agor orea eann heo heore rem m → standard 'orm
.x 3 h/$ % .8 3 @/ $ ' r $ +,h
?enter at the origin ?0$ 0
x$ % 8$ ' r $
o 'ind the + inter)ept o' a )&rve$ set # = 0$ then solve 'or +.
y*intercept , , the dire)ted distan)e 'rom the origin to the point (here the )&rve )rosses
2
#,F
0
+
"eneral orm +panding the 'orm + , h 2 # , F 2 = r 2 be)omes +2 #2 , 2+h , 2F# h2 F2 , r 2 = 0 his is o' the 'orm: x$ % 8$ % Dx % E8 % 1 ' (
general form
(here P$ $ are )onstants not all Qero at a time. !ote: 8# e%&a e%&atition on o' )oe' )oe''i'i)i )ien ents ts:: - 2h = P * h = -X P → abs)issa o' )enter -2F = * F = -X → ordinate o' )enter h2 F2 , r 2 = * r = √h2 F2 ,
P. ;ll tangents dra(n dra(n to t(o )ir)les 'rom 'rom a point on their radi)al a+is have e%&al lengths.
Radical Axis of Two Circles
?onsider the t(o non-)on)entri) )ir)les +2 #2 P1+ 1# 1 = 0 +2 #2 P2+ 2# 2 = 0 he e%&ation:
P. ;ll tangents dra(n dra(n to t(o )ir)les 'rom 'rom a point on their radi)al a+is have e%&al lengths. #
Aadi)al ;+is
x$ % 8$ %D4x % E48 % 14 % @ .x $ % 8$ %D$x % E$8 % 1$/ ' (
represent a )ir)le 'or an# val&e o' F e+)ept 'or F = -1
1
i' F = -1$ the e%&ation o' the 'amil# o' )ir)les above be)omes: P1 , P2 + 1 , 2 # 1 , 2 = 0 his represents a straight line )alled the ;"6:"9 "65 o' t(o )ir)les.
1 Y 2 are points o' tangen)#
2
PT4 ' PT$
+
Properties of the Radical Axis
;. 9' t(o )ir)les interse)t interse)t at t(o distin)t distin)t points$ their their radi)al a+is is the the )ommon )hord o' the )ir)les.
#
%8:8 ?oard 1roblem – ct &/&) Ans; &/&) Ans; C*#, %
+
0 :ondition for rt!ogonality
r ( '
2. ind the the area area o' the the )ir)le (hose e%&ation e%&ation +2 #2 = 6+ , # ? 8oard roblem , Mar. 1!1 ;ns: 25π s%. &nits
?ommon ?hord
he t(o non-)on)entri) non-)on)entri) )ir)les : +2 #2 P1+ 1# 1 = 0 +2 #2 P2+ 2# 2 = 0$ meet at right angles orthogonal i' : D4D$ % E4E$ ' $.14 % 1$/
8. 9' t(o )ir)les )ir)les are tangent$ tangent$ their their radi)al a+is a+is is the )ommon tangent tangent to the )ir)les )ir)les at their point o' tangen)#.
#
-#pplementar8 Pro&lems
1. ind the )enter )enter and radi&s radi&s o' the the )ir)le (hose (hose e%&ation e%&ation is +2 #2 , 4+ ,6# ,12 = 0
3. ind the e%&ation e%&ation o' the )ir)le (hose (hose )enter is at 3$ -5 and (hose radi&s is 4 &nits. Ans; *" $ %# 0 *& 0 ' # ( -2
or roblems 4 , !$ determine the e%&ation o' the )ir)le given the 'ollo(ing )onditions 4. asses asses thro&gh thro&gh the point point 2$ 2$ 3$ 6$ 1 1 and 4$ -3 -3 Ans; " # 0 & # $ -)& ( ) 5. ?enter ?enter on the # , a+is$ and passes passes thro&gh thro&gh the origin origin and point point 4$ 2. 2. Ans; " # 0 & # $ -)& ( )
Aadi)al ;+is
6. asses asses thro&gh thro&gh the points points o' interse)t interse)tion ion o' the )ir)les )ir)les +2 #2 = 5$ +2 #2 ,+ # = 4$ and thro&gh the point 2$ -3 Ans; " # 0 & # $#" 0 #& $% ( ) 7. ?enter on the line line + , 2# ,! = 0 and passes thro&gh the the points 7$ -2 and 5$ 0 Ans; " # 0 & # $ -)" 0 1& 0#' ( )
+
0
. ?ir)&ms)ribe ?ir)&ms)ribe the the triangle triangle determin determinee b# the lines lines + , & , = -# and # = -1.
?. he radi)al a+is a+is o' t(o )ir)les is perpendi)&lar perpendi)&lar to their their line o' )enters.
# ;
Ans; " # 0 & # $6" 0 #& 0 6 ( )
!. "iven the the endpoint endpointss o' the the diameter diameter 5$ 5$ 2 -1$ -1$ 2 Ans; " # 0 & # $ 1" $ 1& $ - ( )
Aadi)al ;+is
;8 , line o' )enters
8
10. ind the e%&ation o' the line ne tangent to the )ir)le +2 #2 , + , # 7 = 0 at the point 1$ 0 Ans; %" 0 1& $ % ( )
0
+
PARA*LA
he lo)&s o' a point that moves in a plane s&)h that its distan)e 'rom a 'i+ed point e%&als its distan)e 'rom a 'i+ed line.
0 a(
+
PARA*LA
he lo)&s o' a point that moves in a plane s&)h that its distan)e 'rom a 'i+ed point e%&als its distan)e 'rom a 'i+ed line.
!otes:
• • • • •
i+ed point is )alled focus i+ed line is )alled directrix "xis , the line passing thro&gh the 'o)&s and perpendi)&lar to the dire)tri+ @ertex , , he midpoint o' the segment o' the a+is 'rom the 'o)&s to the dire)tri+. 9atus rectum , a segment passing thro&gh the 'o)&s and perpendi)&lar to the a+is o' the parabola. • ocal distance , distan)e 'rom verte+ to 'o)&s = a
0 a(
+
?. \erte+ \erte+ at the rigin$ rigin$ \erti) \erti)al al ;+is x$ ' Ba8
i' a is positive a ----- )on)ave &p(ard i' a is negative -a ----- )on)ave do(n(ard
!o !otes:
Standard E78ations o9 ara4o:a
;. \erte+ at \h$F$ \h$F$ \erti)al ;+is ;+is .x>h/$ ' Ba.8>@/
1. ;+is : the # a+is 2. o)& o)&s: s: 0$ 0$a a 3.
i' a is positive a ----- )on)ave &p(ard i' a is negative-a ----- )on)ave do(n(ard
#
<-2a$a
a+is
A2a$a
\0$0 dire)tri+
P. \erte+ \erte+ at the rigin$ rigin$ GoriQo GoriQontal ntal ;+is ;+is
!otes:
1. %&ation o' a+is : +=h 2. o)& o)&ss : h h$F $F a 3. nd o'
8$ ' Bax
# a+is <
i' a is positive a ----- )on)ave to the right i' a is negative -a ----- )on)ave tot he le't
A
\h$F 0
!o !otes:
dire)tri+
+
8. \erte+ \erte+ at \h$F \h$F$$ GoriQonta GoriQontall ;+is ;+is $
.8>@/ ' Ba.8>@/
i' a is positive a ----- )on)ave to the right i' a is negative -a ----- )on)ave to the le't
1. ;+is : the +-a+is 2. o o)&s: 'a$0 3.
# dire)tri+ 7
< 1
+ a+is A
Remar@s:
1. he verte+ verte+ and 'o)&s 'o)&s al(a#s al(a#s lie on the a+is o' the the parabola. parabola. 2. o)&s is al(a#s al(a#s lo)ated lo)ated on the )on)ave )on)ave side o' the the parabola. parabola.
!otes:
1. %&ation o' the a+is: #=F 2. o)&s: ha$ F 3. nds nds o' o'
-#pplementar8 Pro&lems
#
dire)tri+ < \ h$F
Genera: E78ations o9 ara4o:a
a+is
A
1. ind the verte+$ 'o)&s$ and end points nts o' the the
& @ert @ertic ical al "xis "xis • Ax$ % Dx % E8 % 1 ' ( $ or ; m&st not be Qero 2 Aori AoriBo Bont ntal al "xi "xis s • C8$ % Dx % E8 % 1 ' ( $ P or ? m&st not be Qero
Dajor axis , the segment )&t b# the ellipse on the line )ontaining the 'o)i a segment Coining the t(o verti)es o' an ellipse o' length e%&al to -
92a9
-#pplementar8 Pro&lems
1. ind the verte+$ 'o)&s$ and end points nts o' the the
b.
-
5 6 6
Dajor axis , the segment )&t b# the ellipse on the line )ontaining the 'o)i a segment Coining the t(o verti)es o' an ellipse o' length e%&al to -
iameter s, s, the )hords o' an ellipse that pass thro&gh the )enter @ertices , the endpoints o' the diameter thro&gh the 'o)i the endpoints o' the maCor a+is 9atus ;ectum , the segment )&t b# the ellipse passing thro&gh the 'o)i and
8ccentricity , , meas&re the degree o' 'latness o' an ellipse
92a9
2#2 , 5+ 3# - 7 = 0 Ans; V*+ -% 5 6 6, + % 5 1 , <*+-, + % 5 1 , EL*+-, + % 5 1
'
5 1
2. ind the e%&ation o' the parabola determined determined b# the the given )onditions a. 'o)& 'o)&ss at -114$ 1 and the endpoint o' lat&s re)t&m is - 114$ 52 Ans; & # 0 %" $ #& 0 / (
perpendi)&lar to the maCor a+is
)
b. verte+ verte+ at 1$ 1$ -1 -1 and 'o)&s at 1$ 1$ -34 Ans; " # $ #" $ & ( ) ). verte+ verte+ aatt 0$ 3 3 dire)t dire)tri+ ri+ + = -1 -1 Ans; & $ 1" $ 2& 0 = ( ) #
d. a+is verti)al verti)al$$ verte+ -1$ -1$ -1 and passing passing thro&gh thro&gh 2$ 2 Ans; " # 0 #" $ %& $ # ( ) 3. ; )hord passing passing thro&g thro&ghh the 'o)&s 'o)&s o' the the parabola parabola #2 = 16+ has one end at the point 1$ 4. here is the other end o' the )hordT Ans; *1, 6 4. ind the the e%&ation on o' the line line tangent tangent to the parabola parabola +2 2+ 3# , 1 = 0 "ns3 2x C y – & = 0
5. ind the e%&ation o' the )ir)le that that passes thro&gh thro&gh the verte+ verte+ and the endpoints o' lat&s re)t&m o' the parabola #2 = +. Ans; " # 0 & # $ -)" ( ) 6. ind the e%&ation o' parabola (hose a+is is horiQontal$ horiQontal$ verte+ is on the # , a+is and (hi)h thro&gh 2$ 4 and $ -2 Ans; & # ( #)& $-6 0-)) ( ) $ & # ( 1& 0 #" 0 1 () 7. ;n ar)h in the 'orm o' paraboli) )&rve$ (ith a verti)al verti)al a+is is 60 m$ a)ross the bottom. bottom. he highest point is 16 m above the horiQontal base. hat is the length o' a beam pla)ed horiQontall# a)ross the ar)h 3m belo( the topT Ans; #2 m . ;ss&me that (ater iss&ing 'rom the end o' a horiQontal pipe$ 25 't. 't. above the the gro&nd des)ribe a paraboli) )&rve$ the verte+ o' the parabola being at the end o' the pipe$ the 'lo( o' (ater has )&rve o&t(ard 10 't. be#ond a verti)al line thro&gh the end pipe$ ho( 'ar be#ond this verti)al line (ill the (ater striFes the gro&ndT Ans; -/. 26 9t. ELLIP-E 8llipse is the lo)&s o' a point +$ # in a plane (hi)h moves s&)h that the s&m o' its
distan)es 'rom t(o 'i+ed points is )onstant.
-tandard E"#ation of Ellipse Center at C.h6 @/6 )oriontal +aor Axis
#
.x 3 h/ $ >>>>>>>>>>>> a$ !otes:
.8 3 @/$ % >>>>>>>>>>> ' 4 &$
1. MaCor a+is : # = F 2. Minor a+is * + = h 3. \ert \erti) i)es es : \1h-a$ F \2ha$ F 4. o)i : 1h-)$ F 2h)$ F
\1
b
1
?h$F
2
b
)
)
a
0
Center C*>, ?, Vertica: Ma@or A"is
a
y
@ &
1
a
.x 3 h/ $ .8 3 @/$ 33333333 % 33333333 ' 4 a$ &$ !otes:
1. MaCor a+is : + = h 2. Mino Minorr a+is a+is * # = F 3. \ert \erti) i)es es : \1h$ Fa \2h$ F-a 4. o)i : 1h$ F) 2h$ F-)
) b
2 \2
0
+
Center at t>e Oriin, Borionta: Ma@or A"is
+2
he t(o 'i+ed points are )alled foci
-- ------ ------ = 1 a2 b2
#
b )
?h$F
Notes:
\2
#2
0eneral E"#ation of an Ellipse Ax$ % C8$ % Dx % E8 % 1 ' ( (here ; ≠ ? b&t o' the same sign
a
-- ------ ------ = 1 a2 b2
#
0eneral E"#ation of an Ellipse Ax$ % C8$ % Dx % E8 % 1 ' ( (here ; ≠ ? b&t o' the same sign
!otes:
5. MaCor o r a+is a+is : + = a+is a+is 6. Minor a+is * # = a+is 5. \ert \erti) i)es es : \1-a$ 0 \2a$ 0 7. o)i : 1-)$ 0 2)$ 0
\1
1 ?0$0
2
-#pplementar8 Pro&lems:
\2
)
1. 9n the ellipse ellipse belo( determine determine the 'ollo(ing: a ?enter ?enter b \erte+ \erte+ )o)i d MaCor MaCor ;+is ;+is e MaCor ;+is '
)
a
a
Ans; A. a C*-, +-3 4 V-*-, =, V#*-, V#*-, +--3 c <-*-, ', <#*-, <#*-, +/ d #) e -2 9 -#.6 ).2 !. a C*+%, )3 4 V-*+-2, V-*+-2, ), V#*-), )3 c <-*+6, <-*+6, ), <#*#, ) ' d #2 e #1 9 #66 5 -% -% 5 -% -%
2. 9n ea)h o' the 'ollo(ing ng 'ind the the e%&ation o' o' ellipse satis'#ing satis'#ing the given )onditions A. )enter at 0$ 0$ 'o)&s at ±√3$ 0$ and b = 1
\1
Center at the rigin6 7ertical +aor Axis
Ans; " # 0 1& # ( 1 B.
+2 #2 ------- ------ = 1 a2 b2
a
1
) b
!otes:
. MaCor o r a+is a+is : # = a+is a+is !. Minor a+is * + = a+is 10. \erti)es \erti)es : \10$ a \20. -a 11. o)i o)i : 10$ ) 20$ -)
b
'o)&s at 0$ -1$ -4$ -1$ a = √6 a = 52 b = 2
Ans; -2" # 0 #'& # 0 -2" 0 1 ( -))&
)
a
2 \2
\erti)es )es and 'o)i lie lie on the the maCor or a+is a+is 9a9 is the the distan)e distan)e 'rom 'rom the )enter )enter to the the verte+ verte+ 9)9 is the distan) distan)ee 'rom the )enter )enter to the the 'o)i 'o)al distan) distan)e e he ellipse pse is s#mmetri) s#mmetri)al al to the maCor$ maCor$ minor minor a+es and the the )enter. )enter.
Important Relations
1. 2. 3. 4.
C.
P. )ent )enter er at -12$ 2
?0$0 ?0$0
e= √32
Ans; " # 0 %& # 0 1" 0 2& 0- ( )
0eneral Remar@s
1. 2. 3. 4.
)enter at 1$ 0 'o)&s at 1$ √3
Ans; 1& # 0 & # ( 6"
a V b$ a V ) a2 = b2 )2 e = e))en e))entri tri)it )it## = )a )a K 1
. ;n ar)h in the 'orm o' a semi semi , ellipse has a span at 45 m and and its greatest height ght is 12m. here are t(o verti)al s&pports e%&idistant 'rom ea)h other and the ends o' the ar). ind the height o' the s&pport.
. )enter )enter at at 0$ 0$ verte+ verte+ 0$ 0$ 4
e =X
Ans; 1" # 0 %& # $ 16 ( )
3. ; satellite orbits aro&nd the earth in an ellipse ellipse orbit o' e))entri)all# o' 0.0 and semi , maCor a+is o' length 20$000 Fm. 9' the )enter o' the earth is at one 'o)&s$ 'ind the ma+im&m altit&de apogee o' the satellite. Ans; %2, ))) ?m 4. ind the e%&ation on o' the lo)&s o' o' a point (hi)h )h moves so that the the s&m o' its distan)e 'rom -2$ 2 and 1$ 2 is 5. Ans; -2" # 0 #'& # 0 -2" $ -))& 01 ( ) 5. ind the e))entri)it# o' an ellipse (hose (hose maCor a+is a+is is thri)e a long as its minor a+is Ans; # # 5 %
6. hat is the the %&adrilateral 'ormed b# Coining oining the 'o)i o' an ellipse to the endpoints o' the minor a+isT Ans; r>om48s 7. ind the distan)e o' the point 3$ 4 to the 'o)i o' the ellipse ipse (hose e%&ation e%&ation 4+2 !#2 = 36 Ans; %
'
1. ransverse a+is: # = F 2. ?onC&g ?onC&gat atee a+is: a+is: + = h transverse
. ;n ar)h in the 'orm o' a semi semi , ellipse has a span at 45 m and and its greatest height ght is 12m. here are t(o verti)al s&pports e%&idistant 'rom ea)h other and the ends o' the ar). ind the height o' the s&pport.
1. ransverse a+is: # = F 2. ?onC&g ?onC&gat atee a+is: a+is: + = h transverse 3. \ert \erti) i)es es:: \1 h , a$ F \2 h a$ F 4. o)i: 1 h , )$ F 2 h )$ F 5. ;s#mptotes: # , F = ± ba + , h
Ans; 6 # m.
!. Petermine the lo)&s o)&s o' a point +$ # so that the prod&)t o' the the slopes Coining +$ +$ # to 3$ -2 and -2$ 1 is ,6. Ans; 2" # 0 & # $ 2" 0& $ #) ( ) 10.
a+is
)onC&gate a+is 0
+
hat is the area o' the ellipse (hose e%&ation is 25+ 2 16#2 = 400. Ans; #) s7. 8nits
)PER*LA
G#perbola is the lo)&s o' point +$ # in a plane (hi)h moves s&)h that the di''eren)e o' its distan)es 'rom t(o 'i+ed points is a positive )onstant.
Center at C.h6 @/6 7ertical Transerse Axis
!otes:
.8 3 @/$ F a$
he t(o 'i+ed points are )alled foci rans#erse axis , a line segment Coining the t(o verti)es o' h#perbola the length o' the transverse a+is is 9 2a 9 :onjugate "xis , the perpendi)&lar bise)tor o' the transverse a+is. the length o' the )onC&gate a+is is 92b 9 :enter , , point o' interse)tion o' transverse and )onC&gate a+is :entral ;ectangle , the re)tangle re)tangle (hose area is 2a 2b 2b and (hose diagonals are as#mptotes o' h#perbola @ertices , the endpoints o' the transverse a+is "symptotes of t!e !yperbola , t(o interse)ting lines )ontaining the diagonal o' the )entral re)tangle ,o 'ind the e%&ation e%&ation o' the as#mptote$ set the the right side o' the the e%&ation o' h#perbola in standard 'orm to Qero then solve 'or #. :entral :ircle , the )ir)le o' radi&s ) (ith )enter at the )enter o' the h#perbola )ir)&ms)ribing the )entral re)tangle 8quilateral Ayperbola , h#perbola (hose transverse a+is e%&als its )onC&gate a+is :onjugate Ayperbolas , h#perbolas (hose transverse a+is o' one is the )onC&gate a+is o' the other
.x 3 h/$ &$
ransverse a+is
' 4
!otes:
1. 2. 3. 4. 5.
;s#mptote
1
a+is ransv ransvers ersee a+is: a+is: + = h ?onC&g ?onC&gate ate a+is: a+is: # = F \ert \erti) i)es es:: \1 h$ F a \2 h$ F , a o)i: 1 h$ F a 2 h$ F , ) ;s#mptotes: # , F = ± ab + , h
?onC&gate
\1
?
\2 2
Center at the rigin6 )oriontal Transerse Axis x$ a$
F
8$ &$
#
' 4
!otes:
-tandard E"#ations of )8per&ola
?enter at ?h$F$ GoriQontal ransverse ;+is .x 3 h/ h /$
a2
F .8 3 @/ $ ' 4
b2
8 1
;s#mptote
1. ransv ransvers ersee a+is: a+is: +,a+ +,a+is is 2. ?onC&g ?onC&gate ate a+is: a+is:1 #,a+ #,a+is \1 is \2 3. \erti) \erti)es: es: \1 -a$ -a$ 0 ? \2 a$ 0 4. o)i o)i:: 1 1 -)$ -)$ 0 2 )$ 0 ? # = ± ba + \5. ;s#mptotes: \ 1
2
2
+
2
Center at the origin 6 7ertical Transerse Axis
!otes:
8$ a$
F
x$ ' 4 &$
# c. 1
C*),)3 V- *$1,),V # * 1,)3 <*$ 1- ,), < # * 1- ,)3 #*1,)3 #* As&mtote * & ( ' 5 1 " 3 TA ( 63 CA ( -)3LR ( #' 5 # #3 e (
1-
5 1
8$ a$
F
#
x$ ' 4 &$
c. 1
!otes:
\1 ?
1. ransv ransvers ersee a+is: s : # , a+is a+is 2. ?onC&g ?onC&gate ate a+is: a+is: + , a+is a+is 3. \erti) \erti)es: es: \1 0$ a \2 0$ -a 4. o)i o)i:: 1 1 0$ 0$ ) 2 0$ -) 5. ;s#mptotes: # = ± ab +
\2
C*),)3 V- *$1,),V # * 1,)3 <*$ 1- ,), < # * 1- ,)3 #*1,)3 #* As&mtote * & ( ' 5 1 " 3 TA ( 63 CA ( -)3LR ( #' 5 # #3 e (
1-
5 1
2. ind the e%&ation e%&ation o' the h#perbola satis'#ing the )onditions given in ea)h )ase )ase a. ?enter3$, ?enter3$,1* 1* verte+ verte+1$,1* 1$,1* 'o)&s0$,1 'o)&s0$,1 Ans. '" # $1& # $%)"$6&0#-()
2
b. verti)es verti)es at0$4 at0$4 and 4$4* 4$4* 'o)i 'o)i at 5$4 5$4 and and ,1$4 ,1$4 Ans. '" # $1& # $#)"0%#&$21()
). ?enter ?enter at 1$1$ 1$1$ verte+1 verte+1$3$ $3$ e))entr e))entri)it# i)it#=2 =2 Ans. " # $%& # $#" 0-) ()
d. Pire Pire)t )tri ri)e )es: s: #= #=±4* as#mptotes: #= ±32+
0eneral Remar@s
1. 2. 3. 4.
\erti)es )es and 'o)i are are on the the transvers transversee a+is 9a9 is the the distan)e distan)e 'rom 'rom the )enter )enter to the the verte+ verte+ 9)9 is the the distan)e distan)e 'rom 'rom the )enter )enter to the the 'o)&s. 'o)&s. he h#perbola is s#mmetri)al to the transverse transverse and )onC&gate )onC&gate a+is and to the )enter. )enter.
Ans. =& # $%#1" # $#)6 ( )
e. ;s#m ;s#mpt ptot otes es:: 3#= 3#=±4+ * 'o)i ±6$0 Ans. 1))" # $##'& # $'-61()
'. 1. ) V a$ a$ ) V b a = b or a K b or a V b 9' a = b$ then the h#perbola is )alled ed e%&ilateral h#perbola $ 2. c ' a$ % &$ 3.
o)i0 o)i0$0 $0$$ 0$10* 0$10* as#mpto as#mptote: te: +#=5 +#=5 Ans. #" # $#& # 0#)&$#'()
0eneral Relations
a
0eneral E"#ation of )8per&ola Ax$ % C8$ % Dx % E8 % 1 ' (
g. ;s#mptotes ;s#mptotes:: +#=1 and +,#=1 +,#=1 and passing thro&gh thro&gh ,3$4 ,3$4 and 5$6 5$6 Ans. " # $& # $#"$#()
h. ;+es along along the )oordina )oordinates tes a+es$ a+es$ passing passing thro&gh thro&gh ,2$5 Ans. 1& # $'" # (-=
i.
\ert \erti) i)es es at 0$ 0$ ±4 passing thro&gh ,2$5
Ans. 1& # $=" # ( 21
(here ; and ? are o' opposite signs
3. ind the e))entri)it# e))entri)it# o' a h#perbola (hose (hose transverse a+is a+is and )onC&gate a+is are e%&al e%&al in length Ans. e( #
-#pplementar8 Pro&lems
1. ind the )enter$ verti)es$ 'o)i and as#mptotes$ as#mptotes$ transverse transverse a+is$ )onC&gate )onC&gate a+is$ lat&s lat&s re)t&m and e))entri)it# o' h#perbola belo(. a. !+2 ,16#21+64#=!1 b. 16+2 ,4#2 ,62+24#!2=0 ). 25+2 ,16#2=400 Ans. a. C*$-,#3 V*$- #, #3 <*$- ' 5 # *%"01&$'(), %"$1&0--()3 # , #3 as&mtote *%"01&$'(), TA ( 13 CA ( %3LR ( = 5 13 e ( ' 5 1 4.
C*#,%3 V -*#,-,V # #*#,/3 *#,/3 <*#,%0 #) , < # #*#, *#, %$ #) 3 as&mtote*#"$&$-(), as&mtote*#"$&$-(), #"0&$/()3 TA ( 63 CA ( 13LR ( #3 e ( ' 5 # #
T)E C!IC -ECTI!- %a -ECTI!- %a summary) ; conic section is the lo)&s o' a point (hi)h moves s&)h that its distan)e 'rom a 'i+ed point )alled focus is in )onstant ratio and )alled eccentricity to to its distan)e 'rom a 'i+ed straight line )alled directrix A2 0eneral 0eneral 1orm 1orm of a Q#adratic Q#adratic E"#ati E"#ation on in x and 8 Ax$ % C8$ % Dx % E8 % 1 ' (
1. llips l ipsee : ; ≠ ?$ same signs 2. ?ir ?ir)l )lee : ; = ?$ same same sign signss 3. G G#pe #perbo rbola la : ; and and ? have have oppos opposite ite signs signs
*2 Eccent Eccentric ricit8 it8 e ' cGa cGa
1. ?ir) le : e = 0 2. arabola :e=1 3. lli llips psee : e K 1
1. he s&m o' the digits o' a t(o-digit n&mber is 11. 9' the digits are reversed$ the res<ing n&mber is seven more than t(i)e the original n&mber. hat is the original n&mberT a. 3 b . 53 ) . 3 d. 44
1. ?ir) le : e = 0 2. arabola :e=1 3. lli llips psee : e K 1 4. G#pe G#perb rbol olaa : e V 1
Diameter of a Conic , the lo)&s o' the midpoints o' a s#stem o' parallel )hords. Direction; Encirc:e t>e :etter corresondin corresondin to t>e correct anser.
1. he s&m o' the digits o' a t(o-digit n&mber is 11. 9' the digits are reversed$ the res<ing n&mber is seven more than t(i)e the original n&mber. hat is the original n&mberT a. 3 b . 53 ) . 3 d. 44 2. ; metal (asher 1-in)h in diameter is pier)ed b# X-in)h hole. hat is the vol&me o' the (asher i' it is 1 in)h thi)F. a. 0.074 b. 0.047 ). 0.02 d. 0.02 3. 9' a reg&lar pol#gon has 27 diagonals$ then it is a$ a. nonagon b. pentagon ). he+agon d. heptagon 4. ind the probabilit# o' getting e+a)tl# 12 o&t o' 30 %&estions on a tr&e or 'alse %&estion. a. 0.12 b. 0.0 ). 0.15 d. 0.04 5. ind the area bo&nded b# the )&rve y = 9 − x 2 and the +-a+is a. 1 &nits 2 b. 25 &nits2 ). 36 &nits2 d. 30 &nits 2 6. 9t is a meas&red o' relationship bet(een t(o variables. a. &n)tion b. Aelation ). ?orrelation d. %&ation 7. ; )entral angle o' 45 degrees degrees s&btends an ar) o' 12 )m. (hat is the radi&s o' the the )ir)leT a. 15.2 )m b. 12.2 )m ). 12.5 )m d. 15.2 )m . (o posts$ one m and the other 12 m high are 15 )m apart. 9' the posts are s&pported b# a )able r&nning 'rom the top o' the 'irst post to a staFe on the gro&nd and then ba)F to the top o' the se)ond post$ 'ind the distan)e 'rom the lo(er post to the staFe to &se minim&mT a. 6 m b. m ). ! m d. 4 m !. ; reg&lar o)tagon is ins)ribed in a )ir)le o' radi&s 10. ind the area o' the o)tagon. a. 22.2 b. 2.2 ). 23.2 d. 22.2 10. he vol&me o' the t(o spheres is in the ratio 27:343 and the s&m o' their radii is 10. ind the radi&s o' the smaller sphere. a. 5 b. 4 ). 3 d. 6 11. ind the appro+imate )hange in the vol&me o' a )&be o' side + in)hes )a&sed b# in)reasing its side b# 1[. a. 0.30+2 in3 b. 0.02 in3 ). 0.1+3 in3 d. 0.03+3 in3 12. he time re%&ired 'or the e+aminees to solve the same problem di''er b# t(o min&tes. ogether the# )an solve 32 problems in one ho&r. Go( long (ill it taFe 'or the slo(er problem solver to solve a problemT a. 5 min&tes b. 2 min&tes ). 3 min&tes d.4 min&tes 13. 9' a = b$ then b = a. his ill&strates (hi)h a+iom in ;lgebraT a. ransitive a+iom b. Aepla)ement a+iom ). Ae'le+ive a+iom d. #mmetri) a+iom 14. ind the distan)e o' the dire)tri+ 'orm the )enter o' an ellipse i' its maCor a+is is 10 and its minor a+is is . a. .5 b. .1 ). .3 d. .7 15. ; reg&lar he+agonal p#ramid has a slant height o' 4 )m and the length o' ea)h side o' the base is 6 )m. ind the lateral area. a. 62 )m2 b. 52 )m2 ). 72 )m2 d. 2 16. ;t the s&r'a)e o' the earth ! = !.06 ms 2. ;ss&ming the earth to be a sphere o' radi&s 6.371 + 106 m$ )omp&te the mass o' the earth. a. 5.12 + 10 24 Fg F g b. . 5.!7 + 1023 Fg ). 5.!7 + 10 24 Fg Fg d. . 5.62 + 10 24 Fg 17. he perimeter o' an isos)eles right triangle is 6.624. 9ts area is
a. 4 b. 2 ). 1 d. X 1. Petermine the verti)al press&re d&e to a )ol&mn o' (ater 5-m high. a. .33 + 105 Nm2 b. .33 + 10 4 Nm2
32. ind the s&m o' the rots o' 5 x 2 − 10 x + 2 = 0. a. -12 b. - 2 ). 2 d. X 33. 9n a bo+ there are 25 )oins )onsisting o' %&arters$ ni)Fels and dimes (ith a total amo&nt
!ote:
he )ir)le$ parabola$ ellipse and h#perbola are )alled )oni) se)tions or )oni)s be)a&se an# one o' them )an be obtained geometri)all# b# )&tting a )one (ith a plane.
:ircle
8llipse
1arabola
Ayperbola
9' the )&tting plane is perpendi)&lar perpendi)&lar to the a+is o' the )one$ the se)tion se)tion is a circle 9' the )&tting plane is maFing an angle othe otherr than !0o (ith the a+is o' the )one$ the se)tion is an ellipse 9' the )&tting plane is parallel to one o' the elements o' a )one$ the se)tion is a parabola 9' the )&tting plane is parallel b&t not )oin)ident to the a+is o' the )one$ the se)tion is a !yperbola
9n ea)h o' the )ases$ the )&tting plane sho&ld not pass thro&gh the verte+ o' the )one$ other(ise the se)tion hat (ill be 'ormed is a deenerate conic. Degenerate Conic one point$ one line$ t(o t(o lines is a )oni) 'ormed i' the )&tting plane plane is
passing thro&gh the verte+ along one o' its elements
Principal Axis of a Conic , is the line thro&gh the 'o)&s and perpendi)&lar to the
dire)tri+
6
2
3
2
a. 4 b. 2 ). 1 d. X 1. Petermine the verti)al press&re d&e to a )ol&mn o' (ater 5-m high. a. .33 + 105 Nm2 b. .33 + 10 4 Nm2 ). .33 + 10 6 Nm2 d. .33 + 103 Nm2 1!. a 40-gm ri'le (ith a speed o' 300 ms striFes into a ballisti) pend&l&m o' mass 5 Fg s&spended 'rom a )ord 1 m long. ?omp&te the verti)al height thro&gh (hi)h the pend&l&m rises. a. 2.7 )m b. 2!.42 )m ). 2!. )m d. 2.45 )m 20. hat is the area o' the largest re)tangle that )an be ins)ribed in a semi-)ir)le o' radi&s 10T 2 a. 2 50 b. 100 &nits2 ). 1000 &nits2 d. 50units 2 units 21. he amo&nt o' heat needed to )hange solid to li%&id is a. )ondensation b. )old '&sion ). latent heat o' '&sion d. solid '&sion 22. Mr Z. Ae#es borro(ed mone# 'rom the banF. Ge re)eived 'rom the banF 1$42 and promise to repa# 2$000 at the end o' 10 months. Petermine the simple interest. a.1!.45[ b. 15.70[ ). 16.10[ d. 10.2![ 23. ind the length o' the ve)tor 2$4$4 a. 7.00 b. 6.00 ). .50 d. 5.1 24. ;))ording to this la($ he 'or)e bet(een t(o )harges varies dire)tl# as the magnit&de o' ea)h )harge and inversel# as the s%&are o' the distan)e bet(een them. a.
−1
d 2 y dx 2
b.
x
. 1
−
x
2
).
1 x
d.
1 x 2
27. he integral o' an# %&otient (hose n&merator is the di''erential o' the denominator is the LLLLLLLLLLLL. a. )ologarithm b. b . prod&)t ). logarithm d. derivative 2. ind the nominal rate$ (hi)h i' )onverted %&arterl# )o&ld be &sed instead o' 12[ )ompo&nded semi-ann&all#. a. 14.02[ b. 21.34[ ). 11.2![ d. 11.3[ 2!. val&ate the e+pression 1 i2 10 (here 9 is an imaginar# n&mber. a. 1 b. 0 ). 10 d. - 1 30. ; \M has a selling pri)e o' 400. 9' its selling pri)e is e+pe)ted to de)line at a rate o' 10[ per ann&m d&e to obsolen)e$ (hat (ill be its selling pri)e a'ter 5 #earsT a. 213.10 b. 24!.50 ). 200.00 d. d. 236.20 31. 9' tan 4 A = cot 6 A, then (hat is the val&e o' angle ;. a. !0 b. 120 ). 100 d. 140
46. hat is the distan)e in )m bet(een t(o verti)es o' a )&be (hi)h are 'arthest 'rom ea)h other$ i' an edge meas&res )mT a. 13.6 b. 16.!3 ). 12.32 d. 14.33
32. ind the s&m o' the rots o' 5 x 2 − 10 x + 2 = 0. a. -12 b. - 2 ). 2 d. X 33. 9n a bo+ there are 25 )oins )onsisting o' %&arters$ ni)Fels and dimes (ith a total amo&nt o' ^ 2.75. 9' the ni)Fels (ere dimes$ the dimes (ere %&arters and the %&arters (ere ni)Fels$ the total amo&nt (o&ld be ^ 3.75. Go( man# %&arters arethereT a . 12 b. 16 ) . 10 d. 5 34. ; point moves so that its distan)e 'rom the point 2$-1 is e%&al to its distan)e 'rom the a+is. he e%&ation o' the lo)&s is. a. x 2 − 4 x + 2 y + 5 = 0 b. x 2 − 4 x − 2 y + 5 = 0 2 ). x + 4 x + 2 y + 5 = 0 d. x 2 + 4 x − 2 y − 5 = 0 35. Ro& loan 'rom a loan 'irm an amo&nt o' 100$000 (ith a rate o' simple interest o' 20[ b&t the interest (as ded&)ted 'rom the loan at the time the mone# (as borro(ed. 9' at the end o' one #ear #o& have to pa# the '&ll amo&nt o' 100$000 (hat is the a)t&al rate o' interest. a. 25.0[ b. 27.5[ ). 30.0[ d. 1.[ 36. 9t is a pol#hedron o' (hi)h t(o 'a)es are e%&al pol#gons in parallel planes and the other 'a)es are parallelograms. a. etrahedron b. rism ). r r&st&m d. r rismatoid 37. ; railroad is to be laid ,o'' in a )ir)&lar path. hat sho&ld be the radi&s i' the tra)F is to )hange dire)tion b# 30 0 at a distan)e o' o' 157.0 mT a.150 m b . 2 00 m ). 250 m d. 300 m 3. ; 200-gram apple is thro(n 'rom the edge o' a tall b&ilding (ith an initial speed o' 20 ms. hat is the )hange )hange in Feneti) energ# energ# o' the apple i' it striFes striFes the gro&nd 50 msT a. 130 Zo&les b. 210 Zo&les ). 100Zo&les d. 2 Zo&les 3!. ; ma)hine )osts $ 000 and an estimated li'e o' 10 #ears (ith a salvage val&e o' 500. hat is its booF val&e a'ter #ears &sing straight-line methodT a. 2$000 b. 4$000 ). 3$ 000 d. 2$ 500 40. 40. he he dist distan an)e )e bet( bet(ee eenn the the poin points ts ;8 de'i de'ine nedd b# A( cos A,−sin A) and B ( sin A, cos A) is e%&al to a. )os ; b. 1 ). 2 d. 2 tan A 41. hat nominal nominal rate$ )ompo&nded )ompo&nded semi-ann&all semi-ann&all#$ #$ #ields #ields the same amo&nt amo&nt as 16[ )ompo&nded %&arterl#T a. 16.64[ b. 16.16 ). 16.32 d. 16.00[ 42. 9' (2log x to the base 4) , (log9 to the base 4) = 2, 'ind +. a . 10 b. 13 ) . 12 d. 1 1 43. he point o' interse)tion o' the planes x + 5! − 2 = 9,3x − 2! + = 3 and x + y + z = 2 is at a. (1,2,1) b. ( 2,1,-1) ). (1,-1,2 ) d. ( - 1,-1,2) 44. o )omp&te the val&e o' n 'a)torial$ in s#mboli) 'orm nE* (here n is large n&mber$ (e &se a 'orm&la )alled a. Ai)har ) hards dson on-P -P&) &)hm hman an or orm&la &la b. Piopha o phant ntiine orm orm&la &la ). t tirling@s ;p ;ppro+imation d. Ma Matheson@s o orm&la 45. ; boat )an travel miles per ho&r in still (ater. hat is it velo)it# (ith respe)t to the shore i' it heads 350 ast o' NorthT a. 6$743 b. $!63 ). 5$400 d. 4$5
Degree of Differential E"#ation: he po(er to (hi)h the highest - order derivative is raised $ i' the e%&ation is (ritten as a -
46. hat is the distan)e in )m bet(een t(o verti)es o' a )&be (hi)h are 'arthest 'rom ea)h other$ i' an edge meas&res )mT a. 13.6 b. 16.!3 ). 12.32 d. 14.33
1 = π , then the val&e o' + is 3 4
Degree of Differential E"#ation: he po(er to (hi)h the highest - order derivative is raised $ i' the e%&ation is (ritten as a -
pol#nomial in the &nFno(n '&n)tion .
47. 9' a"ctan
a. 13 b. 12 ). 14 d. 15 4. he energ# stored in a stret)hed elasti) material s&)h as a spring is a. me)hani)al energ# b. elasti) potential energ# ). internal energ# d. Fineti) energ# 4!. "iven the points 3$ 7 and -4$ -7. olve 'or the distan)e bet(een them. a. 15.65 b. 17.65 ). 16.65 d. 14.65 50. hat rate o' interest )ompo&nded ann&all# is the same as the rate o' interest o' [ )ompo&nded %&arterl#T a. .4[ b. .42[ ). .24[ d. .6[
Linearit8 of a Differential E"#ation -
; di''erential e%&ation is linear i' it has the 'orm: n+ dn# n-1 + dn-1# DD 1+ d# o+# = /+ d+n d+n-1 d+ (here: # , the &nFno(n '&n)tion
+ , the independent variable /+ $ n+$ n-1+D0+ - pres&med Fno(n and depend onl# on the
variable +
A!-,ER-A #A %A 1! 'C 2C /A 6 A = D -) C --D -#D -%D -1C -'C -2C -/! -6A -=D #)!
_ Pi''erential e%&ations that )annot be red&)ed or p&t in this 'orm are nonlinear.
#-C ##D #%! #1! #'! #2C #/C #6D #=D %)D %-A %#C %%D %1A %'A %2! %/D %6!
-ample Pro&lems:
%=A 1)C 1-C 1#C 1%! 11C 1'D 12A 1/! 16! 1=A ')C Differential E"#ation - an e%&ation involving di''erential )oe''i)ient ent or di''erentials.
?onsider the e%&ation : d # d+ + =/ + 2
_ hen an e%&ation involves ves one or more derivatives (ith th respe)t to a parti)&lar variable$ that variable is )alled independent variable. or the given e%&ation$ the independent variable is +. _ 9' a derivative o' a variable o))&rs$ that variable is a dependent variable. or the given e%&ation* dependent variable is +. T8pes of Differential E"#ation:
1. Ordinar& Di99erentia: E78ation , an e%&ation (hi)h all di''erential )oe''i)ient have re'eren)e to a single independent variable.
Petermine the order$ degree$ linearit#$ &nFno(n '&n)tion$ and independent variable o' the di''erential e%&ation. 1. #@@@ , 2+# ` = ln + 2 2. 5+ d2# d+2 !+3 d# d+ tan+ # = 0 3. d2b dp2 p db dp 2 = 0 Ansers;
1. third order $ 'irst 'irst degree$ degree$ linear linear 3+ =1 $ 1+ = -2+ $ 2+ = 0 + =0 $ /+ = ln+ 2$ the &nFno(n '&n)tion is #$ and the independent variable is +. 2. se)ond order order $ 'irst 'irst degree$ degree$ linear$ linear$ &nFno(n &nFno(n '&n)tion '&n)tion is #$ and the the independen independentt variable is +. 3. se)ond order order $ se)ond se)ond degree degree $ nonlinear near one o' o' the derivat derivatives ives is raised raised to a po(er other than the 'irst &nFno(n &nFno(n '&n)tion is b and its derivatives$ derivatives$ and the independent variable is p.
-#pplementar8 Pro&lems:
Petermine aorder b degree ) linearit# linearit# d &nFno(n '&n)tion on and eindependent variable able 'or the 'ollo(ing di''erential e%&ations. 1. #@@@3 - 5+#@2 = e-+ 1 2. 5# d2Q d#
3#2 dQd# - sin# Q = 0
2. artia: Di99erentia: E78ation , an e%&ation (hi)h there are t(o or more independent variables and partial di''erential )oe''i)ients (ith respe)t to an# o' them.
3. d4+ d#45 7d+ d#10 +7 ,+5 = #
rder of Differential E"#ation:
4. t#@@ t 2#@ - sin t √ # = t2 , t 1
- he order o' the highest derivative derivative appearing ng in the e%&ation. on.
1amilies of C#res:
amilies lies o' )&rves )&rves ma# be represente representedd b# e%&ation involving involving parameter parameters. s. 9' the )onstants o' this e%&ation is treated as an arbitrar# )onstant$ the res< is )alled di''erential
2. arabolas arabolas (ith th 'o)i at at the origin origin and a+is a+is +. Ans. &* &FF# 0 #"&F $& ()
1amilies of C#res:
amilies lies o' )&rves )&rves ma# be represente representedd b# e%&ation involving involving parameter parameters. s. 9' the )onstants o' this e%&ation is treated as an arbitrar# )onstant$ the res< is )alled di''erential e%&ation o' the 'amil# represented b# e%&ation.
Ans. &* &FF# 0 #"&F $& ()
3. ?ir)les ?ir)les tangen tangentt to to the the +-a+is. +-a+is. Ans. -0 *&F # H % ( && FF && FF 0- 0 *&F # H #
-ample Pro&lems :
1. btain the di''erent di''erential ial e%&ation e%&ation o' o' the 'amil# l# o':
4. ;ll tangents tangents to the parabola parabola # 2 =2+
a. straight lines (ith slope and #- inter)ept e%&al.
Ans. #" *&F # $#&&F 0 $#&&F 0 - ( )
ol&tion:
-ol#tion of Differential E"#ations
# = m+ b b - the slope inter)ept 'orm 9n these )ase$ m = b # = m+m - the )onstant to be eliminated di''erentiating d# = m d#d+ = m s&bstit&ting to the original e%&ation # = d# d+ +1
2. arabolas arabolas (ith th 'o)i at at the origin origin and a+is a+is +.
; '&n)tion # = '+ is a sol&tion o' di''erential e%&ation e%&ation i' it is identi)all# satis'ied (hen # and its derivatives are repla)ed thro&gho&t b# '+ and its )orresponding derivatives. ; di''erential e%&ation o' order n (ill$ in general$ possesses a sol&tion involving n arbitrar# )onstants. his his sol&tion is )alled general general sol&tion. 9t (ill be ne)essar# to to assign spe)i'i) val&es to these arbitrar# ) onstants in order to meet the pres)ribed initial )onditions.
Anser
b. )ir)les (ith )enter on the +-a+is # +
; 'irst order di''erential e%&ation )an be solved b# integration i' it is possible to )olle)t all # terms (ith d# and all + terms (ith d+. hat is$ i' it is possible to (rite it in the 'orm: '# d# g+ d+ = 0 $ hen the general sol&tion is ∫ '# '# d# ∫ g+ g+ d+ = ? (here ? is an arbitrar# )onstant.
-ample Pro&lems:
1. ind the general sol&tion sol&tion o' the 'ollo(ing 'ollo(ing di''erential di''erential e%&ation e%&ation a. #@ =+ =+1 1 # b. ds dsdt = s2 2s 2
ol&tion: ? h$0 +-h2 # 2 = r 2 2+-h 2##@ = 0 +-h ## ` = 0 +##@ = h 1##@@ # `2 = 0
the arbitrar# )onstants are h and r$ there'ore (e m&st di''erentiate the e%&ation t(o times. 'irst di''erentiation di''erentiation
2. btain a parti)&lar sol&tion &tion (hi)h satis'ies the given initial al )ondition. d#d+ = 3+3 √ # $ # = 4 (hen + = 1 ol&tion: 1. a. he e%&ati e%&ation on ma#be ma#be (ritten (ritten in the the 'orm d#d+ = +1#
Anser
#d# = +1 d+ -#pplementar8 Pro&lems :
or ea)h o' the 'ollo(ing$ obtain the e%&ation o' the 'amil# o' plane )&rves. 1. traight ght lines lines (hose dista distan)e n)e is a 'rom the origin. origin. Ans. *"&F $& # ( a# * -0 *&F #
b. eparate eparate the variables variables to obtain obtain ds = dt 2
he sol&tion is
∫ #d# #d# = ∫ +1 +1 d+ X #2 = X +2 + ? #2 , +2 , 2+ = ?
b. d# d#d+ d+ = sse) e) # tan tan +
Ans.
+=0$ +=0$ #=0 #=0 Ans. sin & 0 :n cos " ( )
b. eparate eparate the variables variables to obtain obtain ds = dt s2 2s 2 (hi)h is the same as ds = dt 2 s 1 1 he sol&tion is ds ∫ = ∫ dt dt s 12 1 ? t = ar)tan s1 s = tan ?t , 1
b. d# d#d+ d+ = sse) e) # tan tan +
+=0$ +=0$ #=0 #=0 Ans. sin & 0 :n cos " ( )
#.
ol#nomials in (hi)h all terms are o' the same degree s&)h as + 2 #2 and +2 sin #+ are )alled homogeneo&s pol#nomial. E78ations it> >omoeneo8s coe99icient.
?onsider M +$# d+ N +$# d# = 0 1 9' the )oe''i)ients M and N are o' the same degree in +$# $ the# are )alled homogeneo&s '&n)tions.
Anser -ample Pro&lem:
2. he variables ables are separab separable$ le$ (hi)h (hi)h )an be (ritten (ritten in the 'orm* 'orm*
√ # d# = 3+ d+ 3
1. olv olvee #@ = # # + +
ol&tion:
integrating both sides
∫ √ ∫ √ # d# = ∫ 3+ 3+3 d+
or simpl#
i' # = 4$ + = 1
+ d\d+ d\d+ = 1 \ = ln + ?
? = 23 432 , ] 1 4
\ = ln ? + _
2
3 #32 = ] +4 ?
this
is the general sol&tion
8&t v = #+
55
? = 12 2
d# = \d+ +d\
d#d+ = # + + = \d+ +d\ d+ = \+ + +
here'ore$ 8 ' x ln C x
3 #32 = ] +4 5512
or simpl#
E !e ln of a constant is still a constant
H 85G$ 3 xB ' JJ
-#pplementar8 Pro&lems:
Anser
ind the general sol&tion:
-#pplementar8 Pro&lems:
1. btain the general general sol&ti sol&tion on o' the 'ollo 'ollo(ing: (ing: Ans. "& 0 C ( ) a. +#@ # = 0 b. + d+ = # d# = 0 3
2
). + d+ # 1 d# = 0 d. dsdt = t2 s2 6s ! e. # d# #2 1 d+
1. #@ = 3+# # 2 , +2
Ans. " # 0 & # ( C
2. %
%
Ans. *& 0 - 0 " ( C Ans. *s 0 %% $ t % ( C Ans. :n *- 0 & # 0 #"
2. btain the the parti)&lar parti)&lar sol&tion sol&tion satis'#ing satis'#ing the given initial tial )onditions. tions. a. d# d#d+ 2# = 3 +=0$ #=1 Ans. :n *% $ #& 0 %" ( )
o obtain the general sol&tion (e m&st 'ind a '&n)tion ∅ = ∅ + s&)h that i' the e%&ation is m<iplied m<iplied b# this '&n)tion '&n)tion derivative derivative o' the prod&)t ∅#. his '&n)tion is )alled the integrating 'a)tor ∅.
Ans. *& # 0 #" # % ( C& #
#@ = + 2 #2 4+#
Ans. %& # + " # ( C "
3. #@ = # + √ +#
Ans. $ # "5& "5& 0 :n & ( C
%.
; di''erential e%&ation o' 'irst rst order$ (hi)h is also linear$ )an be (ritten in the 'orm: d#d+ +# =/ +
1
r + d+ 9ntegrating 'a)tor φ = e∫ r here: r+ = 1-n + /r+ = 1-n /+
#e$%ce$ &ol!no'ial in x
o obtain the general sol&tion (e m&st 'ind a '&n)tion ∅ = ∅ + s&)h that i' the e%&ation is m<iplied m<iplied b# this '&n)tion '&n)tion derivative derivative o' the prod&)t ∅#. his '&n)tion is )alled the integrating 'a)tor ∅. Pdx ' e ∫ Pdx
;nd the sol&tion is Pdx Pdx 8 e∫ Pdx ' Q e∫ Pdx dx % C
2
r + d+ 9ntegrating 'a)tor φ = e∫ r here: r+ = 1-n + /r+ = 1-n /+ Q = #1-n "eneral ol&tion: ' Qr .x/dx % c
-ample Pro&lem:
-ample Pro&lem:
1. btain the general general sol&tion sol&tion o' the di''erenti di''erential al e%&ation. e%&ation. + d# # d+ = sin + d+ ol&tion: he di''erential e%&ation m&st be 'irst red&)ed in the 'orm o' e%&ation 1$ hen)e$ d#d+ #+ = sin + +
olve d# d+ , 2# + = 4+3 #3 ol&tion: d# d+ # -2+ = #3 4+3 n=3 Q = #1-3 Q = # ,2 + = -2 + r + = 1-n +
+ = 1+ * /+ = sin + + olving 'or the integrating 'a)tor$
r + = 1-3 +
d++ ∅ = e∫ d++
r + = -2 -2+ = 4+
∅ = eln +
/ + = 4+ 3
∅ = +
/r + = 1-n /+ /r + = 1-3 4+ 3
&sing the 'orm&la$
/r + = -+3
#+ = ∫ sin+ sin+ + + d+ ?
4+ d+ φ = e ∫ rr + d+ = e ∫ 4+ e 4 ln+ = +4
simpli'#ing$ x8 % cos x ' C
Anser
#-2+4 = -∫ + +7 d+ )
btain the general sol&tion: 1. d#d+ + # = 0
Ans2 8 ' c e
2. e d+ 2 +e , # d# -2#
2#
#-2+4 = - + )
.>xKB/GB $8
$
Ans2 x e ' 8 % C
3. ddθ = )os θ # )ot θ 4. #@ = + , 2# )ot 2+
"eneral sol&tion: ∫ φ /r +d+ ) Qφ = ∫ φ #-2+4 = ∫ + +4 -+3 d+ )
-#pplementar8 Pro&lems: 3
#e$%ce$ &ol!no'ial in x
Ans2 T ' c sin > cos
Ans2 B8 sin $x ' c % sin $x 3 $x cos $x
5. #@ = +3 2+#* (hen + = 1$ # = 1
+4 = ) ,+ #2 1.
;n# e%&ation that that )an be (ritten (ritten in the 'orm
Ans2 $8 ' x $ 3 4 % $ e .4>x/ M*",&d" 0 N*",&d& ( )
%. !erno !erno8:: 8::iF iFs s E78ati E78ation on
tandard orm:
dxGd8 % 8 P.x/ ' 8 n Q.x/
provided$ n ≠ 0$ 1
∂#
∂+
he te)hni%&e is to 'ind a '&n)tion '+$# s&)h that
;nd (e have the propert#
∂M
∂N ------- = ------- is said to be an e+a)t e%&ation.
F K 0 , e+ponential de)a# -ample Pro&lem:
1. ; )ertain radioa)tive material is Fno(n to de)a# at a rate proportional to the amo&nt
∂#
∂+
F K 0 , e+ponential de)a# -ample Pro&lem:
he te)hni%&e is to 'ind a '&n)tion '+$# s&)h that
1. ; )ertain radioa)tive material is Fno(n to de)a# at a rate proportional to the amo&nt present. 9' initiall# there is 50 mg o' the material present and a'ter 2 hrs. it is observed that the material has lost 10[ o' its original mass. Petermine mass o' the material a'ter 4 ho&rsT
'@# , the )ommon )ommon term o' '+$# '+$# and N+$# g@# g@# , the the )omm )ommon on ter term m o' '+$ '+$# # and and M+ M+$# $#
ol&tion:
-ample Pro&lem:
Petermine (hether the e%&ation 2+# d+ 1+2 d# = 0 is e+a)t. 9' so$ obtain the general sol&tion. ol&tion: "iven in the e%&ation that M = 2+#$ N = 1 + 2
∂M ∂# = ∂N ∂+ = 2+
∴ he e%&ation is e+a)t
= ∫ 2+# 2+# d+ '#
olving 'or F$ 45 = 50 e 2F F = -0.053
9n the problem t = 4 hrs.
'@# = 1
/ = 50 e -0.053 t
'# = # )
Q ' B(2J mg2
x$8 % 8 % c ' ( +
Answer
Answer
-#pplementar8 Pro&lems: 2
#. Neton NetonFs Fs La La o9 Coo:i Coo:in n #
1. 2+ e e d+ + 1 e d# = 0
x
2. 2+# # d+ + 2+# - # d# = 0
$
$
$
$
Ans2 $x 8 %$x8 3 8 ' c
3. + sin sin # d+ + + )os # , 2# d# = 0 4. d# #- sin + + d+ = 0
8
Ans2 e % e .x %4/ ' c
2
Ans2 M x $ % x sin 8 3 8 $ ' c
Ans2 x8 % cos x ' c
Elementar8 Applications: -. La o9 E"on E"onenti entia: a: C>ane C>ane *Grot> *Grot> 5Deca& 5Deca&
he rate o' (hi)h the amo&nt o' a s&bstan)e )hanges$ is proportional to the amo&nt present or remaining at an# instant. / = amo&nt o' s&bstan)e at an# time t d/ dt = rate o' )hange o' the amo&nt / d/ dt ∝ / d/ dt = F/ Q ' Qo e@t
/2 = 45 mg
/ = 50 e -0.053 t
∂ ∂# = +2 '@# = + 2 1
2
/2 = 50 mg 1-0.1
∴ amo&nt o' material at an# time t
= +2# '#
#
"iven: /o = 50 mg
F V 0 , e+ponential gro(th
o = 36o? o
5 mm = 40 ?
he time rate o' )hange o' the temperat&re o' a bod# is proportional to the temperat&re di''eren)e bet(een bod# and its s&rro&nding medi&m. = temperat&re o' the bod# at an# time t o = initial temperat&re m = temperat&re o' the s&rro&nding medi&m d dt = time rate o' )hange o' the temp. o' the bod# d dt = - F , m d dt F = Fm $ F is al(a#s positive$ d dt = negative 'or )ooling T ' Ce>@t % Tm
or
T ' .T o 3 Tm/ e>@t % Tm
-ample Pro&lem:
; bod# at a temperat&re temperat&re o'' 36o? is pla)ed o&tdoors (here the temperat&re is 60o?. 9' a'ter 5 mm$ the temperat&re o' the bod# is 40 o?. Go( long (ill it taFe to rea)h a temperat&re o' 45o?T ol&tion: "iven: m = 60o?
into the tanF at A i = galmin. galmin. $ litermin litermin. . (hile (hile sim<aneo sim<aneo&sl#$ &sl#$ the (ell-stirr (ell-stirred ed sol&tion leaves the tanF at the rate A o galmin.
?$A
o = 36o?
into the tanF at A i = galmin. galmin. $ litermin litermin. . (hile (hile sim<aneo sim<aneo&sl#$ &sl#$ the (ell-stirr (ell-stirred ed sol&tion leaves the tanF at the rate A o galmin.
o
5 mm = 40 ?
?i $ Ai
olve 'or F:
dsdt = ? i Ai - ?o Ao ?o = H\ o Ai $ Ao tI
40 = 36-60 e ,F5 60
Mi+t&re
F = 0.03646 olving 'or t at = 45 o? 45o? = 36-60 e ,0.03646 t 60 t ' 4$2 min2
Answer
%. Sim:e Sim:e C>emica: C>emica: Conversio Conversion n
9n )ertain rea)tions in (hi)h a s&bstan)e ; is being )onverted into another s&bstan)e the time o' )hange o' the amo&nt + o' &n)onverted s&bstan)e is proportional to +.
@t -ample Pro&lem:
&ppose that a )hemi)al rea)tion pro)eed s&)h that the time rate o' )hange o' the &n)onverted s&bstan)e is proportional to the amo&nt o' it. 9' hal' o' s&bstan)e ; has been )onverted at the end o' 10 se). ind (hen !10 o' the s&bstan)e (ill have been )onverted. ol&tion: "iven: + = X +o
-ample Pro&lem:
; 50 gal. tanF )ontains 10 gal. o' 'resh (ater. ;t t = 0$ a brine sol&tion )ontaining 1 lb. ' salt per gallon is po&red into the tanF at the rate o' 4 galmin. $ (hile the (ell-stirred mi+t&re leaves the tanF at the rate o' 2 galmin$a. 'ind the amo&nt o' time re%&ired 'or over'lo( to o))&r$ b. ind the amo&nt o' salt in the tanF at the amo&nt o' over'lo(. ol&tion: ;t t = 0$ \o = 0 $ ) i = 1$ Ai = 4$Ao = 2 he vol&me o' brine in the tanF at an# time t e%&als to \o Ai t - Ao t = 10 4t , 2t = 102t over'lo( (ill o))&r i' the vol&me o' the brine in the tanF e%&als to the vol&me o' tanF no(. = X 50-10 = 20 min. b. d/dt /102t = 4
X +o = +o e F-10
d/dt 2/102t = 4
F = 0.06! (hen !10 is )onverted onl# 110 is &n)onverted
∴ + = 110 +
dt 5t integrating 'a)tor = e ∫ dt = e ln 5t = 5t
/ 5t = 4 ∫ 5t 5t dt
10 +o = +o e ,0.06! t
/ 5t = 4 H 5t 12 t 2I )
solving 'or t
/ = 20t 12 t 2 ) 5t
1
t ' 55 sec2
&tgoing
9n general$ d-Gdt Ai V Ao $ then \ o (ill over'lo( \o'$-oo G N7o % .R? i 6 Ro/ tO ' Ci Ri Ai =o $AAo $o \o = ) Ao K A9 $ then \ o↓
Answer
;t t = 0 / = a = 0 0 = H200 202 )I 5t
1. Di:8ti Di:8tion on 5 <:o <:o ro4: ro4:em; em;
he tanF initiall# hold \ o gal.$ liter$et). o' sol&tion that )ontains o lb.$Fg.$N o' a s&bstan)e. ;nother sol&tion )ontaining a s&bstan)e at ? 9 = lbgal.* lbgal.* N < is po&red po&red
/ = H2020 2202 520 Q ' BH l&2
Answer
) = 0* no( / = 20t 2 t 2 5t Ae%@d: / =T ;t t = 20 min.
5. 9' a pop&lation pop&lation o' a )o&ntr# )o&ntr# do&bles do&bles in 50 #ears$ in ho( ho( man# #ears #ears (ill it treble &nder the ass&mption that the rate o' in)rease is proportional to the no. o' inhabitantsT Ans2 6. ; )#lindri)al )#lindri)al tanF tanF )ontains )ontains 40 gal. o' a salt salt sol&tion sol&tion )ontainin )ontainingg 2 lb. o' salt per gallon. gallon. ;
/ = H2020 2202 520 Q ' BH l&2
Answer
'. Ort>o Ort>oon ona: a: Tra@ect Tra@ectori ories es
"iven a 'amil# o' )&rves given b# '+$#$) = 0 he )&rves that interse)ts a 'amil# '+$#$) = 0 at right angles$ (henever the# do interse)t is given b# the 'amil# o' another )&rve g+$#$F and are )alled orthogonal traCe)tories. his t(o )&rves are said to be orthogonal to ea)h other$ be)a&se ea)h point o' interse)tion$ the slopes o' the )&rves are negative re)ipro)als to ea)h other.
5. 9' a pop&lation pop&lation o' a )o&ntr# )o&ntr# do&bles do&bles in 50 #ears$ in ho( ho( man# #ears #ears (ill it treble &nder the ass&mption that the rate o' in)rease is proportional to the no. o' inhabitantsT Ans2 6. ; )#lindri)al )#lindri)al tanF tanF )ontains )ontains 40 gal. o' a salt salt sol&tion sol&tion )ontainin )ontainingg 2 lb. o' salt per gallon. gallon. ; salt sol&tion o' )on)entration 3 lbgal 'lo(s into the tanF at 4 galmin. Go( m&)h salt is in the tanF at an# time t i' the (ell-stirred sol&tion 'lo(s at 4 galminT Ans. -#) + 1)e $t5-) 7. ;n ind&)tan)e o' 1 henr# and a resistan)e resistan)e o' 2 ohms ohms are )onne)ted in series (ith an em' o' 100e-t volts. 9' the )&rrent is initiall# Qero$ (hat is the ma+im&m )&rrent attainedT Ans2 $JA
'c82 Ans. #& # 0" # (c . ind the the orthogo orthogonal nal traCe) traCe)torie toriess o' x$'c82 Ans.
DI<
LI+IT 1 1U!CTI!-
No( the P o' the orthogonal traCe)tories is d#d+ = NM
Pe'initions
•
-ample Pro&lem:
x
• ol&tion: 2
2
he limit o' a '&n)tion o' + or '+ as + approa)hes a is <* (ritten as lim f.x/ ' L
ind the orthogonal traCe)tories o' all )ir)les (hose )enter is at the origin.
a
lim f.x/ = < i' and onl# i'$ 'or an# )hosen positive n&mber ∈$ x
a
2
+ # = )
ho(ever small$ there e+ists a positive n&mber δ s&)h that$ (henever 0 K +-a K δ $ then ƒ+ , < K ∈
Pi''erentiating impli)itl#$ impli)itl#$ 2+d+ 2#d# = 0
Theorems on Limits
impli'#ing
& 9imit 9imit of of a consta constant nt c
d#d+ = - +# = - MN
∴ NM = d#d+ the P. . o' traCe)tories ∫ #d# #d# = ∫ +d+ then integrating 8 ' @x
Answer
lim ) = ) +→a
2 9imit 9imit of of a #ariab #ariable le x
lim + = a +→a
9imit 9imit of sum of t(o t(o functi functions ons
lim H ƒ+ g+ I = lim ƒ+ lim g+ +→a
-#pplementar8 Pro&lems:
1. ; bod# at an &nFno(n &nFno(n temper temperat& at&re re is pla)ed pla)ed in a room (hi)h (hi)h is held at a )onsta )onstant nt temperat&re o' 30o? and a'ter 10 min.$ the temperat&re o' the bod# is 0 o? and a'ter 20 min.$ the temperat&re o' the bod# is 15 o?. ind the e+pression 'or the temperat&re at an# ( + 2) e $).)2=t 0 %) time t. "ns ( 2. ;t e+a)tl e+a)tl## 10:30 10:30 pm$ a bod# at a temper temperat& at&re re o' 50 o is pla)ed 9n an oven (hose temperat&re is Fept at 150 o$ 9' at 10:40$ the temperat&re o' the bod# is 75 o$ at (hat .m. time (ill it rea)h 100 o. Ans.-);'1 .m. 3. ind the re%&ire re%&iredd 'or a radioa)tive radioa)tive s&bstan) s&bstan)ee to disintegrat disintegratee hal' o' its its original original mass i' >rs. three %&arters o' it are present a'ter hrs. Ans. -=.% >rs. 4. ;'ter ;'ter 2 da#s$ log o' a radioa)tive ve material al is present. hree hree da#s later$ later$ 5g. is present. present. Go( m&)h o' the )hemi)al (as present initiall#$ ass&ming the rate o' disintegration is proportional to the amo&nt present. Ans. -'.6/ .
Limit to Infinit8 or 9ero
"iven a )onstant ) and variable + 1. lim )+ = ∞ 'or positive )
+→a
+→a
4 9imit 9imit of of product product of t(o functions functions
lim H ƒ+ g+ I = lim ƒ+ H lim g+ I +→a
+→a
+→a
+ 9imit 9imit of of a quotie quotient nt
lim ƒ+ = lim ƒ+ +→a +→a g+ lim g+ +→a
. 9imi 9imitt of a radi radica cal l
lim n √ ƒ+ = n√ lim ƒ+
+→a
+→a
I!TER+EDIATE 1R+-
;. 00 or ∞∞ <@Gospital@s A&le is appli)able 8. ∞ - ∞ $ 0. ∞ <@Gospital@s Gospital@s A&le is not dire)tl# appli)able
Limit to Infinit8 or 9ero
I!TER+EDIATE 1R+-
"iven a )onstant ) and variable + 1. lim )+ = ∞ 'or positive ) +→∞ = - ∞ 'or negative ) 2. lim )+ = 0
;. 00 or ∞∞ <@Gospital@s A&le is appli)able 8. ∞ - ∞ $ 0. ∞ <@Gospital@s Gospital@s A&le is not dire)tl# appli)able ?. 00$ ∞0$ 1∞ !ote:
+→∞
3. 4.
lim ) = ∞ 'or positive ) +→∞ = - ∞ 'or negative ) 4. lim )+ = ∞ 'or positive ) +→0 = - ∞ 'or negative ) +
9' the eval&ated '&n)tion t&rned o&t to be in the 'orm o' those in 8 and ?$ )hange the 'orm o' the given '&n)tion to obtain an eval&ated '&n)tion in the 'orm o' that in ;.
•
The Indeterminate 1orm (2
9' '+ approa)hes Qero and g+ approa)hes in'init# as + approa)hes a or +
→ ± ∞ $ the prod&)t ƒa. ga is &nde'ined and (ill be o' the 'orm 0. ∞. 9' the limit '+ . g+ e+ists as + → a or as + → ± ∞$ it ma# be 'o&nd b# (riting the
L;)ospitals R#le
9' the '&n)tions '+ and g+ are )ontin&o&s in an interval )ontaining + = a$ and i' their derivatives e+ist and g@+ ≠ 0 in this interval e+)ept possibl# at + = a$ then (hen 'a = 0 and ga = 0 lim '+ = lim ' @+ +→a +→a g+ g @+ rovided that the limit on the right side e+ists. Eal#ation of Limits
dire)t s&bstit&t s&bstit&tion ion obvio&s obvio&s limit limit ration rationali aliQin Qingg N + + or P + + e+pand e+panding ing N+ N+ or P + + )ombin )ombining ing term termss in N + + or P + 'a)tor 'a)toring ing N + + or P + appl#i appl#ing n g <@Gospi <@Gospita tal@s l@s A&le
Limits to Infinit8 of a 1raction
N+ P+
+→∞
"ns3 e
4. lim 3 , + +→3 +,3 1
The Indeterminate 1orm
1
"ns3 & '
>
9' ƒ+ and g+ both in)rease (itho&t bo&nd as + → a or + → ± ∞$ the di''eren)e ƒa , ga is &nde'ined and (ill be o' the 'orm ∞ - ∞ 9' the limit o' ƒ+ , g+ e+ists as + → a or + → ± ∞$ then b# algebrai) means. 1 lim H ƒ+ , g+I = lim 1g+ - '+ ' + LL +→a +→a 1LLL g+ '+
•
+ethods of Eal#ation
1. 2. 3. 4. 5. 6.
•
prod&)t as a 'ra)tion lim ƒ+ g+ = lim ƒ+ = lim g+ +→a +→a 1 g+ +→a 1'+ hen appl# <@Gospital@s A&le
he 'orm 00$ ∞0$ 1∞ 9' the limit o' ƒ+g+ e+ists as + → a or as + → ± ∞$ then b# logarithm$ lim ƒ+g+ =# )an be eval&ated +→a
< = lim ƒ+g+ +→a
aFing the logarithm o' both sides ln < = ln lim ƒ+g+ = lim ƒ+g+ lim g+ ln ƒ+ = F +→a
+→a
+→a
9n < = F $ then e = e or < = eF ln<
F
-#pplementar8 Pro&lems
val&ate the limits given belo( 1. lim lim si sin 5+ 5+ +→0 + 2. lim 1 , )os θ θ →∞ 2θ 3. lim lim 1 1+3+
"ns3 + "ns3 0
lope o' tangent line =- 1 d#d+
+→∞
4. lim 3 , + +→3 +,3 5. lim sin θ θ →0
lope o' tangent line =- 1 d#d+ =- 1 '@+o
"ns3 e 1
1
"ns3 & '
θ
"ns3 &
ANGLE !ETEEN !ETEEN TO CURVES ;ngle ;ngle o' 9nterse)tion , is de'ined as the angle bet(een
6. lim 1+ , 13sin+ +→0
their tangents at their point o' interse)tion. o determine the angle o' interse)tion o' t(o )&rves$ '+ and g+ 1. olve the the e%&ations sim<aneo&sl# to 'ind the points points o' interse)tion. 2. ind ind the the slo slope pess m1 = '@ + o and m2 = g@ + o
"ns3 0
7. 7. lim lim + +1 1
ln +
+→0
"ns3 &
. . lim lim + )s) )s) 5+ 5+ +→0
hen the a)&te angle θ o' interse)tion is given b#
"ns3 & ' + +
!. ? 8oard 8oard e+am e+am 1!7 1!7 val&ate val &ate lim li m +3 , 2+ ! +→∞ 2+3 ,
"ns3 F
10. ? 8oard 8oard +am 1!7 1!7 val&ate val &ate lim 2+4 , 2+3 !+ 2 , + 7 +→∞ +3 , 9nterpretation o' Perivative
m4 3 m$ ' >>>>>>>>>>>>>>> 4 % m4m$
Tan "ns3 ∞
$2 Deriatie as Rate of Change
d# d+ = lim ∆# ∆+
42 Deriatie as -lope
+→0
#
he val&e o' the derivative o' the '&n)tion is the instantaneo&s rate o' )hange o' the '&n)tion (ith respe)t to the independent variable.
#=ƒ+ angent
;ectilinear Dotion , motion in a straight line
;ss&mption: 1. Motion (ill al(a#s be ass&med to taFe pla)e along ong straight line$ altho&gh altho&gh the obCe)t in motion ma# go either dire)tion. 2. he bod# bod# in motion motion is idealiQ idealiQed ed to be a point nt or a parti)le. parti)le.
o+o$#o
θ 0
+
De9initions
he tangent to the )&rve (ith e%&ation # = '+ at o+o$ #o is the line thro&gh o+o$ #o (ith slope '@+o lope o' angent
I. !a sic
1.
d (c ) dx
=0
1
4.
d dx d
(cotu)
= − csc
2
u
du dx
du
I. !a sic
1. 2. 3. 4. 5. 6. 7.
.
d (c ) dx d dx d dx
(x)
4.
=0
5. =
1
6. (x n ) = nx n-1
d du (cu) = c dx dx d du dv (u + v) = + dx dx dx d udv vdu (uv) = + dx dx dx
d (u / v) dx d (k / u) dx
v
=
=
du dx
−u
dv dx
v2 du −k dx u2
d du (u n ) = nu n−1 dx dx d 1 du ( u) = 10. dx 2 u dx d - n du 11. (1/u (1/u n ) = n+1 dx dx u
12.
dy
dx dy 13. dx
=
=
dy dx
•
du dx
he ?hain A&le
1 dx / dy
II2 Differentiation of Trigonometric Trigonometric 1#nction
1. 2. 3.
d du (sin (sinu) u) = cosu cosu dx dx d du (cos (cosu) u) = -sin -sinu u dx dx d du 2 (tan (tanu) u) = sec sec u dx dx
VI. Di99erentiation o9 B&er4o:ic B&er4o:ic <8nction <8nction
1.
d (sinu (sinu)) dx
du dx
= cosu cosu
(cotu)
= − csc
2
u
du dx
du (secu) (secu) = sec u tan u dx dx d du (csc (cscu) u) = −cscu cscuco cotu tu dx dx
III. Di99erentiation o9 Inverse Trionometric Trionometric <8nction
1. 2. 3. 4. 5.
!.
d dx d
6.
d 1 du (Arcsinu) = dx 1 − u 2 dx d -1 du (Arcosu) = dx 1 − u 2 dx 1 d du
(Arctanu) = 2 dx 1 + u dx −1 d du (Arcotu) = 2 dx 1 + u dx
d 1 du (Arcsecu) = 2 dx dx u u −1 d -1 du (Arcscu) = 2 dx u u − 1 dx
I72 Differentiation of Logarithmic Logarithmic 1#nctions
1.
d dx
(lnu)
1 du =
2.
d (logu) dx
3.
d (logbu) dx
u dx
=
1 du (log e) u dx log b e du
=
u
dx
72 Differentiation of Exponential Exponential 1#nctions
d du (e u ) = e u dx dx d du 2. (a u ) = a u (ln a ) dx dx d v du dv v 3. + lnu (u ) = u dx dx dx
1.
6. he he tang tangen entt to # = +3 , 6+2 + at 3$ -3 interse)ts the )&rve at another point. Petermine this point. ;ns: 0$ 0 7. or or the the )&rv )&rvee # = +2 +$ at (hat point does the normal line at 0$ 0 interse)t the tangent
VI. Di99erentiation o9 B&er4o:ic B&er4o:ic <8nction <8nction
d du (sinu (sinu)) = cosu cosu dx dx d du 2. (cosu (cosu)) = sinu sinu dx dx d du 2 3. (tan (tanu u)) = sec sec u dx dx d du 2 4. (cot cotu) u) = −csc sc u dx dx d du 5. (secu (secu)) = −secu secutan tanu u dx dx d du 6. (csc (cscu u)) = −csc cscuc ucot otu u dx dx
1.
VII. Di99erentiation o9 Inverse B&er4o:ic <8nction
d 1 du (sin (sin −1u) = 1. dx u 2 + 1 dx d 1 du (cos (cos −1u) = 2. dx u 2 − 1 dx d 1 du 3. (tan (tan 1u) = 2 dx − 1 u dx d - 1 du 4. (cot (cot 1u) = 2 dx 1 − u dx d -1 du (sec (sec 1u) = 5. 2 dx dx u 1−u −
−
−
6.
d -1 du (csc (csc 1u) = dx u 1 + u 2 dx
6. he he tang tangen entt to # = +3 , 6+2 + at 3$ -3 interse)ts the )&rve at another point. Petermine this point. ;ns: 0$ 0 7. or or the the )&rv )&rvee # = +2 +$ at (hat point does the normal line at 0$ 0 interse)t the tangent line at 1$ 2T "ns3 1%'&0, *&'&0) . ind the the angle o' o' interse)ti interse)tion on bet(een bet(een the )&rves )&rves # = +2 2 and = .40 # = + + -1 1. "ns3 θ = !. ; bo# 3 't. tall (alFs a(a# 'rom alight (hi)h is on top o' post 7.5 't. high. ind the rate o' )hange o' the length o' his shado( (ith respe)t to his distan)e 'rom the lamp post. "ns3 2' ft'ft
ind the rate o' )hange the area o' an e%&ilateral triangle (ith respe)t to the side o' this triangle (hen the latter is 2 't. "ns3 √ ft 2 'ft 11. 9' the parti)le moves a))ording to the the la( = t 2 , t3 3$ 'ind the velo)it# (hen "ns3 *&' a))eleration is Qero. "ns3 12. 9' the motion motion o' the bod# is des)ribed b# = 3t 5 , 30t2 5$ (hen (ill the a))eleration be QeroT ;ns: 1 13. (o parti)les have position at time me t given b# the e%&ation 1=t3 6t2 , 7t 1 and 2 = 2t3 = , 3t2 , t ind their position (hen the# have the same a))eleration. "ns3 5& = .&- 5 2 =
10.
2+
*&-yI = 0 14. ind #@ and and #$ given given +3# +#3 = 2 and + = 1. "ns3 yH = *&-yI 15. ind the e%&ation o' the tangent tangent and normal to +2 3+# #2 = 5 at 1$ 1. "ns3 angent3 x C y – 2 = 0, 0, 7ormal3 x = y ther Applications of Deriaties 6ncreasing and ecreasing unctions3
; '&n)tion # = '+ is said to be in)reasing i' its val&e val&e in)reases as # in)reases ; '&n)tion # = '+ is said to be de)reasing i' # de)reases de)reases as + in)reases "iven a '&n)tion '+ di''erentiable in the interval a ≤ + ≤ b 1. 9' '@+ '@+ V 0$ then then '+ '+ is in)reas in)reasing ing 2. 9' '@+ '@+ K 0$ then then '+ '+ is de)reasi de)reasing ng 3. 9' '@+ '@+ = 0$ then then '+ '+ is station stationar# ar#
:onca#ity, :ritical 1oints, 6nflection 1oints :onca#e Jp(ard
−
he graph o' a '&n)tion '&n)tion is said to be )on)ave &p(ard i' the '&n)tion '&n)tion is de)reasing then in)reasing.
:onca#e o(n(ard
"ns3 * 1. ind the sslope lope o' o' the )&rve # = + 2+-2 at point 1$ 2. "ns3 2. ind ind the poin pointt on the the )&rve )&rve # = +2 , 6+ 3 (here the tangent is horiQontal. "ns3 1%, *.) 3. ;t (hat (hat point point on the the )&rve )&rve # = + 4 1 is the normal line parallel to 2+ # = 5T "ns3 1%&'2, &G'&.) 4. ind the ppoint oint on the the )&rve )&rve # = 7+ 7+ , 3+2 2 (here the in)lination o' the tangent is ."ns3 1%&, .) 450 "ns3 5. ind the the point point (here (here the normal normal to # = + +12 1 at 4$ 7 )rosses the #- a+is. "ns3 1%0,&0 & ' + + )
Daximum 1oint
:ritical 1oint
6nflection 1oint
&bstit&te in the e+pression 'or the 'irst derivative a val&e slightl# less than and then a val&e slightl# greater than the )riti)al val&e &nder )onsideration. 1. 9' '@+ )hanges )hanges 'rom positive positive to negative negative as + in)reases thro&gh thro&gh the )riti)al )riti)al val&e$ val&e$
5. Peterm Petermine ine a$ b$ b$ ) and d so that that the the )&rve )&rve # = a+ 3 b+ 2 )+ d (ill have horiQontal tangents at the points 1$ 2 and 2$ 3 6. ind the the e%&at e%&ation ion o' the line line normal normal to the )&rve )&rve # = 3+ 5 , 10+ 3 15+ 3 at its point o'
-#pplementar8 Pro&lems:
he graph o' a '&n)tion is )on)ave do(n(ard i' the '&n)tion is in)reasing then de)reasing.
; point (here the '&n)tion 'rom in)reasing to de)reasing and the '&n)tion is said to have a relative minim&m val&e. he point at (hi)h #@ = 0 and val&e o' + at this point )riti)al val&e.
; point at (hi)h (hi)h the )&rve )hanges its its dire)tion o' )on)avit#. )on)avit#.
1irst Deriatie Test
&bstit&te in the e+pression 'or the 'irst derivative a val&e slightl# less than and then a val&e slightl# greater than the )riti)al val&e &nder )onsideration. 1. 9' '@+ )hanges )hanges 'rom positive positive to negative negative as + in)reases thro&gh thro&gh the )riti)al )riti)al val&e$ val&e$ then the )riti)al is a maximum point 2. 9' '@+ '@+ )hanges 'rom 'rom negative negative to positive positive as + in)reases in)reases thro&gh thro&gh the )riti)al )riti)al val&e$ val&e$ then the )riti)al is a minimum point 3. 9' '@ + does not )hanges )hanges sign$ sign$ the )riti)al )riti)al is neither neither a ma+im&m ma+im&m or a minim&m minim&m point. point. inflection (ith horiQontal tangent. 9t is the point of inflection
5. Peterm Petermine ine a$ b$ b$ ) and d so that that the the )&rve )&rve # = a+ 3 b+ 2 )+ d (ill have horiQontal tangents at the points 1$ 2 and 2$ 3 6. ind the the e%&at e%&ation ion o' the line line normal normal to the )&rve )&rve # = 3+ 5 , 10+ 3 15+ 3 at its point o' in'le)tion. "ns3 x C &+y – 4+ = 0 7. ind a s&)h s&)h th that at the the )&rve )&rve # = 2+3 , 3a+2 12+ , 1 (ill have are o' its )riti)al points (here + = 2 "ns3 a = . ind the all val&es o' + (here the )&rve # = + 2 , 2+ 5 is in)reasing. "ns3 x > & +A
ma+im&m point #@= 0 9n'le)tion oint
# = '+
#@=0 minim&m point
)on)ave do(n(ard
+=a )on)ave &p(ard
tep in olving roblems roblems 9nvolving Ma+ima and Minima 1. 9dent 9denti'# i'# the %&anti %&antit# t# to be ma+imiQed miQed or minimiQed nimiQed 2. Ose the in'ormati in'ormation on in the problem problem to eliminate eliminate all %&antiti %&antities es so as to have a '&n)tion '&n)tion variables. Petermine the possible domain o' this '&n)tion. 3. Pi''erent Pi''erentiate iate this '&n)tion '&n)tion (ith respe)t respe)t to the variable variable (hose ma+im&mm ma+im&mminim inim&m &m val&e is to be determined 4. %&ate %&ate the derivative derivative o' step step 4 to Qero and solve solve 'or the &nFno(n &nFno(n.. -#pplementar8 Pro&lem & 8:8 ?oard ?oard 8xam 8xam &G &G
;n open re)tang&lar tanF (ith s%&are base is to have a vol&me o' 10 )&. m and the material material 'or the bottom bottom is to )ost per s%&are meter meter and that 'or the sides 6 )ents per s%&are meter. ind the height o' the tanF i' the )oil o' maFing the tanF is to be a cm minim&m. "ns3 2+ cm
2 8:8 ?oard ?oard 8xam 8xam eb eb &G &G
9n problem no. 1 above$ 'ind the most e)onomi)al dimension 'or the tanF. "ns3 2m x 2m x
2+m 8:8 ?oard ?oard 8xam 8xam &/& &/& -econd Deriatie Test
1. he '&n)tion '&n)tion # = '+ has has a ma+im&m ma+im&m val&e at + = a i' '@a '@a = 0 and 'a 'a K 0. he )&rve is conca#e do(n(ard at at that point. 2. he '&n)tion # = '+ has a minim&m val&e at + = a i' '@a = 0 and 'a 'a V 0. he )&rve )&rve is conca#e up(ard at at that point. 3. 9' at '@a a = 0$ 'a$ then his test 'ails. Ose the the 'irst derivative derivative test.
Third Deriatie Test
; '&n)tion # = '+ has an in'le)tion in'le)tion point at + = a i' 'a = 0 and '@a ≠ 0. -#pplementar8 Pro&lems
1. ind the ma+im&m ma+im&m$$ minim&m minim&m and in'le)tio in'le)tionn point o' the )&rve )&rve +0'2G)- 6nflection 6nflection %2', +2'2G) #=-+3 2+2 , + 2 "ns3Dax%&, 2)- Din %&', +0'2G)2. ind the the val&e val&e o' F s&)h s&)h that that the )&rve )&rve # = + 3 , 3F+2 5+ , 10 "ns3 = & 3. Petermine Petermine the the e%&ation e%&ation o' o' the parabol parabolaa # = a+2 b+ ) passing passing thro&gh thro&gh 2$ 1 and be tangent to the line # = 2+ 4 at point 1$ 6 "ns3 y = *Gx2 C &.x – 4. hat hat )&rve )&rve o' o' the the 'orm 'orm # = a+ a+ 3 b+ 2 )+ d (ill have )riti)al points at 0$ 4 and 2$ 0.
Pivide 60 into t(o parts so that the prod&)t o' one part and the s%&are o' the other is a ma+im&m. "ns3 20, 40
4 8:8 ?oar ?oard d 8xam 8xam "pril "pril &// &//
ind the altit&de o' the largest )ir)&lar )#linder that )an be ins)ribed in a )ir)&lar )one o' radi&s r and height h. "ns3 &' !
+ 8:8 ?oard ?oard 8xam 8xam &/ &/
ind the greatest vol&me o' a right )ir)&lar )#linder that )an be ins)ribe in a sphere o' 24&/ r radi&s r. "ns3 r = 24&/
. 8:8 ?oard ?oard 8xam 8xam 7o# 7o# &+ &+
ind the radi&s o' a right )ir)&lar )#linder that )an be ins)ribe in a )one o' a radi&s A and 2' ; height G. "ns3 r = 2'
G 8:8 ?oard ?oard 8xam 8xam eb eb &G/ &G/
he s&m o' the t(o n&mbers is 36. hat are these n&mbers i' their their prod&)t is to be the largest possible. "ns3 &/ and &/
/ 8:8 ?oard ?oard 8xam 8xam eb eb &G/ &G/
; s%&are sheet o' galvaniQed iron 100)m + 100)m (ill be &sed &sed in maFing an open open top )ontainer )ontainer b# )&tting )&tting a small s%&are 'rom ea)h ea)h )orners and bending bending &p the sides.
"ns3 y = x – x2 C 4
Petermine ho( large the s%&are sho&ld be )&t 'rom ea)h )orner in order to obtain the largest possible vol&me. "ns3 &..Gcm x &..G 8:8 ?oar ?oard d 8xam, 8xam, Dar Dar &/ &/
; baseball diamond is a s%&are$ 27 m on ea)h side. he instant nstant a r&nner is hal'(a# 'rom home to 'irst base$ he is giving to(ards 'irst base at ! ms. Go( 'ast is his distan)e 'rom the se)ond base )hanging at this instantT
Petermine ho( large the s%&are sho&ld be )&t 'rom ea)h )orner in order to obtain the largest possible vol&me. "ns3 &..Gcm x &..G 8:8 ?oar ?oard d 8xam, 8xam, Dar Dar &/ &/
; re)tang&lar 'ield )ontaining )ontaining a given area is to to be 'en)ed o'' along ong a straight river. 9' no 'en)ing is needed along the river$ (hat sho&ld be the dimension o' the 'ield so that 2K least amo&nt o' 'en)ing materials (ill be &sedT "ns3 9 = 2K
&0 8:8 ?oard ?oard 8xam, 7o# &/
ind the minim&m vol&me o' a right )ir)&lar )#linder that )an be ins)ribe in a sphere having a radi&s e%&al to r.
&& 88 ?oard 8xam , "pril &/&
; telephone )ompan# agrees to p&t &p a ne( e+)hange 'or 100 s&bs)ribers or less at a &ni'orm )hange o' 40 ea)h. o en)o&rage more s&bs)riber the )ompan# agrees to ded&)t 20 )entavos 'rom their &ni'orm rate 'or ea)h s&bs)riber in e+)ess o' 100. 9' the )ost to serve ea)h s&bs)riber is 14$ (hat n&mber o' s&bs)riber (o&ld give the telephone )ompan# the ma+im&m net in)ome. "ns3 &&+ 12. ind the e%&ation on o' the tangent tangent line to the the )&rve # = +3 , 3+2 5+ = 2 that has the least slope "ns3 2x – y C = 0 13. ind the area o' the largest largest re)tangle (ith sides parallel to to the )oordinate a+es a+es (hi)h )an be ins)ribe in the bo&nded b# + 2 = 2 , 4 and + 2 = # , 4.
"ns3 .4 sq units units
14. ;n isos)eles isos)eles trapeQoid trapeQoid is 6 )m long on ea)h side. Go( long m&st m&st be the longest side i' the area is ma+im&m. "ns3 &2 cm 15. ind ind the dimensio dimensionn o' the right )ir)&lar )ir)&lar )one o' minim& minim&m m vol&me vol&me (hi)h )an be √ 2 cm - !eig!t = 2 cm )ir)&ms)ribed abo&t a sphere o' radi&s )m. "ns3 radius = / √ 16. ind the dimension o' the )#linder o' ma+im&m lateral area (hi)h )an be ins)ribe in a cm sphere o' radi&s 6√2 )m. "ns3 radius = . cm - !eig!t = &2 cm 17. ind the ratio o' the vol&me o' the right )ir)&lar )#linder o' ma+im&m ma+im&m vol&me to that o' the '@ cone )ir)&ms)ribing )one. "ns3 @ cyl cyl cone = 4' 1. ind the e%&ation on o' the line passing passing thro&gh the point 3$ 4 (hi)h )&ts 'rom the 'irst %&adrant o' a triangle o' minim&m area. "ns3 4x C y – 24 = 0 19. ind the dimension o' the right )ir)&lar )one o' ma+im&m vol&me (hi)h )an be ins)ribe in a sphere o' radi&s 12 )m. "ns radius=/ √ √ 2 cm - !eig!t = &.cm 20. ind the area o' the largest re)tangle re)tangle that )an be ins)ribe in an ellipse π square square units !+2 4#2 = 36. "ns3 &2 π
RELATED RATE-
9' a %&antit# + is a '&n)tion o' time t$ the time rate o' + given e+pressed as d+dt. hen t(o or more time var#ing %&antities are related b# an e%&ation$ the relation bet(een their rates o' )hange ma# be obtained b# di''erentiating both members o' the e%&ation (ith respe)t to time t. -#pplementar8 Pro&lems: & 8:8 ?oar ?oard d 8xam, 8xam, 5ept 5ept &/. &/.
12. ;t a )ertain instant instant the semi maCor maCor a+is and semi minor minor a+is o' an ellipse are 12 and respe)tivel# and the semi maCor a+is is in)reasing X &nit ea)h min&te. ;t (hat rate is the unit'min semi maCor a+is de)reasing i' the area remains )onstantT "ns3 &' unit'min
; baseball diamond is a s%&are$ 27 m on ea)h side. he instant nstant a r&nner is hal'(a# 'rom home to 'irst base$ he is giving to(ards 'irst base at ! ms. Go( 'ast is his distan)e 'rom the se)ond base )hanging at this instantT "ns3 *402+ m's m's 2 8:8 ?oar ?oard d 8xam 8xam , 5ept 5ept &/ &/
; boat is being to(ed to a pier. he pier is 20 't. above the boat. he remaining length length o' the rope to be p&lled is 25 't. 9t is being p&led at 6 't. per se)ond. Go( 'ast does the boat approa)hes the pierT "ns3 &0 ft's
8:8 ?oar ?oard d 8xam, 8xam, Darc Darc! ! &/& &/&
"iven a )oni)al '&nnel o' radi&s 5 )m and height 15 )m. he vol&me is de)reasing at the rate o' 15 )&. )ms. ind the rate o' )hange in height (hen the (ater is 5 )m 'rom the top.
"ns3 04 cm's cm's 4 8:8 ?oard ?oard 8xam, 8xam, "pril &/ &/ ' 8:8 ?oard ?oard 8xam, 8xam, ct &/+ &/+
and is po&ring 'rom a hole at the rate o' 25 )&$ 't. per se)ond and is 'orming a )oni)al pile on the gro&nd. 9' the )oni)al 'ormation has an altit&de al(a#s o' the diameter o' the base$ ho( 'ast is the altit&de in)reasing (hen the )oni)al pile is 5 't. highT "ns3 &'&2G+
ft's increasing + 8:8 ?oar ?oard d 8xam, 8xam, eb eb &G/ &G/
; heli)opter is rising verti)all# verti)all# 'rom the gro&nd at a )onstant rate o' 4.5 ms. ms. hen it is 75 m o'' the gro&nd$ Ceep passed beneath the heli)opter travelling in a straight line at a )onstant speed o' 0 Fph. Petermine ho( 'ast the distan)e bet(een them )hanging a'ter m's %increasing) 1 se)ondT "ns3 &02 m's
. 8:8 ?oar ?oard d 8xam, 8xam, eb eb &GG &GG
(o boats starts at the same point. ne sail d&e east starting 10 ;.M. at a )onstant rate o' 20 Fph. he other sail d&e so&th starting 11 ;.M. at a )onstant )onstant rate o' ! Fph. Go( 'ast are the# separating at noonT
"ns3 2&4 p! G 8:8 ?oar ?oard d 8xam, 8xam, "ugus "ugustt &G.
; dive bomber loss altit&de at a rate o' 400 mph. Go( 'ast is the visible s&r'a)e o' the earth de)reasing (hen the bomber is 1 mile highT "ns3 2G2 . ; man li'ts li'ts a bag o' sand sand to a s)a''old s)a''old 30 30 m above his head b# means means o' a rope rope (hi)h (hi)h passes over a p&lle# on the s)a''old. he rope is 60 m long. 9' he Feeps his end o' the rope horiQontal horiQontal and (alFs a(a# 'rom beneath the p&lle# at 4 ms$ ho( 'ast is the bag m's rising (hen he id 22.5 m a(a#T "ns3 24 m's !. ater is passing passing thro&gh thro&gh a )oni)al )oni)al 'ilter 'ilter 24 )m deep deep and 16 )m a)ross a)ross the top top into a )#lindri)al )ontainer o' radi&s 6 )m. ;t (hat rate is the level o' (ater in the )#linder rising i' (hen the depth o' the (ater in the 'ilter is 12 )m its level is 'alling at the rate o' 1 cm'min )mminT "ns3 4' cm'min )mse). (o 10. ; parti)le starts at the origin and travel &p the line # = √3 + at a rate o' 5 )mse). se)onds later$ another parti)le starts at the origin and travels &p the line # = + at the rate o' 10 )ms. ;t (hat rate are the# separating separating 2 se)onds se)onds a'ter the last parti)le parti)le startedT "ns3 0G ft's ft's
11. ; parti)le travels along along a parabola # = 5+2 + 3. ;t (hat point do its abs)issa abs)issa and ordinate )hange at the same rateT "ns3 1%0, )
?h$F$ ?enter o' ?&rvat&re
12. ;t a )ertain instant instant the semi maCor maCor a+is and semi minor minor a+is o' an ellipse are 12 and respe)tivel# and the semi maCor a+is is in)reasing X &nit ea)h min&te. ;t (hat rate is the unit'min semi maCor a+is de)reasing i' the area remains )onstantT "ns3 &' unit'min 13. ; )lo)F )lo)F hands are 1 and 5 in)hes in)hes long long respe) respe)tiv tivel# el#.. ;t (hat rate are the the ends ends in'min approa)hing ea)h other (hen the time is 2@o )lo)FT "ns300+ in'min 14. ;n elevated train on a tra)F 30 't above the gro&nd gro&nd )rosses a street at the rate o' 20 't. per se). ;t the instant that the )ar approa)hing at the rate o' 30 'ts and the )ar are ft'sec separating 1 se) later "ns3 2.G ft'sec
?h$F$ ?enter o' ?&rvat&re
θ 0 Pe'inition:
Differential Approximation Approximation Approximation Approximation of Error
-
9' # = '+$ then d# = '@+ d+ d+ = )hange or error in + d# = )hange or error in # d++ = relative error in + d## = relative error in # d++ 100 = per)entage error in + d## 100 = per)entage error in #
# U = -------------------H1 #@2I32 Notes:
#
tangent
he c#rat#re = o' a )&rve # = '+$ at an# point +$ # on it $ is the rate o' )hange indire)tion that is$ the angle o' in)lination - θ o' the tangent line per &nit ar) length . ∆s→0
-#pplementar8 Pro&lems:
CUR7ATURE
+
U = dθds = lim ∆θ∆
1. hat is the the ma+im&m ma+im&m allo(able allo(able error error in the edge o' a )&be to to be &sed to )ontain )ontain 10 )&bi) meters i' the error in the vol&me is not to e+)eed 0.015 )&bi) meterT "ns3 000&0/
2. he semi maCor maCor a+is and semi semi minor a+is a+is o' an ellipti)al ellipti)al plate plate is meas&re to be )m and 6 )m respe)tivel#. 9' there is an appro+imate error o' 0.01 )m and .02 )m in meas&ring error in )omp&ting 'or the area2 "ns3 0.2/ 3. he altit& altit&de de o' a )ertai )ertainn )ir)&lar )ir)&lar )one )one is the same same as the radi&s radi&s i' the base base and is meas&red as 12 )m (ith a possible error ' 0.04 )m. ind appro+imatel# the per)entage error in the )al)&lated val&e o' the vol&me. "ns3 &L 4. ind the allo(a allo(able ble per)entag per)entagee error in the radi&s &s o' a )ir)le i' the area area is to be )orre)t to (ithin 5[. "ns3 2+L 5. hat is the per)ent per)entage age error made made in the )omp&ted )omp&ted s&r'a)e s&r'a)e area area o' a sphere i' the error error "ns3 .L made in meas&ring the radi&s is 3[. "ns3
Normal ?ir)le o' ?&rvat&re
- + or U = ---------------------H1 +@ 2I32
9' U > 0$ the point is on the ar) that is )on)ave &p(ard 9' U < 0$ the point is on the ar) hat is )on)ave do(n(ard
he )&rvat&re U is given b#: g@ h - g h@ U = -----------------------H g@2 h2I32 9n olar 'orm$ r = ' θ here r@ = drdθ $ r = d2r dθ2 r 2 2r@ 2 , rr U = ------------------------ Hr 2 r@ 2I32
# = ƒ+ Radi#s of C#rat#re the radi&s )&rvat&re A 'or a point on the )&rve is the re)ipro)al o' its )&rvat&re
Notes:
+$#
A
R ' 4G=
9' A > 0$ the )&rve is )on)ave &p(ard
at that point
1. he h#poten&se h#poten&se o' a right triangle triangle is 34 34 )m )m.. ind ind the lengths lengths one leg is 14 )m longer than the other.
o' the t(o legs i'
Notes:
R ' 4G=
9' A > 0$ the )&rve is )on)ave &p(ard 9' A <$ the )&rve is )on)ave do(n(ard
Circle of C#rat#re or scillating Circle he )ir)le o' )&rvat&re o' a )&rve at a point on it is the )ir)le o' radi&s A l#ing -
on the )on)ave side o' the )&rve and tangent to it at #
?
# = ƒ+
A
+
Center of C#rat#re
he )enter o' )&rvat&re 'or a point +$ # o' a )&rve # = '+ is the )enter ?h$ U o' the )ir)le o' )&rvat&re at # H1 #@ 2I h = + - --------------------------#
1 #@ 2 F = # - ---------------------#
E7LUTE he eol#te of a c#re is the lo)&s o' the )enter o' )&rvat&re o' the given )&rve. -#pplementar8 Pro&lems:
1. ind the the )&rvat&re )&rvat&re o' e)h e)h o' the 'ollo( 'ollo(ing ing )&rves )&rves Ans; +a # = sin + Ans; +b +2 = 12# at + = 6 2. ind the the point point o' ma+im&m ma+im&m )&rvat )&rvat&re &re o' the the )&rve # = ln ln + Ans; * - 5 # # #, + 5 # # :n :n #
3. ind the the radi&s radi&s o' )&rvat )&rvat&re &re o' the the )&rve )&rve o' +3 +#2 , 6#2 = 0 at point 3$ 3 Ans; ' '
4. ind the the e%&ation e%&ation o' the the )ir)le )ir)le o' )&rvat&re )&rvat&re o' the parabo parabola la #2 = 12+ at point 3$ 6 Ans; *" $ -'# 0 *& 0 2# ( #66
%*G, /) 5. ind the the )enter o' the the )&rvat&re o' +3 +#2 , 6#2 = 0 at 3$ 3 / Ans; C %*G,
+AT)E+ATIC- +DULE B
8. 360 ?. 460 P. 560
1. he h#poten&se h#poten&se o' a right triangle triangle is 34 34 )m )m.. ind ind the lengths lengths o' the t(o legs i' one leg is 14 )m longer than the other. ;. 15 and 2! )m. 8. 16 and 30 )m. ?. 17 and 31 )m. P. 1 and 32 )m. 2. he area o' a rhomb& rhomb&ss is 132 132 s%. s%. m. m. i' its its shorter shorter diagonal diagonal is 12 m$ 'ind the longer diagonal. ;. 20 m 8. 22 m ?. 36 m P. 2 m 3. ne side o' the parallelogram lelogram is 10 m and its diagonals diagonals are 16 m and 24 m respe)tivel#$ 'ind its area. ;. 15.7 s%. m 8. 120 s%. m ?. !6 s%. m P. 1!2 s%. m 4. he diame diameter ter o' t(o spheres spheres is in the ratio ratio t(o is is to three three and the the s&m o' their their vol&mes is 1260 )&. m. ind the the vol&me o' the larger sphere in )&. m. ;. !0 8. !72 ?. !60 P. !3 5. he side o' the the triangle triangle are 5$ 7 and 10 respe)tive respe)tivel#. l#. ind the radi&s radi&s o' the )ir)&ms)ribed )ir)le. ;. 5.3! m 8. 6.40 ?. 7.20 P. 4.0 6. he vol&me vol&me o' o' a sphere sphere is 36c 36c )&. )&. m. he he s&r'a)e s&r'a)e area o' this sphere sphere in s%. m is: ;. 25c 8. 42c ?.36c P. 54c 7. Go( man# man# side does a pol#gon has has i' the s&m o' the the interior or angles is 2520oT ;. 14 8. 16 ?. 12 P. 10 . he 'irst 'irst term o' an arithmeti) progression is 3 and the the 15th term is 45. ind the s&m o' the 'irst 15 terms. ;. 260
!6. (o 'err#boats 'err#boats pl# ba)F ba)F and and 'orth 'orth a)ross a)ross a river river (ith (ith )onst )onstant ant b&t b&t di''erent di''erent speeds$ t&rning at the riverbanFs (itho&t (itho&t loss time. he# leave opposite shores at the same instant$ meet 'or the 'irst time me at !00 meter 'rom one shore and meet meet 'or the se)ond time
8. 360 ?. 460 P. 560 !. he 'irst 'irst term o' an arithmet arithmeti) i) progressi progression on is -2 and and the s&m oo'' the 'irst 11 terms terms is is . he )ommon di''eren)e is: ;. 4 8 .3 ?. 2 P. 1 10. ;n ellipse ellipse (ith maCor maCor a+is and a minor minor a+is 6 is revolved revolved abo&t its minor minor a+is. ind the vol&me o' the solid revol&tion. ;. 201.06 8. 150.0 ?. 160.50 P. 1206.37 11. ind the the e))entri) e))entri)it# it# o' an ellips ellipsee (ith maCor maCor a+is a+is and minor minor 6. ;. ] 8.42 ?. 43 P. 0.66 E2. ind the length o' the lat&s re)t&m o' an ellipse (ith maCor a+is and minor a+is a+is 6. ;. 3.5 8. 4.5 ?. 4 P. 5 !4. ind the the smallest smallest n&mber n&mber (hi)h (hi)h (hen #o& divide divide b# 2$ the remainder remainder is 1* (hen (hen #o& divide b# 3$ the remainder remainder is 2* (hen #o& divide b# 5$ the remainder remainder is 4 and (hi)h (hen #o& divide b# 6$ the remainder is 5. ;. 3! 8. 4! ?. 5! P. 6! !5. ; man man bo&gh bo&gh 20 p)s p)s o' o' assorte assortedd )al)&lators )&lators 'or 20$ 000. hese )al)&lator )al)&latorss are are o' three t#pes namel#: a. programmable at 3$000p) b. s)ienti'i) at 1$500p) ). ho&sehold t#pe Go( man# programmable )al)&lators did the man bo&ghtT ;. 5 8. 2 ?. 13 P. 15
11. P 12. 8 13. P
31.? 32.8 33.P
51.8 52.? 53.;
71.; 72.P 73.;
!1.8 !2.; !3.;
!6. (o 'err#boats 'err#boats pl# ba)F ba)F and and 'orth 'orth a)ross a)ross a river river (ith (ith )onst )onstant ant b&t b&t di''erent di''erent speeds$ t&rning at the riverbanFs (itho&t (itho&t loss time. he# leave opposite shores at the same instant$ meet 'or the 'irst time me at !00 meter 'rom one shore and meet meet 'or the se)ond time 500 meters 'rom the opposite shore. hat is the (idth o' the riverT ;. 2000 8. 2200 ?. 2020 P. 2002 !7. Go( m&)h m&)h lead m&st m&st be added added to n allo# lo# (hi)h (hi)h is 50[ tin tin and 25[ lead lead to maFe maFe an allo# (hi)h is 60[ tin and 20[ leadT ;. 0 8. 1 Fg ?. 2 Fg P. 3 Fg !. ; pipe )an 'ill &p a tanF (ith drain open in 3 hrs. 9' the pipe r&ns (ith the drain open 'or 1 hr. and then the drain drain is )losed$ it ill taFe 45 more min&tes min&tes 'or the pipe to 'ill &p the tanF. 9' the drain (ill be )losed right at the the start o' 'illing ing ho( long (ill it taFe 'or the the pipe to 'ill &p the tanFT ;. 1.1 hrs. 8. 1.125 hrs. ?. 1.25 hrs. P. 1.3 hrs. !!. Ping )an 'inish nish the the Co Cobb in hrs. ito )an do it in 5 hrs. 9' Ping Ping (oF (oF 'or 3 hrs. and then ito ito (as asFed to help him 'inish it$ it$ ho( long ito (illll have to (orF (ith PingT ngT ;. 25 hrs 8. 2513 min ?. 1.!23 hrs P. 30 hrs 100 ind the vol&me o' the solid generated b# revolving revolving the area bo&nded b# + = #2 and + = 2- # 2 abo&t the # , a+is. ;. pi3 8. 16 pi3 ?. 10 pi3 P. 7 pi3 18 2. 8 3. ; 4. 8 5. ; 6. ? 7. 8 . 8 !. ? 10 ;.
). ,ba d. ,)a
21.? 22.; 23.8 24.8 25.P 26.; 27.; 2.; 2!.? 30.
41.8 42.? 43.; 44.8 45.P 46.? 47.; 4.; 4!.8 50.?
61.P 62.P 63.8 64.? 65.; 66.8 67.P 6.? 6!.8 70.P
1.; 2. ? 3. 4.; 5.P 6.; 7.8 .8 !.; !0.8
11. P 12. 8 13. P 14. P 15. ? 16. ; 17. P 1. 8 1!. ; 20. P
31.? 32.8 33.P 34.8 35.8 36.; 37.? 3.8 3!.? 40.;
51.8 52.? 53.; 54.; 55.; 56.? 57.? 5.? 5!.8 60.;
71.; 72.P 73.; 74.; 75.? 76.; 77.; 7.; 7!.P 0.? 1
!1.8 !2.; !3.; !4.? !5.8 !6.8 !7.; !.8 !!.? 00.8
). ,ba d. ,)a
+AT)E+ATIC- +DULE $
1. hen the )orrespondi )orresponding ng elements elements o t(o t(o ro(s o' determina determinant nt are proportio proportional$ nal$ then the val&e o' the determinants is: a. M<iplled b# the ratio b. ero ). OnFno(n d. ne 2. hen t(o ro(s are inter)hang inter)hanged ed in in positi position$ on$ the val&e o' the dete determina rminant nt (ill be: a. On)hanged b. 8e)omes Qero ). M<iplied -1 d. Onpredi)table 3. 9' ever# element o' a ro( or )ol&mn &mn are m<iplied b# a )onstant F$ then the val&e o' the determinant is: a. M<iplied b# ,nF b. F to the n ). M<iplied b# F d. ;n#one o' the above ma# be tr&e 4. 9' the the %&a %&adr drat ati) i) e%&a e%&atition on a+2 b+ ) = 0$ (hen b 2 is e%&al than 4a). then the root are: a. %&al b. Aeal and &ne%&al ). 9maginar# d. +traneo&s 5. 9n the %&adrati) e%&ation a+2 b+ ) = 0$ (hen b 2 is greater than 4a)$ then the root are: a. %&al b. Aeal and &ne%&al ). 9maginar# d. +traneo&s 6. 9n the %&adrati) e%&ation a+2 b+ ) = 0$ i' r 1 r 2 represent the roots$ the r 1 = r 2 is e%&al to: a. ba b. )a
5!. he energ# (hi)h bod# possess b# virt&e o' its positions$ me)hanism is )alled LLLLLLLL. a. potential energ#
)on'ig&ration or internal
7. he# are e%&ation (hose members are e%&al onl# 'or )ertain 'or possibl# no val&es o' &nFno(n. a. ?onditional e%&ation b. 9ne%&alities ). 'i+ e%&ation d. emporar# e%&ation . Aoots Aoots (hi) (hi)hh are e%&al e%&al to Qero Qero are )all )alled ed the: the: a. rivial roots b. 9denti)al ). #mmetri) d. Aational !. hen all + are repla)ed repla)ed b# # and all # are repla)ed repla)ed b# + and the e%&ation e%&ation remains the same$ then the e%&ation is said to be: a. %&ivalent b. 9denti)al ). s#mmetri) d. Aational 10. he n&mber 0.123123123 D is: a. 9rrational b. &rd ). rans)endental d. Aational 11. o eliminat eliminatee the s&rd$ (e (e m<ipl# m<ipl# it b# its LLLLLLL LLLLLLLLLLLL LLLLLLL: LL: a. %&are b. ?&be ). Ae)ipro)al d. ?onC&gate 12. he letter letter P in the the Aomans Aomans N&merals N&merals is e%&ivale e%&ivalent nt to: a. 50 b. 500 ).5000 d.50000 13. ; statement statement (hi)h is a))epted (itho&t proo': a. ost&late b.
a. 8o#le@s la( b. Ro&ng@s la( ). "a# l&ssa) la(
5!. he energ# (hi)h bod# possess b# virt&e o' its positions$ )on'ig&ration or internal me)hanism is )alled LLLLLLLL. a. potential energ# b. Uineti) energ# ). Me)hani)al energ# d. ele)tri)al energ# 60. 9' the mass o' the bod# is e+pressed e+pressed in grams and the velo)it# velo)it# in )mse)$ )mse)$ the Fineti) energ# is e+pressed in: a. rgs b. Zo&les ). ?o&lombs d. l&gs 61. nerg# nerg# is given given to a bod# bod# or s#stems s#stems o' bodies bodies (hen (hen (orF (orF is done &pon &pon it. 9n 9n this pro)ess there is merel# merel# a trans'er o' energ# 'rom one bod# to another. another. 9n s&)h trans'er$ trans'er$ no energ# is )reated or destro#ed* it merel# )hanges 'rom one to another. his statement is Fno(n as: a. he la( o' )onservation o' energ# b. nerg# trans'ormation ). ?o&lomb@s la( d. la( o' po(er 62. he de'ormation de'ormation o' elasti) asti) bod# is dire)tl# dire)tl# proportional proportional to the applied applied 'or)e$ 'or)e$ provided that the elasti) limit is not e+)eeded. his theor# is Fno(n as: a. GooFe@s la( b. Ro&ng theor# ). Ueppler@s la( d. 8&lF mod&l&s 63. 9' an e+ternal e+ternal press&re press&re is applied applied to a )on'ined ned 'l&id$ 'l&id$ the press&re press&re (ill be in)reased in)reased at ever# point in the 'l&id 'l&id b# the amo&nt amo&nt o' e+ternal e+ternal press&re. press&re. his theor# theor# is Fno(n as: a. as)al@s la( b. G#dra&li) la( ). G#drostati) la( d. 8rangg@s la(
64. ; bod# (holl# (holl# or parti partiall# all# s&bmerge s&bmergedd in a 'l&id e+perien)es en)es an &p(ard &p(ard 'or)e e%&al e%&al to the (eight o' the 'l&id displa)ed. his theor# is Fno(n Fno(n as: a. ;r)himedes prin)iple b. 8o#le@s la( ). hird la( o' Ne(ton d. l&id theor# 65. 9' the temperat&re o' a )on'ined )on'ined gas does not )hange the the prod&)t prod&)t o' the press&re and vol&me is )onstant. his statement is Fno(n as:
2. hen t(o planes interse)t (ith ea)h other other the the amo&nt amo&nt o' divergen)e bet(een the t(o planes planes is e+pressed b# meas&ring the: a. pol#hedral angle
a. 8o#le@s la( b. Ro&ng@s la( ). "a# l&ssa) la( d. ?harles la( 66. ;t an# t(o points nts along a streamline streamline in an ideal 'l&id in stead# 'lo($ the s&m o' the press&re$ the potential energ# per &nit vol&me$ and the Fineti) energ# per &nit vol&me has the same val&e. his )on)ept is Fno(n as: a. 8erno&lli@s theorem b. l&id theor# ). G#dra&li) theorem d. as)al theorem ;ns(ers: 1. 8 2. ? 3. ? 4. ; 5. 8 6. ? 7. ; . ; !. ? 10.P 11.P 12.8 13.; 14.? 15.; 16.P 17.8 1.P 1!.; 20 ;
21 8 22.8 23.8 24.? 25.8 26.8 27.; 2.? 2!.P 30.? 31.8 32.; 33.8 34.8 35.; 36.; 37.P 3.8 3!.; 40.P
41.; 42.; 43.? 44.8 45.P 46.? 47.8 4.8 4!.; 50.P 51.P 52.; 53.P 54.? 55.; 56.? 57.8 5.; 5!.; 60.;
61.; 62.; 63.; 64.; 65.; 66.;
+AT)E+ATIC- +DULE 4
.
1.
he solid 'ormed 'ormed b# revolving revolving the ellipse ellipse abo&t is
a. spheroid b. oblate spheroid ). prelate spheroid d. ellipsoid
a. he prod&)t o' the 're%&en)# and (avelength b. Pistan)e o' the )rest to the ne+t )rest ). ;l(a#s e%&al to 16.00 miles per. se).
minor minor a+is is )alled )alled LLLLLLLL LLLLLLLL
2. hen t(o planes interse)t (ith ea)h other other the the amo&nt amo&nt o' divergen)e bet(een the t(o planes planes is e+pressed b# meas&ring the: a. pol#hedral angle b. plane angle ). re'le+ angle d. dihedral angle 3. 9' the the prod&)t o' the the slope o' an# t(o straight ght lines lines is negative. ne o' this are said to be: a. parallel b. sFe( ). perpendi)&lar d. Non-interse)ting 4. he logarithm o' 1 to an# base is: a. Qero b. one ). in'init# d. interse)ting 5. he ma+im&m displa)ement o' vibration 'rom the e%&ilibri&m is )alled: a. 're%&en)# b. speed ). amplit&de d. period 6. 9' the velo)i velo)it# t# o' o' the the bod# bod# is do&b do&bled led$$ a. he Fineti) energ# is %&adr&pled b. the Fineti) energ# is halved ). the potential energ# is halved d. the potential energ# is do&bled
7.
; bod# traveling to a )ir)le (ith )onstants speed: a. 9s a))elerated b. Gas )onstant velo)it# ). Poes no move . 9vor# 9vor# soap soap 'loa 'loats ts in (ater (ater be)a be)a&se &se:: a. ;ll matter has mass b. he densit# o' ivor# soap is &nit# ). he spe)i'i) gravit# o' ivor# soap is greater than that o' (ater d. he spe)i'i) gravit# o' ivor# soap is greater than that o' (ater !. hen t(o (ave o' the the same same 're%&en)#$ 're%&en)#$ speed and amplit&de t&de traveling traveling in opposite te dire)tions are s&perimposed: a. Pestr&)tive inter'eren)e al(a#s res<s b. ?onstr&)tive inter'eren)e al(a#s res<s ). tanding (aves are prod&)ed d. he phase di''eren)e is al(a#s Qero 10. he velo)it# o' a (ave is:
5. ; 6. P 7. ;
25. ; 26. 8 27. ;
45. P 46. P 47. P
65. ; 66. ; 67. ;
5. ; 6. 8 7. 8
a. he prod&)t o' the 're%&en)# and (avelength b. Pistan)e o' the )rest to the ne+t )rest ). ;l(a#s e%&al to 16.00 miles per. se). d. he ratio o' the 're%&en)# to (avelength 11. 11. ;))e ;))ele lera ratition on is: is: a. the same as velo)it# b. the same as displa)ement ). the rate o' )hange o' velo)it# d. al(a#s Qero 12. Mass is the %&antitative meas&re o a. inertia b. gravit# ). (eight d. moment&m 13. ; se%&en)e o' n&mber (here ever# term is obtain b# adding all the pre)eding terms o' a s%&are n&mber series s&)h as 1$ 5$ 14$ 30$ 55$ !1 Dis )alled LLLLLLLLL. a. riang&lar n&mber b. etrahedral n&mber ). &ler@s n&mber !. %&are root o' the prod&)t o' t(o t(o terms o' a geometri) progression. progression. a. Median b. Mean ). "eometri) mean d. "eometri) term !!. hi)h o' the 'ollo(ing )annot be probabilit#T a. 0.1 b. 0 ). 1 d. 0.232323 100. 9' a is the n&mber o' times that an event (ill taFe pla)e and b is the n&mber o' times that it (ill not taFe pla)e$ then the probabilit# that it (ill taFe pla)e is: a. ab b. ba ). aba d. abab
;ns(ers: 1. 8 2. P 3. ? 4. ;
21. ? 22. ; 23. ; 24. ;
41 . ; 42. ? 43. ? 4 4. ;
61. P 62. ? 63. 8 64. P
1. ? 2. ? 3. P 4. ?
Concentric circles , )ir)le having the same )enter (ith &ne%&al radii. Tangent circles 3 )ir)les tangent to the same line at the same point. Inscri&ed circle , in a pol#gon (hen the sides o' the pol#gon are tangent to it.
5. ; 6. P 7. ; . P !. ? 10. ; 11. ? 12. ; 13. P 14. ; 15. ; 16. ; 17. ; 1. ; 1!. ; 20. ;
25. ; 26. 8 27. ; 2. ; 2!. ; 30. ; 31. ; 32. 8 33. 8 34. ? 35. P 36. ; 37. P 3. P 3!. ; 40. 8
45. P 46. P 47. P 4. 4!. ; 50. 8 51. ; 52. ; 53. ; 54. ; 55. 8 56. ; 57. 8 5. ; 5!. 8 60. ;
65. ; 66. ; 67. ; 6. ; 6!. ; 70. 8 71 . 8 72. P 73 . ? 74 . ; 75 . 8 76. ? 77 . 8 7 . ; 7!. 8 0. ;
5. ; 6. 8 7. 8 . ; !. 8 !0. P !1. P !2. ; !3. ; !4. 8 !5. ; !6. ? !7. ? !. ? !!. ? 100.?
Concentric circles , )ir)le having the same )enter (ith &ne%&al radii. Tangent circles 3 )ir)les tangent to the same line at the same point. Inscri&ed circle , in a pol#gon (hen the sides o' the pol#gon are tangent to it. Circ#mscri&ed Circ#mscri&ed circle , abo&t a pol#gon (hen it passes thro&gh the verti)es o' the
pol#gon.
Circ#lar ringGann#l#s ringGann#l#s , area in)l&ded bet(een t(o )on)entri) )ir)les o' &ne%&al radii.
OLYGON Pol8gon , a plane )losed b# broFen lines Reg#lar pol8gon 3 pol#gon (hose angles are e%&al and all o' (hose sides are -imilar pol8gon , pol#gon (hose )orresponding angles are e%&al and their
e%&al
)orresponding sides are proportional.
Center of a pol8gon , )ommon )enter o' its ins)ribed and )ir)&ms)ribed )ir)le. Diagonal of a Pol8gon , line Coining an# t(o non-)onse)&tive verti)es. Apothem .of a reg#lar pol8gon/ , the perpendi)&lar line dra(n 'rom the )enter o' the
ins)ribed )ir)le to an# one o' the sides. 9t is the radi&s o' the ins)ribed )ir)le.
PLA!E 0E+ETR
C:assi9ications o9 o:&on
N&mber o' ides 3 4 5 6 7 ! 10 11 12
Definitions of Terms Axiom , a statement o' tr&th o' (hi)h is admitted (itho&t proo'. Theorem , a statement o' tr&th (hi)h m&st be established b# proo'. Corollar8 , a statement o' tr&th o' (hi)h 'ollo(s (ith little or no proo' 'rom a theorem. Post#late , 9n )onstr&)tion or dra(ing o' lines and 'ig&res o' (hi)h is admitted (itho&t proo'. )8pothesis , part o' a theorem (hi)h is ass&med to be tr&e. Concl#sion , part o' a theorem (hi)h is to be proved. Conerse of a Theorem , another theorem (herein the h#pothesis and )on)l&sion o' the
'irst are reversed$ i.e. the h#pothesis be)omes the )on)l&sion and the )on)l&sion be)omes the h#pothesis.
CIRCLE Circle , lo)&s o' points (hi)h are at the same distant 'rom a point (ithin$ )alled the )enter. Diameter , , a line passing thr& the )enter$ terminating at both ends on the )ir)le. Radi#s , a line dra(n 'rom the )enter to the )ir)le Arc , a part o' the )ir)le. Chord , a line Coining t(o points on a )ir)le. -ecant , an inde'inite line interse)ting the )ir)le in t(o points. Tangent , an inde'inite line to&)hing a )ir)le at onl# one point. -egment , position o' a )ir)le bet(een )hord and its ar). -ector , , position o' a )ir)le bet(een t(o radii and ar). Inscri&ed angle , an angle (hose verte+ is a point in the )ir)le$ the sides o' (hi)h are
)hords.
ol#gon
Trapeoid , a %&adrilateral (ith onl# t(o sides o' (hi)h are parallel. Parallelogram , a %&adrilateral (hose opposite sides are parallel. Rhom&#s ,a parallelogram (ith e%&al sides and obli%&e angles. Rectangle , a parallelogram (hose angles are right angles. -"#are , a re)tangle (ith e%&al sides. Isosceles trapeoid , one (hose non-parallel sides are e%&al. Isometric fig#res , 'ig&res (hose parameters are e%&al.
Properties of Plane 1ig#res
1.
β α
Central angle , an angle (hose verte+ is at the )enter o' the )ir)le the sides o' (hi)h are
φ
he e+terior angle o' a triangle is greater either nonadCa)ent interior angle and is e%&al to their s&m.
radii.
d
2. ;
φ = α β 8
P he di diagonal o' o' a parallelogram di divides th the
triangle %&adrilateral pentagon he+agon heptagon o)tagon nonagon de)agon &nde)agon dode)agon
;
8
P ;8 P = X ;8
φ = α β 8
2. ;
he di diagonal o' o' a parallelogram di divides th the parallelogram into t(o )ongr&ent triangles.
∆ ;?P ≅ ∆ ;8? P
? 3.
P
;
4.
;P = 8?P ∠8;P = 8?P
;
;? P ⊥8P ∠ ;?P = ∠ ;?8 = X ∠ ;8? ∠ ;P8 = ∠?P8 = X ∠ ;P?
?
he median o' a trapeQoid is parallel to the bases and e%&al to one-hal' o' their s&m. ;8 = ;8?P2
P
?
; !.
?
o
;
he interse)tion o' the three angle bise)tors meet at a )omm )ommon on poin pointt is )all )alled ed in)e in)ent nter er$$ (hi) (hi)hh is e%&idistant 'rom the three sides o' a triangle. he ins)ribed )ir)le is )alled in)enter.
/ 8
A
he diagonals o' a rhomb&s are perpendi)&lar to ea)h other and bise)t the angles thro&gh (hi)h the# pass.
8
P ;8 P = X ;8
8
;8 = ?P ∠ ;P? = ∠ ;8?
?
; .
he opposite sides and opposite angles o' a parallelogram are e%&al.
8
= / = A
8
10.
he interse)tion o' the three perpendi)&lar bise)tors meet at a )ommon point )alled )ir)&m)enter$ (hi)h is e%&idistant 'rom the three verti)es o' the triangle. he )ir)le (hose )enter is 0 to&)hing to&)hing the three verti)es is )alled )ir)&m)enter.
o ;
?
8 = ; = ?
8 5.
;
he diagonals o' a parallelogram bise)t ea)h other.
8
?
P
;
? 6.
7.
he interse)tion o' the three altit&des o' a triangle meet at a )ommon point )alled ortho)enter.
;⋅> = >? = X ;? 8> = >P = X 8P
>
he median o' the h#poten&se o' a right triangle have e%&al distan)es 'rom the three verti)es.
;
11.
12.
8
8
M)
;M) = 8M) = ?M) he line segment (hi)h Coins the midpoint
?
?
;
M
/
23 ? = ?M 23 ;/ =? ;M A ?
he three medians o' an# triangle meet at a )ommon point (hi)h is t(o-thirds o' the distan)e 'rom ea)h verte+ to the midpoint o' the opposite side. he point o' interse)tion is )alled the )entroid. 23 8A = 8M
8
8
;P = √;8;?
;P = √;8;?
8 ? 13 .
he altit&de &pon the h#poten&se o' the right triangle is the mean proportional bet(een the segments o' the h#poten&se.
?
1.
he tangent line is a mean proportional bet(een the entire se)ant and its e+ternal segment.
?P = √;PP8
;
;1;2 = <12 < 22
<1
8
P
<2
;1 14 .
;2
; )entral angle is meas&red b# its inter)epted ar).
θ
1!.
θ
;
8
15 .
∠ ; ∠8 ∠?D = n,210°
he ins)ribed angle is meas&red b# one-hal' o' the inter)epted ar).
; 8
he s&m o' interior angles o' an# pol#gon o' n sides is
?
∠ ;8? = X ;?
θ
P
20. 9nterior angle θ i and e+terior angle θ 0 o' a reg&lar pol#gon o' n sides is
?
θ0
16 .
;n angle 'ormed b# t(o )hords interse)ting (ithin a )ir)le is meas&red b# one hal' o' the s&m o' the ar)s inter)epted b# it and its verti)al angle.
; 8
θi
θ9 = Hn,2nI 10° θo = 360° n
;8 = ?P
P
?
1orm#las for Plane 1ig#res
17 .
he prod&)t o' one entire se)ant and its e+ternal segmen segmentt e%&als e%&als the prod&) prod&)tt o' the other entire entire se)ant and its e+ternal segment.
P ;
here: s ' M .a%&%c/
)
9.
;rea o' riangles 9.0 "iven sides a$ b$ and ) Gero@s 'orm&la ; = √ss-as-bs-)
a
2 sin δ 6. "iven: sides a$ b$ and ) and ins)ribed in a )ir)le o' radi&s A
b
here:
2 sin δ
)
s ' M .a%&%c/
6. "iven: sides a$ b$ and ) and ins)ribed in a )ir)le o' radi&s A ; = ab) 4A
2. "iven base b and altit&de h ; = X bh Gbbh
a
A
h
b
b
b )
r
7. "iven: sides a$ b$ and ) and )ir)&ms)ribed abo&t a )ir)le o' radi&s r ; = rs here: s=ab) 2 3. "iven: e%&ilateral triangle o' side s
b
a
A ' 5 s$ B
r ) s
99.
s
s
;rea o' arallelogram 1. "iven: "iven: base base b and and altit altit&de &de h ; = bh b 2. "iven: sides a and b and their in)l&ded angle θ ; = ab sin θ a
4. "iven: t(o adCa)ent sides and the in)l&ded angle ; = X b) sin α ; = X a) sin β ; = X ab sin δ b a
999.
θ
;rea o' a rapeQoid "iven: bases a and b and altit&de h ; = X ab ab a h
5. "iven: at least t(o angles and a side 2 β sin δ ; = a sin 2 sinα 2 α sin δ ; = b sin 2 sin β ; = ) 2 sin α sin β
s = X ab)d ;? = 10° 8P = 10 °
)
b 9\. 9\.
8
;rea ;rea o' )#)li) )#)li) %&ad %&adril rilate ateral ral 8rama 8ramag&p g&phta hta@s @s orm orm&la &la "iven: sides a$ b$ )$ and d ; ; =√s-as-bs-)s-d a here:
d2 d1
)
>.
;rea rea o' a se)t se)tor or o' a )ir)le r )le
d P
s = X ab)d ;? = 10° 8P = 10 °
8
d2 d1
>.
)
b ? \.
;rea o' Ahomb&s "iven: diagonals d1d2 ; = X d1d2
d1
>9. >9.
d2
;rea rea o' a se)t se)tor or o' a )ir)le r )le "iven: radi&s r and θ s = r θ ; = X r 2θ
r
θ
;rea ;rea o' a seg segme ment nt o' a )ir )ir)l )lee ; = X r 2 θ , sinθ here θ is in radi&s
r
θ \9.
r
;rea o' rapeQi&m 1. "iven: diagonals d1 and d2 and their in)l&ded angleφ ; = X d1d2 sin φ
;
φ
d2
>99. >99.
d1
;rea ;rea and and ?ir ?ir)& )&m' m'er eren en)e )e o' o' lli llips psee "iven: maCor a+is a and minor a+is b ; = πab ? = 2π √a2b22 a
\99. \99.
;rea ;rea o' Aeg& Aeg&la larr ol ol#g #gon on 1. "iven "iven : n sides$ sides$ ea)h o' length length s. ; = X ns2 )ot πn 2. "iven: "iven: n sides sides$$ and apoth apothem em a. ; = na2 tan πn 3. "iven: "iven: n sides sides ins)ribed ins)ribed in a )ir)le )ir)le o' radi&s radi&s A. ; = X nA2 sin 2πn
b
A a a
\999. \999.
9>.
;rea ;rea ; ; and and eri erimet meter er o' o' a ?ir)le ?ir)le 1. "iven: "iven: radi&s radi&s r and and diamet diameter er d ; = πr 2 = πd2 4 = 2πr = πd ;rea o' ;nn&l&s "iven: )ir)le o' radii r 1 and r 2 ; = πr 12 , r 22 here r 1Vr 2
r
d r 1 r 2
S8:ementar& ro4:ems
1. 8ise)tors o' the 3 angles o' a triangle meet at a )ommon point )alled the LLLLLLLLLL
>999 >999..
;rea ;rea o' o' a par parab abol oli) i) seg segme ment nt "iven: base b and height h 2 ; = 3 bh h
b
) a)&te angle d ins)ribed angle 11.(o angles (hose s&m is 360 degrees are said to be a s&pplemen s&pplementar# tar# b )omplime )omplimentar# ntar#
s
S8:ementar& ro4:ems
1. 8ise)tors o' the 3 angles o' a triangle meet at a )ommon point )alled the LLLLLLLLLL a. ortho)enter b. )entroid ). in)enter d. )ir)&m)enter 2. he perpendi)&lar bise)tor o' the sides o' a triangle pass thro&gh a )ommon point )alled the LLLLLLLLLL a. ortho)enter b. )entroid ). in)enter d. )ir)&m)enter 3. hi)h o' the 'ollo(ing is not a propert# o' a )ir)leT a thro&gh 3 points not in the straight line one )ir)le and onl# 1 )an be dra(n b a tangent to a )ir)le is perpendi)&lar to the radi&s at the point o' tangen)# and )onversel#. ) an ins)ribed angle is meas&red b# X o' the inter)epted ar). d the ar) o' 2 )ir)les s&btended b# e%&al )entral angle angle are e%&al. 4. hi)h o' the 'ollo(ing is not a propert# o' a triangleT a the s&m o' the 3 angles o' the triangle is e%&al to t(o right triangles. b the s&m o' the 2 side o' the triangle triangle is less than the 3rd side ) i' the 2 sides o' the triangle are &ne%&al$ the angles opposite are &ne%&al. d the altit&de o' a triangle meets in a point 5. he radi&s o' the )ir)le ins)ribed in a pol#gon is )alled as a internal radi&s b radi&s o' g#ration ) apot apothe hem m d h#dr h#dra& a&lili)) radi radi&s &s 6. ; pol#gon (ith 12 sides is )alled as a bide)agon b dode)agon ) nonagon d pentede)agon 7. ; pol#hedron having bases 2 pol#gons in parallel plane and 'or lateral 'a)es triangles or trapeQoid (ith 1 side l#ing on 1 base and the opposite verte+ or side l#ing on the other base o' the pol#hedron is a p#ramid b )one ) prismatoi prismatoidd d re)tang&la re)tang&larr parallelep parallelepiped iped . ;n angle greater than a straight angle b&t less than 2 straight angles is )alled as a )omplement b s&pplement d )omp )omplle+ d re'l re'le+ e+ !. ; part o' a )ir)le is o'ten )alled as a se)tor b )ord b ar) d segment 10. ;n angle (hose verte+ is appoint on the )ir)le and (hose sides are )ords is Fno(n as a inte interi rior or angl anglee b vert verti) i)al al angl anglee
23. oints that lie on the same plane are said to be a. )ollinear b. )oplanar ). dihedral d. parallel
) a)&te angle d ins)ribed angle 11.(o angles (hose s&m is 360 degrees are said to be a s&pplemen s&pplementar# tar# b )omplime )omplimentar# ntar# b elementar# d e+plementar# 12. ;ll )ir)les having the same )enter b&t (ith &ne%&al radii are )alled as a. e))e e))ent ntri ri)) )ir) )ir)le less b. )on) )on)en entr tri) i) )ir) )ir)le less ). inner )ir)les d. #thagorean )ir) les 13. ; )ir)le is LLLLLLLLL o&tside the triangle i' it is tangent to one side and the other t(o sides prolonged. a. ins)ribed b. es)ribed ). )ir)&ms)ribed d. tangent 14. ; triangle having three sides o' &ne%&al length is Fno(n as a. e%&i e%&ilat l ateral eral tri triangl angle b. s)al s)alene e ne tri triang a ngle ). isos) sos)el eles es tria trianngle gle d. e%&i e%&ian ang& g&la larr tri triaangl ngle 15. 9n a proportion o' 'o&r %&antities$ the 'irst and the 'o&rth terms are re'erred to as the a.means b. e+tremes ). denominators d. a+iom 16. ; statement the tr&th o' (hi)h is admitted (itho&t proo' is a. theorem b. )orollar# ). post&late d. a+iom 17. he part o' the theorem (hi)h is ass&med to be tr&e is the a. )orollar# b. h#pothesis ). post&late d. )on)l&sion 1. 9n "eometr#$the )onstr&)tion or dra(ing o' lines and 'ig&res$ the possibilit# o' (hi)h is admitted (itho&t proo' is )alled the: a. )orollar# b. theorem ). post&late d. h#pothesis 1!. ; statement the tr&th o' (hi)h 'ollo(s (ith little or no proo' 'rom the theorem is a. )orollar# b. a+iom ). post&late d. )on)l&sion 20. ; pol#gon is LLLLLL (hen no side$ (hen e+tended$ (ill pass thro&gh the interior o' the pol#gon. a. )onve+ b. e%&ilateral ). isoperimetri) d. reg&lar 21. ; )ir)le is said to be LLLLLLL to a pol#gon having the same perimeter (ith that o' the )ir)le a. )ongr&ent b.isoperimetri) ).proportional d. similar 22. he interse)tion o' the sphere and the plane thro&gh the )enter is the a. great )ir)le b. small )ir)le ). poles d. polar distan)e
36. ind the area o' the 'olded triangle sho(n belo(.
23. oints that lie on the same plane are said to be a. )ollinear b. )oplanar ). dihedral d. parallel
36. ind the area o' the 'olded triangle sho(n belo(.
24. hat Find o' a %&adrilateral is al(a#s 'ormed b# )onne)ting the midpoints o' the )onse)&tive sides o' a %&adrilateralT ;ns. parallelogram 25. he angles o' a pentagon are in the ratio 3: 3: 3: 4: 5. ind the largest angle. ;ns. 150deg 26. he legs o' a triangle are in the ratio 3: 3 and its area is 10 s%. )m. ind the length o' the legs. ;ns. &2 cm, &/ cm 27. ind the side o' a reg&lar o)tagon ins)ribed in a )ir)le o' radi&s 1! )m. 2. he &pper and lo(er bases o' a trapeQoid are 6)m and 12)m respe)tivel#$ and the altit&de is 4)m. he non-parallel sides o' the trapeQoid are prod&)ed &ntil the# meet at . ind the altit&de o' the triangle (hose verte+ is and (hose base is the lo(er base o' the trapeQoid. ;ns. )m 2!. ; se)ant and a tangent are dra(n to )ir)le 'rom the same point. 9' the internal segment is 1)m longer than the e+ternal segment$ and i' the tangent is 6)m long$ 'ind the length o' the se)ant. "ns√ &4+ &4+ cm. cm. 30. (o parallel )hords o' a )ir)le are 16)m and 30 )m long respe)tivel#$ and the distan)e bet(een them is 23)m long. ind the distan)e ea)h is 'rom the )enter$ and and radi&s o' the )ir)le. &+cm and /cm, r = &G cm 31. he diagonals o' a rhomb&s have the ratio o' 3:4. he area o' the rhomb&s is !6 s%.)m. ind a side o' the rhomb&s. "ns &0cm 32. he areas o' t(o similar triangles are to ea)h other as !:25. he perimeter o' the 'irst is 36 )m. hat is the perimeter o' the se)ond triangleT ;ns. .0cm. 33. he side o' a rhomb&s is 13)m$ and one o' its diagonals is 24)m. inds its area. &20 sqcm
34. 9n a )ir)le (hose radi&s is 10)m$ a )hord bise)ts the radi&s at right angles. ind the area π * * √ )sq )sq o' the smaller o' the t(o segments into (hi)h the )hord divides the )ir)le. 2+'%2 π cm
35. rom a point $ e+terior to )ir)le ?$ the tangents are dra(n to the )ir)le. he distan)e o' 'rom the )enter o' ? is 4)m. he length o' ea)h tangent is 2)m. ind the diameter o' the )ir)le. ind the area bo&nded b# the tangents and the ar) bet(een them. π ) sq cm "ns d = 4√ cm, " = %4 √ * 2 π
a 15 b 20 ) 25 d 40 37. he area o' a triangle (hose angles are ; = 6!.15!$ 8 = 34.246$ ? = 4.5!5 is 60.60 m2. he longest side is a 15.37 b 52.431 ) 37.53 d 64.!74 3. ind the area o' a de)agon that )an be ins)ribed ns)ribed in a )ir)le (ith radi&s 10 )m. a11.3745! b10.066446 )37.53527 d64.!74136 3!. ind the area o' a reg&lar 5-pointed star star that )an be ins)ribed in a )ir)le (ith radi&s 10 )m. a 112.257 b 56.12! ) 114.535 d 124.431 40. ind the radi&s o' the largest )ir)le that that )an be ins)ribed in the triangle (ith sides a = )m$ b = 15)m$ and )= 17)m ;a 3 b 1. ) 30 d .5 41. ind the radi&s o' the smallest )ir)le that )ir)&ms)ribe the triangle in the previo&s problem. a .5 b 10.2 ) !.51 d 7.75 42. ?hords ;8 = 12 )m and 8? = )m o' a )ir)le 'orms 120. ind the radi&s o' the )ir)le. a 10 b 12.164 ) 14.742 d 24.634 43. he radi&s o' the smallest )ir)le that )an be )ir)&ms)ribe a right triangle o' sides a$ b$ and ). a. a b ) b. a b , ) 2 2 ?. a , b ) d. ) 2 2 44. he radi&s o' the largest )ir)le that an be ins)ribed in a right triangle o' sides a$ b and ). a. a b ) b. a b , ) 2 2 ). a , b ) d. ) 2 2 ind ?P.
54. i+ lines are sit&ated in the plane so that no t(o are parallel and no three are )ongr&ent. Go( man# points o' interse)tion
45. ind the area o' the )#)li) %&adrilateral.
; = 50)m$ 50)m$ 8 = 2)m$ 2)m$ P = 56)m$ 56)m$ f = 30degrees 30degrees a 456 456 b 627 627 d 364 364 d52 d5255 46. he internal angle o' a pol#gon is 150 greater than its e+ternal angle$ ho( man# side has the pol#gonT a ! b 6 ) 7 d 47. he 30 angle o' a right triangle is bise)ted. ind the ratio o' (hi)h opposite side is divided. a 1:2 b 1:3 ) S2:2 d 1:4 4. ; )ertain angle has a s&pplement 5 times the )ompliment. ind the angle. a. 67.5 b. 5.5 ). 30 d. 27 4!. ind the s&pplement o' an angle (hose )ompliment is 62 degrees a. 30 b. 2 ).152 d. 11 50. he s&m o' the interior angles o' a pol#gon is 540 degrees. ind the n&mber o' the sides. a. 5 b. 6 ). d. 11 51. i+ lines are sit&ated in the plane so that no t(o are parallel and no three are )ongr&ent. Go( man# points o' interse)tion are thereT ;. 13 b. 14 ). 15 d. 16 52. he area o' a s%&are 'ield e+)eeds another s%&are b# 56 s%&are meters. he perimeter o' the larger 'ield e+)eeds X o' the smaller b# 26 m. hat are the sides o' ea)h 'ieldT ;ns. larger 'ield$ !m or 253m* smaller 'ield$ 5m or 113m 53 . he s&m o' the areas o' t(o &ne%&al s%&are lots is 5$200 s%&are meters. 9' the lots (ere adCa)ent to ea)h other$ the# (o&ld re%&ire 320 meters o' 'en)e to en)lose the )ombined area 'ormed b# them. ind the dimensions o' ea)h lot. ;ns.60m and 40m$ 40m$ or 6m and 24m. 24m.
a. 220
b. 120
). 240
d. 10
6. ; gro&p o' )hildren pla#ing (ith marble pla)e 50 pie)es o' the marble inside the
54. i+ lines are sit&ated in the plane so that no t(o are parallel and no three are )ongr&ent. Go( man# points o' interse)tion are thereT a. 13 b. 14 ). 15 d. 16 55. ; triangle is ins)ribed in a) ir)le s&)h that one side o' the triangle is a diameter o' the )ir)le. 9' one o' the a)&te angles o' the triangle has a meas&re o' 60 and the opposite side o' that angle has a length 12$ then the nearest val&e o' the radi&s o' the )ir)le is. a. 6.!3 b. 1.!3 ). !.6 d. 5. 56. he h#poten&se o' a right triangle is 34 )m. ind the length o' the t(o legs i' one leg is 14 )m longer than the other. a. 1 and 32 )m b. 15 and 2! )m ). 17 and 32 )m d. 16 and 30 )m 57. he s&m o' the interior angles o' a pol#gon is 540 degrees. ind the n&mber o' the sides. a. 5 b. 6 ). d. 11 5. ; )ir)le o' radi&s 6 has hal' its area removed b# )&tting o'' a border o' &ni'orm (idth. ind the (idth o' the border. a. 22 b. 135 ). 375 d. 176 5!. ; )ir)le is ins)ribed in a 3$ 4$ 5 right triangle. Go( long is the line segment Coining the point o' tangen)# o' the 3-side and the 5-sideT a. 1.2 b. 1.35 ). 1.46 d. 1.7! 60. (o 'ig&res having e%&al perimeter are said to a. )ongr&ent b. isoperimetri) ). e%&al d. similar 61. ; right triangle (hose length and side ma# be e+pressed as ratio o' integral &nits a. isos)eles triangle b. s)alene ). pedal triangle d. primitive triangle 62. he perpendi)&lar bise)tor o' the sides o' a triangle interse)t at the point Fno(n as the a. ortho)enter b. )i )i)&m)enter ). )entroid d. in)enter 63. he bise)tors o' the 3 angles o' a triangle meet at a )ommon )alled the a. )ir)&m)enter b. )entroid ). ortho)enter d. in)enter 64. ;n isos)eles trapeQoid ;8?P$ ;8=5$ 8?=;P = 3 and ?P = . ind the length o' the diagonals. a. 7.5 b. ). 7 d. 6 65. he diagonals o' rhomb&s have length 4 and 6 in)hes. ind the area o' the region inside the rhomb&s b&t o&tside the )ir)le that is ins)ribed in a rhomb&s. a. 3.3 b. 3.4 ). 2.! d. 2.5 66. (o e%&ilateral triangles$ ea)h (ith 12-)m sides$ overlap ea)h other to 'orm a 6-point tar o' Pavid. Petermine the overlapping area$ in s%-)m. a. 36.64 b. 41.57 ). 2.7 d. 4!. 67.
Ae)tang&lar arallelepiped , pol#hedron (hose si+ 'a)es are all re)tangles.
a. 220
b. 120
). 240
d. 10
6. ; gro&p o' )hildren pla#ing (ith marble pla)e 50 pie)es o' the marble inside the )#lindri)al )ontainer (ith (ater 'illed to a height o' 20 )m. 9' the diameter o' ae)h marble is 1.5 )m and that o' the )#lindri)al )ontainer 6 )m$ (hat (o&ld be the ne( height o' (ater inside the )#lindri)al )ontainer a'ter the marbles (ere pla)ed insideT a. 23.125 b. 24.125 ). 26.125 d. 25.125
Ae)tang&lar arallelepiped , pol#hedron (hose si+ 'a)es are all re)tangles. rism , pol#hedron o' (hi)h t(o 'a)es are e%&al pol#gon in parallel planes and the other 'a)es are e%&al parallelogram. #ramid , a pol#hedron o' (hi)h one 'a)es is a reg&lar pol#gon and other 'a)es are triangles (hi)h have a )ommon verte+.
6!. ; horiQontal )#lindri)al tanF (ith hemispheri)al ends is to be 'illed (ith (ater to a height o' 762mm. 9' the total inside length o' the )#linder is 3$600mm$ 'ind the vol&me o' (ater in )&bi) meters that (ill have to 'illed into the tanF &p to the re%&ired height. 9' one edge o' a )&be and the vol&me o' the )&be respe)tivel# in s%.)m. and )&.)m. a. 64 and 172 b. 64 and 172 ). 64 and 172 d. 64 and 172!
Aeg&lar rism , prism (hose lateral edges are perpendi)&lar to its bases.
70. ; p#ramid (ith s%&are base has an altit&de altit&de o' 25 )m. 9' the edge o' the base is 15)m$ 15)m$ )al)&late 'or the vol&me o' the p#ramid. a. 15 b. 175 ). 175 d. 1!5
e)tion , pol#hedron is the plane 'ig&re 'ormed b# a plane passing thro&gh the solid.
71. 9' a right )ir)&lar )one has a base radi&s o' 35 )m and an altit&de o' 45)m$ solve 'or the total s&r'a)e area in s%&are )ms. a.10116 b. 10117 ). 11117 d. 12117
r&st&m o' #ramid , se)tion o' the p#ramid bet(een the base and a se)tion parallel to the base.
72. (o )ir)les o' radi&s 4 and 6.!3 are pla)ed in a plane so that their )ir)&m'eren)e interse)t at right angles. ind the area o' the interlapping region. a. 14.1 b. 13.!2 ). 14.1 d. 15.1
Pihedral ;ngle , the divergen)e o' t(o interse)ting planes.
Aeg&lar #ramid , p#ramid (hose base is a reg&lar pol#gon and (hose altit&de passes thro&gh the )enter o' base. lant height , altit&de o' lateral 'a)es.
?onve+ ol#gon , a pol#gon (hose ea)h interior angle o' (hi)h is less than 1 0.
?one , a solid bo&nded b# a )oni) s&r'a)e and a plane interse)ting all the elements.
73. ne side o' a reg&lar o)tagon is 2. ind the area o' the region inside the o)tagon. a. 3.!1 b. 13.!2 ). 14.1 d. 15.1
phere , a solid bo&nded b# a s&r'a)e all points o' (hi)h are e%&idistant 'rom a point )alled )enter. "reat )ir)le , the interse)tion and a plane passing thro&gh the )enter.
-LID 0E+ETR
mall )ir)le , the interse)tion o' a sphere and a plane not passing thro&gh the )enter.
Definition of Terms:
/&adrant , one-'o&rth o' a great )ir)le.
ol#hedron , a solid bo&nded b# planes. Aeg&lar ol#hedron , pol#hedron (hose 'a)es are )ongr&ent reg&lar pol#gons and (hose pol#hedral angles are reg&lar pol#gons and (hose pol#hedral angles are e%&al. here are onl# 'ive reg&lar pol#hedron: etrahedron , 4 'a)es Ge+ahedron , 6 'a)es )tahedron , 'a)es Pode)ahedron , 12 'a)es 9)osahedron , 20 'a)es ?&be , a pol#hedron (hose si+ 'a)es are all s%&ares
pheri)al (edge , portion bo&nded b# a l&ne and the planes o' t(o great )ir)les. o)&s , a solid 'ormed b# revolving a )ir)le abo&t a line not interse)ting it.
one , portion o' o' sphere bo&nded b# a spheri)al pol#gon and and the plane o' its sides. <&ne , portion o' a sphere l#ing bet(een t(o semi , )ir)les o' great )ir)le. pheri)al p#ramid , portion o' sphere bo&nded b# a spheri)al pol#gon and the plane o' its sides. pheri)al e)tor , portion o' sphere generated b# the revol&tion o' )ir)&lar se)tor abo&t an# diameter o' the )ir)le o' (hi)h the se)tor is apart. pheri)al segment - portion o' sphere in)l&ded bet(een bet(een t(o parallel lines.
rismoidal orm&la: \ =
1 6
h( b + 8 + 4M)
pheri)al (edge , portion bo&nded b# a l&ne and the planes o' t(o great )ir)les. o)&s , a solid 'ormed b# revolving a )ir)le abo&t a line not interse)ting it.
1
rismoidal orm&la: \ =
6
h( b + 8 + 4M)
1orm#las in -olid +ens#ration
6. phere o' radi&s r or diameter P:
1. ?&be (ith edge s: \ol&me : \
=
\ol&me : \
s3
&r'a) &r'a)ee ;rea ;rea : ; = 6s2
= 4 πr 3 or 3
= 1 πP 3
\
6
&r'a) &r'a)ee ;rea ;rea : 5 = 4πr 2 or 5 = πP 2 7. Aight )ir)&lar )#linder (ith radi&s r and altit&de h:
2. Ae)tang&lar parallelepiped (ith edges a$ b$ ) and diagonal P: \ol&me : \ = ab)
\ol&me : \
= πr 2h
&r'a) &r'a)ee ;rea ;rea : ; = 2( ab + a) + b) ) Piagon Piagonal al : P = a 2 + b 2 + ) 2
. Aight )ir)&lar )one (ith radi&s r and altit&de h:
3. \ol&me o' a prism (ith base 8 and altit&de h:
\
=
\ol&me : \
= 1 πr 2h 3
8h
= πrl
l = slant slant height
4. \ol&me o' a p#ramid (ith base 8 and altit&de h: \ =
!. r&st&m o' a right )ir)&lar )one (ith base radii r and A and altit&de h: slant height l:
1 8h 3
\ol&me : \ =
5. \ol&me o' a prismatoid (ith bases b and 8$ midse)tion M and altit&de h:
1 3
h(r 2
π
+
A2
+
rA)
\ol&me : \ =
1 3
(
h b + 8 + b8
π
)
\ol&me : \ =
1 dA or \ 6
2 3
= πA 2h
\ol&me : \ =
1 3
(
h b + 8 + b8
π
=
)
\ol&me : \ =
1 6
dA or \
2
= πA 2h 3
(here \ = vol&me o' spheri)al se)tor
1 ( p + 6 )l 2
l = slant height
= area o' the the Qone (hi)h )h 'orms the base o' o' the se)tor
p = perimeter o' base b
A = radi&s o' the sphere
= perimeter o' base 8
h = altit&de o' the sphere
11. ;rea o' a Qone (ith altit&de h on a sphere o' radi&s A: 15. llipsiod d
\ = 43 π ab)
= 2πAh
b
or oblate spheroid \ = 43 π ab2 12. pheri)al segment o' one base and altit&de h on a sphere o' radi&s A: \ol&me : \ = otal ;rea
1 2 πh ( 3A − h) 3
:
)
or prolate pheriod \ = 43 π a2 b
16araboloid
h( 4A − h)
h
\ = X π a2 h
=π
a
a 13. pheri)al segment o' t(o bases (ith radii a and b and altit&de h on a sphere 17. Ong&la \ = 23 r 2h = 2 rh
o' radi&s A: \ol&me : otal ;r ;rea
\
1 =
6
(3a
πh
2
+
3b
2
+
: = π( 2Ah + a 2
h
2
h
) r
+
b2
14. pheri)al ?one = a spheri)al se)tor having onl# one )oni)al s&r'a)e -#pplementar8 Pro&lems:
1.
he slant height o' a reg&lar he+agonal p#ramid is ! )m. and a side o' the base is 6 )m. ind the vol&me o' the ins)ribe )one. ;ns. 27 √6π )&. )m.
13. ind the vol&me o' a prism having an altit&de o' 13 )m and a re)tang&lar base o' )m long and 4 )m (ide. "ns4&. cm 14. he base o' a right prism is a rhomb&s rhomb&s (hose sides are ea)h 10 )m long and (hose
1.
he slant height o' a reg&lar he+agonal p#ramid is ! )m. and a side o' the base is 6 )m. ind the vol&me o' the ins)ribe )one. ;ns. 27 √6π )&. )m.
2.
; re)tangle$ 4 )m (ide and ! )m long is revolved abo&t its longer side. ind the total s&r'a)e area and the vol&me o' the )#linder generated. ;ns. = 104π s%. )m. * \ = 144 π )&. )m
3. ; solid (ooden )one )one is )&t into t(o t(o parts$ the )&t being parallel to the the base and hal'(a# hal'(a# bet(een the base and the verte+. ind the ratio o' the (eights o' the t(o parts. ;ns. 7 : 1 4.
5.
6.
ind the area o' a Qone o' one base$ i' the base is a )ir)le o' radi&s 6 )m and is )m 'rom the )enter o' the sphere. ;ns 40 π s%. )m. hree spheres o' radii a$ 2a$ and 3a are melted and 'ormed into a ne( sphere. ind the s&r'a)e o' this sphere. ;ns. 24 3√6 πa2 ; sphere (hose radi&s is 10 )m is )&t into three parts b# parallel planes )m and 6 )m. 'rom the )enter respe)tivel#$ and on the opposite sides o' the )enter. ind the total s&r'a)e o' the part in the middle. ;ns. 30 π s%. )m.
7. 9n a 'r&st&m 'r&st&m o' a reg&lar p#ramid the bases are are s%&ares (ith sides 6 )m and 12 )m respe)tivel#. 9' the lateral area o' the 'r&st&m is C&st hal' o' its total area$ (hat is the vol&meT ;ns: 420 m3 (.
he vol&me o' t(o similar pol#hedron are 64 and 125 )& )m. respe)tivel#. 9' the total s&r'a)e o' the 'irst is 112 s%&are )entimeters$ 'ind the total s&r'a)e area o' these)ond.;ns.30πs%.)m
13. ind the vol&me o' a prism having an altit&de o' 13 )m and a re)tang&lar base o' )m long and 4 )m (ide. "ns4&. cm 14. he base o' a right prism is a rhomb&s rhomb&s (hose sides are ea)h 10 )m long and (hose shorter diagonal is 12 )m. ind its vol&me i' the altit&de is )m. ;ns. G./ cu cm 15. he diameter o' t(o spheres are in the ratio 2:3 and the s&m o' their vol&mes is 1$260 )& m. ind the vol&me o' the larger sphere. ;ns. G2 cu m 9' the vol&me o' a )&be is 625 m 3$ 'ind the length o' the diagonal. ;ns. &4/ m 16. ind the vol&me o' a reg&lar triang&lar p#ramid (hose slant height is 17 )m and (hose "ns .0 3 cu cm altit&de is 15 )m. "ns 17. ind the )apa)it# in liters o' a pail in the 'orm o' a 'r&st&m o' a )ir)&lar )one i' the radii o' the bases are 10 and 15 )m and the depth o' the pail is 36 )m. "ns &G 9 1. he spa)e o))&pied b# the (ater in a reservoir is the 'r&st&m 'r&st&m o' a right )ir)&lar )one. a)h a+ial se)tion o' this 'r&st&m has an area o' 2 s% m and the diameter o' the &pper and lo(er bases are in the ratio 4:3. 9' the reservoir )ontains 14 Π3 )& m$ 'ind the depth o' the (ater in the reservoir. ;ns. 4m 1!. he lateral s&r'a)e area o' a right )ir)&lar )#linder is 330 Π s% )m. 9' its altit&de altit&de is 20 )m$ 'ind the diameter o' its base. ;ns. &.+ cm 20. ; reservoir is in the 'orm o' the 'r&st&m o' an inverted s%&are p#ramid (ith &pper base 24 m$ lo(er base edge 1 m and altit&de ! m. Go( man# ho&rs (ill it re%&ire 'or an inlet pipe to 'ill the reservoir i' (ater 'lo(s in at the rate o' 5$000 liters per min&teT ;ns. &2 !r 21. ; horiQontal )#lindri)al tanF (ith diameter o' 0.60 m and 3.66 m long is 'illed (ith (ater to a depth o' 0.46 m. ind the n&mber o' liters o' (ater in the tanF. "ns /+& 9
!. ind the total s&r'a)e and the vol&me vol&me o' a tetrahedron (hose (hose edges are e%&al to a. ;ns: = √3 a2 $\ = √212 a3
22. ; sphere is ins)ribed in aright )ir)&lar )one (ith altit&de 15 )m. 9' the slant height o' the )one is e%&al to the diameter o' its base$ 'ind the s&r'a)e area o' the sphere. "ns &00 sq cm
10. ; sphere o' radi&s 5 )m and a right )ir)&lar )one o' bass radi&s radi&s 5 )m and height 10 )m stand on a plane. ind the position o' a plane that )&ts the t(o solids in e%&al )ir)&lar se)tions. ;ns: d = 2 )m and h = 10 )m
23. Ose the prismoidal 'orm&la to 'ind the vol&me o' the )ommon part o' t(o )#linders$ ea)h (ith a radi&s o' 6 )m$ (hi)h interse)t at right angles. "ns &,&+2 cu cm
11. ; )#lindri)al tin )an has its height e%&al to the the diameter o' its base. ;nother ;nother )#lindri)al tin )an (ith the same )apa)it# has its height e%&al to t(i)e the diameter o' its base. ind the ratio o' the amo&nt o' tin re%&ired 'or maFing the t(o )ans (ith )overs. ;ns. 0+24 12. ind the vol&me o' the 'r&st&m o' a reg&lar s%&are p#ramid (hose base edges are 4 )m and 10 )m and (hose slant height is 5 )m. ;ns. 20/ cu cm
a. 17 4. val&ate the limit
b. 0 lim
). in'init# + , sin + 3
d. indeterminate
24. 9' the radi&s o' a sphere is ins)ribed b# 6 )m$ its vol&me is m<iplied b# 27. ind the radi&s o' the sphere. "ns cm
1. ind the slope o' + 2# = at point 2$ 2 a . ,2 b. 2 ). 2. 9' # = + ln+$ 'ind # a. 1 1 + b. ln + ). 1ln+ 3. val&ate the limit imit + - 4+2 , + , 12 as + approa)hes approa)hes 4
19. 20.
d. 4 d. +
ind the area bo&nded b# the )&rves # = 4+ , +2$ # = 0$ + = 1$ + = 3. a. 233 b. 223 ). 214 253 ind the vol&me generated b# revolving the region bo&nded b# # = + 2 and # = +
a. 17 4. val&ate the limit
b. 0
d. indeterminate
+ , sin + +3 a. 13 b. ). 15 d. 16 5. ind the ma+im&m point on the )&rve # = + 3 , 3+2 , !+ 5 a. 1$ 15 b. 3$ -22 ). -1$ 10 d. -3$ 21 6. Petermine the velo)it# t# o' a bod# (hi)h moves a))ording to the the la( = 2t 3 , t2 4 (here is displa)ement in 't. and t is time in se) at t = 1se). a. 2't b. 6't. ). 4't. d. 10't. 7. ind the e%&ation o' the tangent to the )&rve # = + 4 , +2 2 at + = -1. a. 2+ # = 0 b. 2+ , # = 1 ). 2+ , # = 0 d. # + = 1 . Petermine the altit&de o' the largest )ir)&lar )#linder that )an be ins)ribed in a right )ir)&lar )one o' radi&s 6 in)hes and o' height 15 in)hes. a. 12 in)hes b. in)hes ). 5 in)hes d. 10 in)hes 9. "iven a )one o' radi&s A and o' altit&de G$ (hat per)ent is the vol&me o' the largest )#linder (hi)h )an be ins)ribed in the )one to the vol&me o' the )oneT a. 4![ b. 44[ ). 50[ d. 60[ 10. ind the most e)onomi)al proportion 'or a bo+ (ith an open top and a s%&are base. a. b = h b. b = 4h ) . b = 3h d. b = 2h 11. he 5m pi)t&re h&ng on a (all so that its bottom edge is 4m above an observer@s e#e. Go( 'ar sho&ld the observer stand 'rom the (all so that the angle s&btended b# the pi)t&re at the e#e is a ma+im&m. a. 4.! b. 5.5 ). 6.7 d. 7.2 12. hat is the largest area o' a re)tangle that )an be ins)ribed (ithin an ellipse (hose maCor diameter is 10't and (hose minor diameter is 6't. a.32.50 s%. 't. b. 2.00 s%. 't ). 30.00 s%. 't d. 34.00 s%.'t 13. he hands o' the to(er )lo)F are 4 X 't and 6 't long respe)tivel#. Go( 'ast are the ends approa)hing at 4 o@)lo)F in 't per min&teT a. , 0.246 b. , 0.203 ). ,0.264 d. ,0.256 14. ; man on the (har' (har' pier is p&lling a rope tied to a ra't at a time rate rate o' 0.60 mse) i' the hands o' the man p&lling the rope is 3.66m above the level o' the (ater$ ho( 'ast is the ra't approa)hing the (har' (hen there are 6.10m o' rope o&tT a. ,0.7 ,0.755 mse) se) b. ,0.5 ,0.555 mse mse)) ). ,0.45 0.45 mse) se) d. ,0. ,0.65 mse) se) 15. ; balloon is rising verti)all# over a point point ; on the gro&nd at the rate rate 15 'tse). ; point 8 on the gro&nd level is (ith the same horiQontal plane as ; and 30 't 'rom it$ (hen the balloon is 40 't 'rom ;$ at (hat rate is its distan)e 'rom 8 )hangingT a. 12 'tse) b. 15 'tse) ). 1 'tse) d. 21 'tse) 16. hat is the per)entage error made in the )omp&ted s&r'a)e area o' a sphere i' the error made in meas&ring the radi&s is 3[ a. 3 [ b. 4 [ ). 5[ d. 6 [ 1. hat is the allo(able error in meas&ring the edge o' the )&be that is intended to hold )&.m. i' the error o' the )omp&ted vol&me is not to e+)eed 0.03 )&.m.T a. 0.002 b. 0.003 ). 0.0025 d. 0.001 1(. ind the radi&s o' )&rvat&re o' the )&rve # = + 33 at + = 1. a. 2 b. 2 3 ). 3 2 d. 2 2
34.
lim
). in'init# +
19. 20.
0
ater at 100°? is trans'erred to a room (hi)h is at )onstant temperat&re o' 60 °?. ;'ter 3 min&tes min&tes the (ater temperat&re temperat&re is !0°?$ 'ind the (ater temperat&re a'ter 6 min&tes.
21. 22.
23.
24.
25. 26.
27. 2. 2!. 30. 31. 32.
33.
52.
ind the area bo&nded b# the )&rves # = 4+ , +2$ # = 0$ + = 1$ + = 3. a. 233 b. 223 ). 214 253 ind the vol&me generated b# revolving the region bo&nded b# # = + 2 and # = + abo&t the #= a+is. a. c b. c10 ). c6 d. c12 9' the 'irst derivative o' a '&n)tion is )onstant$ then the '&n)tion is said to be: a. )onstant b. linear ). sin&soid d. e+ponential hree sides o' the trapeQoid are ea)h )m long. Go( long is the 4 th side (hen the area o' the trapeQoid has the greatest val&eT a . 16 ) m b . 12 ) m ) . 15 ) m d. 10 )m ater is r&nning o&t 'rom a )anoni)al '&nnel at a rate o' 2 )&. in. per se)ond. 9' the radi&s o' the top o' the '&nnel is 4 in)hes and the altit&de is in)hes$ 'ind the rate at (hi)h the (ater level is dropping (hen it is 2 in)hes 'rom the top. a. 2!π in i nse) b. -2!π inse)). inse)). -32π inse) inse) d. 5! 5!π inse) ind the area bo&nded b# the parabola +2 = # and its lat&s re)t&m. a. 10.67 10.67 s%. &nits b. 32 s%. &nits &nits ). 4 s%. &nits &nits d. 16.67 16.67 s%. &nits hat theorem is &sed to solve 'or )entroidT a. app app&s &s b. \ari \arign gnon on@s @s ). ?ast ?astig igllllia iano no@s @s d. as) as)al al@s @s ind the area in s%. &nits bo&nded b# the parabola + 2 , 2# = 0 and + 2 2# = . a. 11.7 b. 4.7 ). !.7 d. 10.7 ind the e%&ation o' the )&rve )&rve at ever# point o' (hi)h the target line ne has a slope o' 2+. a. + =-#2 ? b. # = -+ 2 ? ). # = + 2 ? d. + = # 2 ? he rate o' )hange o' a '&n)tion o' # (ith respe)t to + e%&als 2 , #$ and # = (hen + = 0. ind # (hen + = ln 2 b. 5 b. 2 ). ,5 d . ,2 olve the di''erential e%&ation +2 #2d+ 2+#d# = 0 2 3 2 ). 3+# + = ? b. 2+# = ? ). 3+# 2 = ? d. 3+2 2# = ? olve the di''erential di''erential e%&ation # , 5#@ 6# = 0 d. R = ;e 2+ 8e3+ b. R = ;e- 2+ 8e- 3+ ). R = ;e 5+ 8e3+ d. R = ;e 2+ 8e5+ ind the e%&ation ' the 'amil# 'amil# o' orthogonal traCe)tories traCe)tories o' the s#stem o' parabolas #2 = 2+ ?. e. # = )e-+ b. # = )e2+ ). # = )e + d. # = )e-2+ he rate o' )hange o' a )ertain s&bstan)e s&bstan)e is proportional to the amo&nt o' s&bstan)e is 10 grams at the start and 5 grams at the end o' 2 min&tes$ 'ind the amo&nt o' s&bstan)e remaining at the end o' 6 min&tes. '. 1.25 grams b. 2.67 grams ). 3.46 grams d. 2.! grams he rate o' pop&lation gro(th gro(th o' a )o&ntr# is proportional to the n&mber n&mber o' inhabitants. 9' the pop&lation o' a )o&ntr# no( is 40 million and is e+pe)ted to do&ble in 25 #ears$ in ho( man# #ears (ill the pop&lation be 3 times the presentT g. 3!.6 #rs b. 3!.5 #rs ). 37.! #rs d. 36.! #rs
a. 1 , i b. 1 i ). -41 i d. 41 i ind the e%&ation o' the 'amil# o' orthogonal traCe)tories o' the s#stem o' the parabolas #2 = 2+ ?. 2+
2+
ater at 100°? is trans'erred to a room (hi)h is at )onstant temperat&re o' 60 °?. ;'ter 3 min&tes min&tes the (ater temperat&re temperat&re is !0°?$ 'ind the (ater temperat&re a'ter 6 min&tes. h. 2.5°? b. 5.2°? ). 0°? d. 75°? 35. ; bod# at !0°? )ools in 10 mins to 70 °? in a room temperat&re o' 25 °?. hen (ill its temperat&re be 40°?T i. 3!.mins b. 3.mins ). 36.mins d. 34.7mins 36. val&ate val&ate ln 3 3 C4 C. 1.77 C0.43 b. 1.61 C0.!27 C0.!27 ). 1.!5 C0.112 0.112 d. 1.46 C0.102 C0.102 37. ind the the val&e val&e o' 1 i12 (here i is an imaginar# n&mber. F . ,64 b. 6 4 ). 4 d. 4i 3. impli'# impli'# the e+pression e+pression i1!!7 i1!!! l. 0 b. ,1 ). 1 i d. 1- i 0.56 3!. +pres +presss e0.32 j 0.56 in re)tang&lar 'orm m. 1.16 1.1677 j 0.732 b. 1.1!3 j 1.163 .163 ). 1.452 .452 , j 0.315 d. 1. 1.64 , j 1.462 1.462 40. val&ate val&ate )os 0.4!2 0.4!2 j 0.!42 0.!42 n. ,1.032 C0.541 b. 1. 1.302-C0.504 ). 3. 3.12C1.54 d. 1. 1.4C0.01 41. val&ate the the val&e o' log -5 log e b. 5πC log e ). log 5C π lo log e d. log 5πC log e o. 5Cπ lo 42. ind the
The P8thagorean Theorem -
states that the s&m o' the s%&ares o' the legs is e%&al to the s%&are o' the
52.
53. 54. 55.
a. 1 , i b. 1 i ). -41 i d. 41 i ind the e%&ation o' the 'amil# o' orthogonal traCe)tories o' the s#stem o' the parabolas #2 = 2+ ?. a. # = ?e-+ b. # = ?e 2+ ). # = ?e+ d. # = ?e 2+ hat is the %&otient (hen 4 i is divided b# i 3. a. -4i b. 4i ). -4i d. --4i val&ate the e+pression 1i210$ (here i is an imaginar# n&mber. a. - 1 b. 10 ). 0 d. 15 olve 'or + in +#i 24i = 14 i. a. 3 b. 4 ) . 14 d.
TRI0!+ETR Trigonometr8 , the bran)h o' mathemati)s that deals (ith the sol&tion o' triangles. Angle , the spa)e bet(een t(o line meeting at a point )alled verte+.
=inds of Angles: 1. "cute angle , an angle (hi)h meas&res bet(een 0° to !0 ° 2.
;ig!t angle , an angle meas&ring e+a)tl# !0 °
3.
btuse angle , an angle (hi)h meas&res bet(een !0° to 10°
4.
5traig!t angle , an angle meas&ring e+a)tl# 10°
5.
;eflex angle , an angle greater than 10 ° b&t less than 360 °
Two 0eneral Classes of Triangles
1. ;ig!t riangle , a triangle (ith a right angle 2. blique riangle , a triangle (itho&t a right angle &li"#e Triangles can &e f#rther classified as:
1. "cute riangle , a triangle (hose all angles are a)&te. 2. btuse riangle , a triangle (ith one obt&se angle.
4. ?s); ?s);=1 =1 in in; ;
4. ?s); ?s);=1 =1 in in; ;
The P8thagorean Theorem
states that the s&m o' the s%&ares o' the legs is e%&al to the s%&are o' the
-
5. \ersed \ersed in;= in;=1-? 1-?os; os;
h#poten&se. P8thagorean Triple , three positive integers satis'#ing the #thagorean prin)iple.
+ample: 3$ 4$ and 5* 5$ 12$ and 13* 20$ 21 and 2!* $ 15$ 17* 7$ 24$ 25* et).
6. ?overs ?oversed ed in; in;=1=1-in in; ; 7. +se)a +se)ant; nt;= =e); e);-1 -1
-#pplementar8 Pro&lems:
1. ; storm broFe a tree tree 50 't high so that that its top to&)hed to&)hed the gro&nd 30 't 'rom the 'oot o' the tree. hat is the height o' the part standingT ;ns. 16't.
III2 P8thagorean Identities
1. ?os2 ;in2 ;=1
2. 3. Go( 'ar 'rom the the )enter )enter o' a )ir)le is its )hord in)hes in)hes long i' its radi&s radi&s is 5 in)hes. ;ns. ;ns.
2. 1an2 ;=e)2 ;
3 4. or (hat positive tive val&e o' + (ill the 'ollo(ing lengths be sides o' a right triangle triangle 2+ 1$ 1$ 5+
3. ?ot2 ;1=?s)2 ;
, 1$ + , 3T he he last being the the longest ;ns. > = 1 I72 -#m and Difference of Angles Identities
1. in;8= in;8=in; in;?os8 ?os8?os; ?os;in8 in8
9. ?ir)&lar &n)tions 1.in;=#r
4. ?ot;=+# #(x,y)
2. in;-8= in;-8=in; in;?os8?os8-?os; ?os;in8 in8
2.?os;=+r
5.e);=r+
3. ?os;8= ?os;8=?os;? ?os;?os8- os8-in; in;in8 in8
3.an;=#+
6.?s);=r#
4. ?os;-8= ?os;-8=?os;? ?os;?os8 os8in; in;in8 in8 5.
!a n( A + " ) =
6.
!an(A − " ) =
II2 The Relation Among 1#nctions
1. an; an;= =in in; ;?o ?os; s;
!anA !anA + !an" !an" 1 − !anA!an" !anA !anA − !an" !an" 1 + !anA!an"
2. ?ot;= ?ot;=?os ?os; ;in in;= ;=1 1an; an; 72 Do#&le Angle Angle Identities
3. e); e);=1 =1? ?os os; ;
1. in2; in2; = 2in;?os; n ;?os;
2. ?os2 ?os2; ; = ?os ?os2 ;-in2 ;
2. 2?os;in8 2?os;in8=i =in; n;8-i 8-in;-8 n;-8
2
= 2)os ;-1 2
3. 2?os;?os8= 2?os;?os8=?os; ?os;8 8?os; ?os;-8 -8
2. ?os2 ?os2; ; = ?os ?os2 ;-in2 ;
2. 2?os;in8 2?os;in8=i =in; n;8-i 8-in;-8 n;-8
2
3.
= 2)os ;-1
3. 2?os;?os8= 2?os;?os8=?os; ?os;8 8?os; ?os;-8 -8
= 1-2in2 ;
4. 2in;in8 2in;in8=?os =?os;-8;-8-?os; ?os;8 8
!an !an2 A
=
2 !anA 2
1 − !an A
7I2 Complementar8 Complementar8 Angle Identities
in; = ?os!0-;
I<2 -#m and Difference of -ine and Cosine Identities
1. in;in in;in8=2 8=2in in X ;8? ;8?os os X ;-8 ;-8 2. in;-in8 in;-in8=2?o =2?oss X ;8i ;8inn X ;-8 ;-8 3. ?os;?os8= ?os;?os8=2?os 2?os X ;8?os ;8?os X ;-8 ;-8
?os; = in !0-;
4. ?os;-?os8= ?os;-?os8=2in 2in X ;8 ;8in in X ;-8 ;-8
an; = ?ot!0-;
<2 -ine Law
?ot; = an!0-;
1. e); = ?s)!0-; ?s); = e)!0-;
=
c sinc
1. a2 = b2)2-2b)?os;
1 + cosA cosA 2
3. !an A / 2 =
b sinb
1 − cosA osA 2
2. %os2 A / 2 =
2
=
2. ;8? = 10
7II2 )alf Angle Identities
1. $in2 A / 2 =
a sina
2. b2 = a2)2-a)?os8
1 − cosA cosA 1 + cosA cosA
3. )2 = a2b2-2ab?os? 4. ;8 ;8? ? = 10 10
7III2 Prod#ct of -ine and Cosine Indentities
1. 2in;?os8 2in;?os8=i =in; n;8 8in;in;-8 8
π ' b 1.
a −b b
=
!an1/ !an1/ 2(a − b) !an1/ !an1/ 2(a
b)
, 'or tangent and )otangent , distan)e o' one )omplete (ave o' the '&n)tion
π ' b 1.
a −b a +b
=
, 'or tangent and )otangent , distan)e o' one )omplete (ave o' the '&n)tion
!an1/ !an1/ 2(a − b) !an1/ !an1/ 2(a + b)
3)
1!ase 5!ift ' b c
, propert# tr&e 'or an# '&n)tions ,
2.
a −c a +c
=
distan)e the graph is shi'ted to the right or to the le't 'rom its
!an1/ !an1/ 2(a − c)
standard position
!an1/ !an1/ 2(a + c)
, i' ) K 0$ the the graph is shi'ted to the right , i' ) V 0$ the the graph is shi'ted to the le't
3.
b −c b +c
=
4 @ertical translation d
!an1/ !an1/ 2(b 2(b − c) !an1/ !an1/ 2(b 2(b + c)
, tr&e 'or all '&n)tions , de'ined as the distan)e the graph is shi'ted &p(ard or do(n(ard
4. ;8 ;8? ? = 10 10
, i' d V 0$ the graph is shi'ted &p(ard , i' d K 0$ the graph is shi'ted do(n(ard
5. Properties of 0raph of Circ#lar 1#nctions
-ample Pro&lem:
Petermine the ma+im&m ma+im&m val&e and the period o' the '&n)tion '&n)tion '+ = ,3 sinπ4 + 5 - 7
?onsider the '&n)tion
ol&tion:
a = 3
# = a sin b+ ) d
d = ,7 he graph has an amplit&de o' 3$ b&t it is shi'ted do(n(ard b# 7 &nits. here'ore the
a
ma+im&m val&e o' the '&n)tion is 3 , 7 = ,4. eriod = 2ππ4 *
d
)
b
2π
b
Period ' H
amplitude a , highest val&e o' the '&n)tion on in standard 'orm$ a propert# is tr&e
'or sine$ )osine '&n)tions onl# 2)
1eriod 2 π π ' , 'or sine$ )osine$ se)ant and )ose)ant '&n)tions. '&n)tions. b
3. ind the the val&e val&e o' ?os ; 8 8 i' tan ; = ] and )s) 8 = 1315$ both angles are a)&te. 4.
9' tan + tan # = √3 and )s) # = √2$ + and # are a)&te$ 'ind +.
Answer
-#pplementar8 Pro&lems
1. 9' tan # = 13 and tan m = X$ # and m being a)&te$ 'ind # m. 2.
1)
sin)e b = π4
9' tan ; = ] and in 8 = 1213$ ; being greater than 10° and 8 is obt&se $ 'ind a in ; 8 8 b ?os ?os ; ; 8 ) an an ; 8
Ans; # 0 # ( A# 0 !#
3. ind the the val&e val&e o' ?os ; 8 8 i' tan ; = ] and )s) 8 = 1315$ both angles are a)&te. 4.
Ans; # 0 # ( A# 0 !#
9' tan + tan # = √3 and )s) # = √2$ + and # are a)&te$ 'ind +. & %8:8 ?oard ?oard 8xam 8xam "pril "pril ., &&) &&)
1 , tan + tan #K
impli'# )os ; )os8
5. an + = 3$ 3$ + is is a)&te a)&te.. ind ind
in ; - sin 8
a sin 2+ b)
sin ; sin 8
"ns3 0
)os ; - )os 8
tan + 45 ° 20 %8:8 ?oard ?oard 8xam 8xam "pril "pril ., &&) &&)
6. 9' tan tan + # = 2 and ttan an + = 1$ 'ind ttan an #. 3
12
7. "iven the a)&te a)&te an angles gles ; and 8$ sin ; = 5 and sin 8 = 13$ 'ind the val&e o' sin ; 8 )os ; 8.
2sin 8 )os 8 , )os 8 1 , sin 8 sin2 8 ,)os2 8
"ns3 2 sin ? – & & – sin ?
2& %8:8 ?oard ?oard 8xam 8xam "pril "pril ., &&) &&)
. ind the the val&e val&e o' tan ; 28 28 i' )ot ; = tan 8 = 2. 2.
impli'#
!. ind ind sin sin 2; i' tan tan ; = -512 and )os ; is negative. +
impli'#
sin 2 150 sin 2 750
"ns3&
+
10. val&ate val&ate sin +)ot +)ot 2 tan 2. 11.
"iven ;$ an angle bet(een 0 ° and 360° s&)h that sin ; = - 45 and (hose tangent is positive$ )onstr&)t ; and 'ind the val&e o' sin 2; , )ot X ;.
12.
22 %8:8 ?oard ?oard 8xam 8xam "pril "pril ., &&) &&)
impli'#
13. 5sin 2+ , 25)os 25)os + = 10 , 4sin 4sin + 14. )ot 2+ , tan + = 1
2 %8:8 ?oard ?oard 8xam 8xam "pril "pril &., &&)
impli'#: sin 20o sin 21o sin 22o . . . sin 2!0o
"ns3 4++
Angles of Eleation and Depression
15. sin 2+ sin + = 0
Angle of eleation , is the angle made (ith the horiontal b# the line o' sight 'rom an
&. %8:8 %8:8 ?oard ?oard 8xam 8xam 7o# 7o# 4, 4, &+) &+)
1 , )os + ,,,,,,,, sin +
"ns3 &
?os 00 )os 10 )os 20 . . . )os !0 0
ind the val&e o' )os 2+ , sin !0 ° + i' tan + = -34 and sin + is positive. ° satisfying ind all #alues of x less t!an .0 ° satisfying t!e equations belo(
sin 00 sin si n 10 sin si n 20 . . . sin !0 0
observer to an obCe)t on the higher level than the observer.
sin +
,,,,,,,,
Ans; #csc"
1 , )os +
&G 8:8 ?oar ?oard d 8xam 8xam "pril "pril , & &
olved 'or θ in the e%&ation: sin 2θ = )os θ
bCe)t
;ngle o' elevation elevation Ans; %) ) and -') )
&/ 8:8 ?oar ?oard d 8xam 8xam "pril "pril , & &
bserver
GoriQontal
"iven the relations = ;sin θ 8)os θ and / = ;)os θ - 8sin θ$ derive another e%&ation sho(ing the relationship bet(een $ /$ ; and 8 not involving an# o' the trigonometri) '&n)tions o' angle θ
Angle of depression 3 the angle made (ith the horiontal b# the line o' sight 'rom an
observer to an obCe)t o' lo(er level that the observer.
angle o' elevation o' the top o' the lightho&se is 24. ind the height o' the lightho&se. ;ns. !5.11't
Angle of depression 3 the angle made (ith the horiontal b# the line o' sight 'rom an
angle o' elevation o' the top o' the lightho&se is 24. ind the height o' the lightho&se.
observer to an obCe)t o' lo(er level that the observer. GoriQontal
;ns. !5.11't 6. 9' in right triangle eight times the prod&)t o' the the legs e%&als the s%&are o' the h#poten&se$
bserver
'ind the angles o' the triangle. "ns M"rcan %4 C 2 √ )N, 0O √&+)N, &+)N, M"rcan % 2 ' 4 C 2 √ √ &+ &+
;ngle o' depression
7. (o ladders$ ladders$ one o' (hi)h (hi)h is t(i)e t(i)e as long as the the other$ other$ rest on the 'loor 'loor and rea)h rea)h the
same verti)al height on the (all. he shorter ladder maFes an angle o' 60 (ith the 'loor. hat angle does the longer ladder maFe (ith the 'loorT ;ns. 2+..O
bCe)t
(.
rom a point mid(a# bet(een t(o obCe)ts$ one being three times as tall as the other$ the
-#pplementar8 Pro&lems:
angle o' elevation o' the taller is t(i)e the angle elevation o' the shorter. ind this angle
1. (o b&ildings b&ildings 450 450 't. and 600 't. 't. in height are are opposite opposite ea)h other. other. rom rom the roo' o' the the
o' elevation. ;ns. 0 ° ° , , .0 ° °
higher b&ilding$ the angle o' depression o' the edge o' the roo' o' the lo(er b&ilding is 340@. Go( (ide is the streetT ;ns. 17.45't
9.
; man standing at a )ertain point on a level 'iled determines determines the angle o' elevation o' the top o' a to(er 50 'eet high$ he then 'inds that b# going !0 'eet nearer the to(er$ the angle
2. ; to(er and a mon&ment stand on a level gro&nd. he angles o' depression o' the top o' the mon&ment vie(ed 'rom the top o' the to(er are 13 and 31$ respe)tivel#* the height
o' elevation is in)reased b# 45. ;t the 'irst observation ho( 'ar (as he 'rom the to(er$ and (hat (as the angle o' elevation. ;ns. &0/44ft, 2GG+ °
o' the to(er is 145 m. ind the height o' the mon&ment. ;ns. !.3 3. ; 'lagpole 'lagpole 25m high high stands on the top o' a to(er$ to(er$ (hi)h (hi)h is 10.5 m high. ;t (hat (hat distan)e distan)e
10. ; garage is 12 'eet high and 'i+ed on its top is a 'lagpole 15 'eet high. high. n the opposite side o' the street 'rom the garage at a given point$ the garage and the 'lagpole s&btend
'rom the base o' the to(er (ill the 'lagpole s&btend an angle o' 320@.
e%&al angles. Go( (ide is the streetT ;ns. streetT ;ns. 36't.
"ns 4+ m
4. he angle angle o' depression depression o' a barge 'rom the top o' a lightho&se lightho&se is 16. ;'ter ;'ter the barge barge has
traveled 100 m to(ard the lightho&se$ ghtho&se$ the angle o' depression o' the barge is 20.
ind the height o' the lightho&se. ;ns. 135.147
11.
he angles o' a triangle are in the ratio 3:4:5. +press the ratio o' the sine o' the smallest 2'2 angle. ;ns. √ 2'2
12. (o parallel )hords o' a )ir)le o' radi&s in)hes are on the same side o' the )enter and 5
5. rom rom a boat boat dire)tl# dire)tl# so&th so&th o' a lighth lightho&s o&se$ e$ the angle angle o' elevat elevation ion o' the top o' the lightho&se is 32. rom a se)ond boat l#ing dire)tl# east o' the 'irst and 150 't. 'rom it$ the
13. ; man standing at a )ertain point nt on a level 'ield determines the the angles o' elevation o' the 2
in)hes apart. ne s&btends a )entral angle t(i)e as large as the other. ind the length o' the shorter )hord.
Two 0eneral 1orms of &li"#e Triangle
13. ; man standing at a )ertain point nt on a level 'ield determines the the angles o' elevation o' the
Two 0eneral 1orms of &li"#e Triangle
top o' a to(er 50 'eet high. Ge then 'inds that b# going 23 o' the distan)e to the base o' the to(er$ to(ard it$ the angle o' elevation has been do&bled. ;t the 'irst observation$ ho( 'ar (as he 'rom the to(er$ and (hat (as the angle o' elevationT
; triangle has three angles and three three sides. 9' (e are given the meas&res meas&res o' three o&t o' this si+ prin)ipal parts$ at least one o' (hi)h is a side$ (e )an possibl# 'ind the meas&res o' the other three parts.
14. ; and ?$ the bases o' t(o to(ers ;8 and ?P standing standing on a horiQontal plane$ are 120 'eet apart. he angle o' elevation o' P as observed 'rom ; is do&ble the angle o' elevation o' 8 as observed 'rom ?. rom a point mid(a# bet(een ; and ? the angles o' elevation o'
Case I2 "iven t(o angles and one side.
Ose
8
8 and P are )omplementar#. ind the height o' the to(ers. a = b = )
"ns "? = 40 ft, : = 0 ft
)
a
sin ; sin 8 sin ? 15. %8:8 ?oard 8xam 7o# 4, &+) he angle (hi)h the line o' sight to the obCe)t maFes (ith the horiQontal is above the
;
b
?
e#e o' the observer. "ns "ngle of 8le#ation Case II2 "iven t(o sides and an in)l&ded angle.
16. %8:8 ?oard 8xam 7o# 4, &+)
Ose
he angle (hi)h the line o' sight to obCe)t maFes (ith the horiQontal is belo( the e#e o' an observer. ;ns. "ngle of epressio
a2 = b2 )2 , 2b) )os ; b2 = a2 )2 , 2a) )os 8 )2 = a2 b2 , 2ab )os ?
-ol#tions of &li"#e Triangles
?
?
Case III2 "iven t(o sides and the angle opposite one o' them.
his is Fno(n as the ambig&o&s )ase6 #o& )an &se sine la( and e+amine the possibilit# b
a
b
o' no sol&tion$ one sol&tion or t(o sol&tions.
a Case I72 "iven three sides.
;
)
8
;
)
8 Ose )osine la($ to solve 'or ea)h angle.
; = )os-1
b2 )2 , a2 2b)
' BJ24H ft2
Answer
; = )os-1
b2 )2 , a2
' BJ24H ft2
Answer
2b) 2. he sides sides o' a triangle triangle are 3$ 5 and 7. ind ind the largest largest angle. angle. ol&tion : 8 = )os
-1
2
2
2
) a , b
Osing ?osine
2a)
here$ a = longest side = 7 * b = 3* ) = 5 ? = )os-1
; = largest angle → &nFno(n
a2 b2 , )2
2ab
hen$ h en$ b 2 )2 , a2 = ?os ; 2b) ?os ; = 3 2 52 , 72 = -0.5
5ample 1roblem 1.
235
(o angles o' a triangle are 52 °15′ and 5!°30′$ and the shortest side is 3.46 't. long.
; = ?os-1-0.5
ind the length o' the largest side.
; = 120 °
Anser.
ol&tion : Osing ine
• he shortest side is opposite o' the smallest angle 3.46 't. is opposite o'
-#pplementar8 Pro&lems
52°15′.
1. 9n an obli%&e triangle angle ;8?$ it is Fno(n that tan tan ; = ]$ )os 8 = 513 and ;8 = 10. ind
• he longest &nFno(n side m&st be opposite o' the largest angle.
a sin ?
b side ;?
10 , 52°15′ 5!°30′ = 6.25°
) side 8 8? ?
h&s : a =
bLL
2.
; )hord$ ;?$ o' a )ertain )ir)le e%&als 13 ″$ and the angle 8 o' the ins)ribed triangle ;8? is 4!.35°. ind the radi&s o' the )ir)le.
in ; in 8 a = longest side$ ; = 6.25 b = shortest side = 3.46 't. 8 = 52 °15′ No($
a = b in; in8 = 3.46in 3.46 in 6.25 6.2 5o
3.
;n observer in a balloon 1 mile high observes the angle angle o' depression o' an obCe)t obCe)t on the gro&nd to be 35 °40′. ;'ter as)ending verti)all# at &ni'orm rate 'or 10 min&tes$ he 'inds the angle o' depression o' the same obCe)t is to be 55 °20′. ind the rate o' as)ent o' the balloon in miles per ho&r.
in52°15@
4.
rom the top o' a lightho&se 5 't. high standing on a ro)F$ the angle o' depression o' a ship (as 7°3′ and 'rom the bottom o' the lightho&se the angle o' depression (as 3 °.
12. he area o' a triangle is 50 s%. in. and the lengths o' t(o o' its sides are 20 in. and 10 in. ind the in)l&ded angle.
4.
rom the top o' a lightho&se 5 't. high standing on a ro)F$ the angle o' depression o' a
12. he area o' a triangle is 50 s%. in. and the lengths o' t(o o' its sides are 20 in. and 10 in. ind the in)l&ded angle.
ship (as 7°3′ and 'rom the bottom o' the lightho&se the angle o' depression (as 3 °. hat (as the height o' the ro)FT 5.
; to(er 51.63′ high maFes an angle o' 113°12′ (ith the in)lined plane on (hi)h it stands.
13.
(o a&tomobiles leave the same to(n at the same time. ne goes north at the rate o' 30
he angle s&btended b# the to(er at the same point do(n the plane 'rom its base is
miles per ho&r$ the other goes in the dire)tion 47 ° o' N at the rate o' 20 miles per ho&r.
23°27′. Go( 'ar is this point 'rom the baseT
Go( 'ar apart are the# at the end o' 5 ho&rsT 14.
he a)&te angle bet(een the diagonals o' a parallelogram is 45 °. he diagonals are and 12 in)hes. ind the lengths o' the sides o' the parallelogram.
6.
; ho&se is sit&ated on a hillside (hi)h is in)lined 2!°3!′ to the horiQontal plane. ; ladder 32.75′ long C&st rea)hes the bottom o' a (indo( 25 ′ 'rom the gro&nd. hat is the
15. he sides o' a triangle are !$ 12$ and 14 and the largest angle angle is bise)ted$ 'ind the length o' the bise)tor o' this angle and the length o' the median to the shortest side.
distan)e o' the 'oot o' the ladder 'rom the ho&se$ meas&red do(n the slopeT Inerse Trigonometric 1#nctions .
; man standing at a point d&e to (est o' a to(er 150′ high 'o&nd the angle o' elevation o' the top o' the to(er to be 6 °. Ge then (alFed to a se)ond point so&th(est o' the 'irst and
1. 9nvers 9nversee ine ine &n) &n)tio tionn # = ;r)s ;r)sin in +
'o&nd the angle o' elevation o' the top o' the to(er to be 3! °. Petermine ho( 'ar he
i' and and onl onl## i'i' + = sin sin # #
#
(alFed. π
1 (.
2
; )ir)le is ins)ribed in a ∆ ;8? (hose sides are a = 25 ″$ b = 33 ″ and ) = 3 ″. Go( long are tangents to the )ir)le 'rom verte+ ;T verte+ 8T verte+ ?T ,1
!. ; )ir)le is ins)ribe ins)ribedd in a triangle triangle (hose sides sides are 5$ 12 and 13. ind ind the radi&s radi&s o' the )ir)le. 10.
+ π
- 2
2
1
he lo(er base o' an isos)eles trapeQoid is 70.23″. a)h o' the nonparallel sides is 35.1″. ea)h lo(er base angle is 1 °30′. Go( long is ea)h diagonalT
11.
+
π
9' the diameter o' a )ir)le is 32.6 ″$ 'ind the angle at the )enter determined b# a )hord
-π2
-1 # = sin +
# = ;r)sin +
12.!″ long. 2. 9nvers 9nversee ?osine ?osine &n) &n)tio tionn
# = ;r))os + #
i' and onl# i' + = )os #
# = tan + #
# = ;r)tan +
# = ;r))os +
i' and onl# i' + = )os #
# = tan +
#
# = ;r)tan +
# (here$ ∞ ≤ + ≤ ∞$ -π2 K # K π2 → prin)ipal val&e
π
1
6n#erse 6dentities
π
π
2
π
+
,1
,1 # = sin +
1
2
+
# = ;r))os +
(here$ ,1 ≤ + ≤ 1$ 0 ≤ # ≤ π → prin)ipal val&e
1. ;r)sin sin θ = θ
'or -π2 ≤ ≤ π2
2. sin sin ;r) ;r)si sinn + = +
'or 'or ,1 ≤ x ≤ 1
3. ;r))os )os θ = θ
'or 0 ≤ ≤ π
4. )os ;r))os + = +
'or ,1 , 1 ≤ x ≤ 1
5. ;r)tan tan θ = θ
'or -π2 ≤ x ≤ π2
6. tan ;r) ta tan + = +
'or all val&es o' +
-#pplementar8 pro&lems
3. 9nvers 9nversee angent angent &n) &n)tio tionn R = ;r)tan ;r)tan +
olve the 'ollo(ing trigonometri) e%&ations:
i' and onl# onl# i' + = tan #
1. 3sin; = 2)os2 ; $ 0 0 ≤ ; ≤ 3600 ans3 0 0 , &+0 0
#
1
#
2. sin2+ 2sin
+ = 1 $ 0 K + K 10 10 0
3 .an -12+ an-13+ =
π
2
1
2 2
Π 4
4. an-1H2sin?os-1+I = 60 0
ans30 0 ,0 0 , &+0 0
ans3 & ' . ans3 & ' 2
0 +
0 -1
-1
+
5. ;r)tan+ ;r)tan2+ ;r)tan2+ = ;r)tan3 ans3 *&, & ' 2
1 6. ;r)sin1 , + ;r))os+ = !00 ans3 & ' 2
-π2 olve the 'ollo(ing problems 1. 9' sin; = 0.0$ )os; V 0$ 'in 'indd )ot;. ;ns. 0.75
2. 9' sin
=
12 13 0
and and )os )os
K 00$$ 'in 'indd tan tan θ . ;ns. -125
3. 9' ; , 8 = 45 and tan; = ] $ 'ind tan8. ;ns. -17
13. rom rom the the top o' the the to(er 33 m high$ the the angles es o' depression o' o' the top
and bottom bottom 0
o' another to(er standing on the same horiQontal plane are 'o&nd to be 2.!3 and 53.60
2. 9' sin
=
12 13
and and )os )os
13. rom rom the the top o' the the to(er 33 m high$ the the angles es o' depression o' o' the top
K 00$$ 'in 'indd tan tan θ . ;ns. -125
o' another to(er standing on the same horiQontal plane are 'o&nd to be 2.!3 and 53.60
0
3. 9' ; , 8 = 45 and tan; = ] $ 'ind tan8. ;ns. -17 4. 9' )ot θ = -24 7$
θ in
/&adr /&adrant ant 99$ 'ind 'ind se)
and bottom bottom 0
respe)tivel#. ind the distan)e bet(een the tops o' the to(ers. ;ns. 27.7m 14. ; s&rve#or at a )ertain distan)e distan)e meas&red meas&red the angle o' elevation elevation o' a )li''. Ge then
. ;ns. ;ns. -25 -2524 24
(alFed 20 m on a level gro&nd to(ard the )li''. he angle o' elevation 'rom this se)ond station (as then the )omplement o' the 'ormer angle. he s&rve#or again (alFed 5 m
5. 9' )os+ = 1 3$ 'ind )os4+. ;ns. 171 6. 9' )os )os
= -1 -1 3 and and 0 0 K
K 360 0$ 'ind tan
1 2
nearer to the )li'' in the same line and 'o&nd the angle o' elevation 'rom the third station
. ;ns. %rt2
to be do&ble the 'irst angle. Go( high is the )li''T ;ns. 22.!1m
9' sin; = 3 5 and ; is in /&adrant 99$ 'ind )os2;. ans. 725 15. he min&te hand o' a )lo)F is 23 )m long (hile the ho&r hand is 15 )m long. he plane o' rotation o' the min&te hand is 5 )m above the plane o' rotation o' the ho&r hand. ind the
9' tan; = 3 4 $ tan8 = -15 $ ; in / 999$ 8 in / 9\$ 'ind
distan)e bet(een the tips o' the hands o' the )lo)F at 2:30 p.m. ;ns. 30.!)m
)os; , 8. ;ns. 135 straight straight tra)Fs
16. ; point (ithin an isos)eles isos)eles right triangle triangle is at a distan)e distan)e o' 6$ 5 and 4 )m 'rom the
diverging at an angle o' 6 degrees. 9' one train r&ns at 32 Fph and the other at 46 Fph$
verti)es ;$ 8 and ? respe)tivel#. ind the length o' the h#poten&se o' the right triangle i'
ho( 'ar apart are the# at the end o' 3 ho&rsT ;ns. 135.4Fm
the right angle is at ?. ;ns. 10.65)m
!. (o (o train trainss start start at at the the same same time 'rom the same station station and &pon
10. he bearing o' 8 'rom ; is N20 0* the bearing o' ? 'rom 8 is 30 0* and the bearing bearing o' ;
17. he angle o' elevation o' the top o' a pole at a point 30 m 'rom the pole is three times the
'rom ? is 40 . 9' ;8 = 10 m$ 'ind the area o' the triangle 'ormed b# ;$ 8 and ?. ;ns.
angle o' elevation o' the top o' the same pole at a point 150 m 'rom the pole. ind the
13.!4s%.m.
height o' the pole. ;ns. 56.7)m
0
11. ;n airplane 'l#ing an altit&de o' one Fm dire)tl# a(a# 'rom a stationar# observer on the
1. ; )orner lot o' land is 35 m on one street and 25 m on the other street$ the angle bet(een
gro&nd$ has an angle o' elevation o' 4 degrees at a )ertain instant and an angle o'
t(o lines o' the streets being 0 degrees. he other t(o lines o' the lot are respe)tivel#
elevation o' 20 degrees one min&te later. ind the speed o' the plane. ;ns. 110.2Fph
perpendi)&lar to the lines o' the streets. hat is the (orth o' the lot at 1$000 per s%&are meterT
12. rom a point on a level gro&nd$ the angle o' elevation o' the top o' a b&ilding is observed
ans3 1G2+,4G+00
to be t(i)e the angle o' elevation on o' a (indo( one third o' the (a# &p the b&ilding. b&ilding. ind the angle o' elevation o' the top o' the b&ilding. ;ns. 60degrees
1!. ;t noon$ a ship ; is sailing on a )o&rse east(ard at the rate o' 20 Fph. ;t the same instant$ another ship 8$ 100 Fm east o' ship ; is sailing on a )o&rse N30 0 at the rate o' 10 Fph. Go( 'ar a(a# 'rom ea)h other are the ships a'ter one ho&rT ;ns 75.5Fm
!ote: N&ll set is a s&bset o' all sets
20. ; pole )asts a shado( 15 )m long (hen the angle o' elevation o' the s&n is 61 degrees.
!ote: N&ll set is a s&bset o' all sets
20. ; pole )asts a shado( 15 )m long (hen the angle o' elevation o' the s&n is 61 degrees. 9' the pole leans 15 degrees 'rom the verti)al to(ard the s&n$ (hat is the length o' the
Onit et , a set (ith onl# one element
poleT ;ns. 54.23)m
9n'inite et , i' it is impossible to list do(n all elements o' a set
inite et , i' it is possible to (rite all o' its elements
21. ind the interior angles o' a triangle (hose sides are 21$ 2 and 17. ans3 4/40 0 ,
440 , G2. 0
•
:ardinal 7umber
22. he sides o' a triangle ;8? are a = 50 )m$ b = 64 )m and ) = 20 )m. ind the length o'
, denoted b# n; → no. o' elements o' set ;
the median dra(n 'rom 8 to ;?. ;ns. 20.64)m 23. (o stations 8 and ? are sit&ated on a horiQontal horiQontal plane 366 m apart. ; balloon is dire)tl# dire)tl# above a point ; in the same horiQontal plane as 8 and ?. ;t 8$ the angle o' elevation o'
perations on -ets
•
Jnion of 5ets
he &nion o' sets ; and 8 is another set denoted b# ; ∪8 (hose elements
the balloon is 62 degrees and the angle at 8 s&btended b# ;? is 53 degrees and at ?$
are in ;$ or in 8 or both ; and 8.
the angle s&btended b# ;8 is 72 degrees. ind the height o' the balloon. ;ns. 7!!.1!m PR*A*ILIT
, re'ers to the n&mber o' elements o' a )ertain set
9n s#mbol s#mbol$$ ;⋃8 = + + ∈ ; or + ∈ 8j
•
6ntersection of 5ets
9ntrod&)tion to et heor#
he inters interse)t e)tion ion o' sets ; and 8 is another another set denote denotedd b# ;⋂8 (hose
Pe'initions
elements are )ommon to both ; and 8.
•
5et , , a )olle)tion o' obCe)ts )alled elements o' an# sort (ith restri)tion to
•
8lement , , an# obCe)t that belong to a set Penoted b# small letters
those obCe)ts )learl# des)ribed. Penoted b# )apital letters
9n s#mbol s#mbol$$ ;⋂8 = + + ∈ ; and + ∈ 8 j
•
he relative )omplement o' the set 8 (ith respe)t to ; is another set denoted b# ; , 8 (hose elements are all ; b&t not in 8.
- s#mbol &sed to indi)ate that an element belong to a given set.
• •
9n s#mbol$ ; , 8 = + + ∈ ; and + ∉ 8 j
5ubset , , 9' all elements o' set ; are in set 8 or i' there are elements o' set 8 not
belonging to set ;$ then ; is a s&bset o' 8 denoted b# ; ⊂ 8.
•
elements do not belong to ; b&t in the &niversal set O
•
N&ll et or mpt# et , a set (itho&t an element denoted b# j or φ
Note: ; + 8 ≠ 8 + ; isjoint 5et
1roduct 5et or :artesian 1roduct
he prod&)t set denoted b# ; + 8 is a set o' ordered pairs +$ # s&)h that + is
5iBe of 5et
•
9n s#mbol ;′ = + + ∈ & and + ∉ ; j
8qui#alent 5et , , (o sets are e%&ivalent i' the# have e%&al n&mber o'
elements. ; ⇔ 8
•
"bsolute :omplement :omplement
he absol&te )omplement o' set ; denoted b# ; ′ is another set (hose
8qual 5ets , (o sets are e%&al i' the# have e+a)tl# the same elements.
; = 8
•
;elati#e :omplement
an element o' set ; and # is an element o' set 8. 9n s#mbol$ ; + 8 = +$ # + ∈ ; and # ∈ 8j
n8⋃ = J5 total
Answer
Note: ; + 8 ≠ 8 + ;
•
n8⋃ = J5 total
Answer
isjoint 5et
(o sets ; and b are )onsidered disCoint sets i' the# have no element in
1#ndamental Co#nting Principle
9' an element $ )an happen in n 1 (a#s$ and 'or ea)h o' these another event 2 )an
)ommon.
happen in n2 (a#s$ and 'or ea)h o' these events$ another event 3 )an happen in n3 (a#s$
9n s#mb s#mbol ol$$ ;⋂8 = j or ; ⋂8 = φ
and so on and so 'orth$ then all in all the events )an happen sim<aneo&sl# in N (a#s e%&al to: ! ' n4 n$ n5 ?n@
7enn Diagram ,
i)torial representation o' set relations relations and operations * 9ntrod&)ed b# ohn 7enn$ an nglish
no2 of wa8s
PER+UTATI! G C+*I!ATI! 1actorial !otation
N&mber o' elements in a &nion o' ets 3 :J767P ;DJ9"
- the 'a)torial n$ (here n is an# positive integer is denoted b# nE
a. a. n; n;⋃8 = nn; ; n8 n8 , n; n;⋂8
actorial n n is de'ined as the prod&)t o' positive )onse)&tive integers 'rom 1 to n
b. n;⋃8⋃? = n; n8 n? , n;⋂8 , n;⋂? , n8⋂? n;⋂8⋂?
in)l&sive hat is$
-ample Pro&lems:
1. 9n a s&rve#$ 45 men liFe Fe basFetball$ 35 liFe so'tball$ so'tball$ 250 liFe both. Go( man# man# men (ere (ere thereT Go( man# liFe so'tball onl#. ol&tion: 9n &sing \enn Piagram$ al(a#s prioritiQe to indi)ate the n&mber o' elements in the interse)tions. &
8
o'
men
(ho
No($ n8 , n8 n8⋂ = 45 , 250 250 = 20 liFe liFe basFetba basFetballll onl# n n - n8⋂ = 35 35 , 250 250 = 45J liFe liFe so't so'tba ballll onl# nl# n8⋃ = n8 n8 n n - n8 n8∩ = 45 35 ,250
:"58 2 : erm&tation o' n di''erent things taFen all at a time
)ase )ase (hre r = n
liFe
- is an# linear ordering o' the elements o' a set
basFetball and so'tball n8⋃ = total total n&mber n&mber o' men in the s&rve# s&rve#
n r
8# de'inition ( ' 4
- is arrangement o' all or part o' a set o' obCe)ts in a de'inite order
n = n&mber o' men (ho liFe so'tball n8⋂=n&mber
bserve that nE = nn-1E = nn-1n-2E
PER+UTATI!
20 250 135
nE = 123 Dn-3n-2n-1n or nE = nn-1n-2 D3.2.1
CA-E- 1 PER+UTATI! :"58 &: erm&tation o' n di''erent things taFen r at a time
- de'ined as an arrangement o' r o&t the n obCe)ts (ith attention given to the order o' Answer
arrangement nr = =
nE
rKn
n-rE
that is n?a = n?b $ i' a b = n
:"58 2 : erm&tation o' n di''erent things taFen all at a time
that is n?a = n?b $ i' a b = n
)ase )ase (hre r = n
n r
= nn =
n r
nE
= nE = n
2. N&mber N&mber o' )ombinati )ombination on o' n things things taFing 1 or 2$ D or n at a time time
n-nE n-nE 0E
n
?1 n?2 n?3 D n?n = $n 3 4
:"58 : erm&tation o' n things not all di''erent
-#pplementar8 Pro&lems
others are aliFe$ r others are aliFe$ and so onD
1. 9n ho( man# (a#s )an )an 7 bo#s and 6 girls rls sit in a ro( i' the bo#s and girls m&st alternate.
N = nE
(here p % r D= n
Ans2 56 S$H6 H((
pE%ErED
:"58 4: N&mber o' erm&tation o' n distin)t obCe)ts arranged in a )ir)le
2. 9n ho( man# man# (a#s )an )an 6 people people be lined lined &p to get on on a )arT b. 9' )ertain 3 persons insist on 'ollo(ing ea)h other
N = n-1E
). 9' )ertain 2 persons re'&se to 'ollo( ea)h other ho( man# (a#s are possible Ans2 Ans2 a2/ a2/ $( $(
C+*I!ATI!
&2/ &2/ 4BB 4BB c2/ c2/ BH( BH(
- sele)tion o' gro&p o' obCe)ts o&t o' a set irrespe)tive o' their order - )ombination o' n things taFen r at a time re'ers to 'illing o&t r pla)e (hen (e have n things at
3. rom the digits 0$ 1$ 2$ 3$ 4$ 5$ 6$ 7$ ho( man# 3 digit digit n&mber )an be 'ormed b. ho( man# n&mbers are odd i' digits are distin)t
o&r disposal denoted b# n?r n
?r =
). ho( man# n&mbers are even i' digits are distin)t
nELL
Ans 2 a2/ $B
n-rE rE
)orre)t
1. he hen r = n ?n =
nE
= nE
n-nE nE n
c2/ 4J(
4. 9n a m<iple m<iple )hoi)e )hoi)e test o' 6 %&estions %&estions (ith (ith 4 possible ans(ers ans(ers o' (hi)h )h onl# one is
5pecial :ases3
n
&2/ 4BB
a. in ho( man man## di''er di''erent ent (a#s (a#s )an a st&den st&dentt )he)F o'' one ans(er ans(er to ea)h
=1
%&estionT
0E nE
b. 9n ho( man# di''erent di''erent (a#s )an a st&dent st&dent )he)F o'' one n&mber n&mber to ea)h %&estion %&estion
?n = 1 )ombination o' n things taFing all at a time
and get all ans(ers (rong. Ans2 a2/ a2/ B6 (S &2/ $
2. he hen r = 1 n?1 =
nE n-1E 1E
=
nn-1E = n n-1E
5. Go( man# triangle triangless )an be 'ormed &sing &sing the verti)es verti)es o' a dode)agonT dode)agonT Ans2 $$(
n?1 = n )ombination o' n things taFing one element at a time
6. 9n ho( man# man# (a#s )an #o& invite invite one one or more o' #o&r #o&r 5 'riends 'riends in a )ertain )ertain sho(TAns2 54 Other important Combination Formulas
1.
n
?r = = n?n-r
7. Go( man# di''erent di''erent three-digit n&mbers )an be made (ith three three 4@s$ 'o&r 2@s and t(o 3@sT Ans2 $S
15. olve 'or n in the e%&ation e%&ation n4 = 30Hn?5I. Ans2 n ' H
7. Go( man# di''erent di''erent three-digit n&mbers )an be made (ith three three 4@s$ 'o&r 2@s and t(o 3@sT
15. olve 'or n in the e%&ation e%&ation n4 = 30Hn?5I. Ans2 n ' H
Ans2 $S
. ; department department pla#s pla#s 12 so'tball so'tball games d&ring d&ring a season. season. 9n ho( man# (a#s )an the team team
PR*A*ILIT , meas&re o' the liFehood o' o))&rren)e o' an event
end the season (ith 7 (ins$ 3 losses and 2 tiesT Ans. /=#) Classical Definition 9.
Go( man# distin)t perm&tations are there o' the letters o' the (ord ?9A?O9.Ans. 1,260
10. Go( man# n&mbers bet(een 3$000 and 5$000 )an be 'ormed 'ormed (ith the digits 7$ 3$ 4$ $ 2
9' an event )an happen in h (a#s )alled )alled the s&))ess s&))ess and 'ail to happen happen in ' (a#s )alled 'ail&re o&t o' n possible e%&all# liFel# (a#s$ then the probabilit# o' o))&rren)e is denoted b# p =
and 5$ no repetitions being allo(ed. Ans. 16
h
h'
11. 9n ho( man# (a#s (a#s )an bo#s and 5 girls girls be arranged in a )ir)le i'
= 'avorable (a#s possible (a#s
he probabilit# % that an event (ill 'ail to happen is given b# % = ' probabilit# o' 'ail&re
a. )ertain )ertain 2 bo#s (ill (ill not be together together b. a )ertain bo# and girl (on@t (on@t be togethe together r
h '
). )ertain )ertain 2 girls girls (on@t be be together together d. )ertain )ertain 3 girls girls (ill be toget together her
Odd Ratios
Ans2 a6 &6 c6 ' 56 4SH6 4SH6 (((6 d ' $46 $6 $6 H((
42 dds dds in fao faor r
p=h 12. here here are 12 ?@s in a )ertai )ertainn telephone e phone )ompan# )ompan# so that that 3 (ill (ill be assigned assigned to
% '
s(it)hing$ 2 (ill be assigned in planning$ 4 (ill be assigned to transmission and the remaining (ill be assigned in some other '&n)tions. 9n ho( man# (a#s the engineer ma# be dividedT
$2 dds dds agai agains nstt
% = ' p h
Ans2 $6 $(( wa8s
!ote: p % = 1
13. ; )l&b has 25 members 4 o' (hom are engineers. 9n ho( man# man# (a#s )an a )ommittee )ommittee o' 3 members be 'ormed in order that at least one member is an engineerT Ans2 (
Axioms of Pro&a&ilit8 1.
14. rom a gro&p o' 6 (omen and men$ a )ommitte )ommitteee o' 5 is to be 'ormed. 9n ho( man# man# (a#s )an this be done. 9' ea)h )ommittee id to )onsist o' a. e+a)tl# 3 men b. at least 4 men ). at most 2 men. Ans2 a2/ HB( &2/ BS c2/ SHS
one 0≤ p ≤ 1 2. he probabili probabilit# t# o' o))&rren)e o))&rren)e o' an# an# events events in a sample spa)e spa)e is e%&al to to one he probabilit# o' o))&rren)e o' )ertain event is e%&al to one = 1 3.
4. he s&m o' probabilit# o' o))&rren)e o' ea)h event in the sample spa)e is e%&al to one.
Σi = 1
he probabilit# o' o))&rren)e o' a parti)&lar event is a positive real n&mber 'rom Qero to
he probabilit# o' impossible event is e%&al to Qero φ = 0
•
1∪2 = 1 2
'or m&t&all# e+)l&sive events
•
1∪2 = 1 2 , 1∩2
'or not m&t&all# e+)l&sive
4. he s&m o' probabilit# o' o))&rren)e o' ea)h event in the sample spa)e is e%&al to one.
Σi = 1
•
1∪2 = 1 2
'or m&t&all# e+)l&sive events
•
1∪2 = 1 2 , 1∩2
'or not m&t&all# e+)l&sive
!ote: 1∪2 means probabilit# that either 1 o))&r onl#$ 2 o))&r onl# or 1 and E7E!T-
2 o))&r both at the same time
•
•
6ndependent 8#ents
(o or more events are said to be independent i' the o))&rren)e or non-o))&rren)e
or n m&t&all# e+)l&sive events 1∪2∪3∪ D ∪n = 1 2 3 D n
o' an# o' them does not a''e)t the probabilities o' the o))&rren)e o' an# o' the others.
•
ependent 8#ents
1%8H 1%8H)) = &* 1%") 1%")
(o or more events are said to be dependent i' the o))&rren)e o' one events a''e)ts
robabilit# o' the )omplement o' an event$ @
the probabilities o' the o))&rren)e o' an# o' the other.
•
Dutually exclusi#e 8#ents
Pro&a&ilit8 in Repeated Trials 3 *erno#lli;s Experiment
(o or more events are said to be m&t&all# e+)l&sive i' the o))&rren)e o' an# one o' them e+)l&des the o))&rren)e o' the other.
P ' nCr p pr "n>r
that is$ not more than one o' them )an happen in a single trial here: here: = the probabili probabilit# t# that the event event (ill o))&r e+a)tl# e+a)tl# r times in trials trials p = the probabilit# that the event (ill o))&r in a single trial
C!DITI!AL PR*A*ILIT
or t(o events 1 and 2$ the probabilit# that 2 o))&rs$ given that 1 has o))&rred is
% = the probabilit# that the event (ill 'ail to happen probabilit# o' 'ail&re
denoted b#
=1,p
12
= the n&mber o' (a#s that the event (ill o))&r n?r =
or )ompo&nd events 12$ 12 = 1 21
+#ltinomial +#ltinomial Distri&#tion
9' events 1$ 2$ 3$ D$ n )an o))&r (ith probabilities 1$ 2$ 3$ D$ n respe)tivel#$ then
LA,- 1 PR*A*ILIT
the probabilit# that 1$ 2$ 3$ D$ n (ill o))&r +1$ +2$ +3$ D$+n times respe)tivel# is given b#:
& Dultiplic Dultiplication ation 9a( %"7 9"K)
robabilit# that events 1and 2 (ill happen both at the same time
•
1∩2 = 1 2
'or independent events
•
1∩2 = 1 21
'or dependent events
o that$ 21 = 1∩2 1
P'
!
P4<4 P$<$ P5<5 ? Pn
x4 x$ x5 ?xn
(here +1 +2 +3 D + n = N
2 "dditi "ddition on 9a( 9a( %; %; 9"K) 9"K)
+athematical Expectation
9' p is the probabilit probabilit## o' s&))ess s&))ess in a single trial o' an event$ the e+pe)ted n&mber n&mber o'
7. ;n &rn )ontains )ontains 3 red pens pens and a n&mber n&mber o' bl&e pens. pens. ; se)ond se)ond &rn )ontains )ontains 5 bl&e pens and an &nFno(n n&mber o' red pens. ne pen is dra(n 'rom the 'irst &rn and
7. ;n &rn )ontains )ontains 3 red pens pens and a n&mber n&mber o' bl&e pens. pens. ; se)ond se)ond &rn )ontains )ontains 5 bl&e
+athematical Expectation
9' p is the probabilit probabilit## o' s&))ess s&))ess in a single trial o' an event$ the e+pe)ted n&mber n&mber o' s&))esses in n trials is given b# :
pens and an &nFno(n n&mber o' red pens. ne pen is dra(n 'rom the 'irst &rn and pla)ed &nseen in the se)ond &rn. No($ t(o pens are to be dra(n 'orm the se)ond &rn. 9' the odds in 'avor o' getting 2 bl&e pens e%&als 3:5 and the odds in 'avor o' getting pens
E ' nP
o' di''erent )olors e%&al 31:2!$ ho( man# are bl&e pens in the 'irst &rn and ho( man# are red pens in the se)ond &rnT
-#pplementar8 Pro&lems
1. 9n a 'amil# 'amil# o' 5 )hildren$ (hat is the probabilit# probabilit# that the 'irst 3 are girls. Ans2 GH 4
2. o&r le)tron le)troni)s i)s booFs and three three ?omm&ni)a ?omm&ni)ations tions booF booF are pla)ed on a shel'. shel'. hat is the probabilit# that the ?omm&ni)ations booF (ill be togetherT Ans2 (24B$HS 3. Petermine the probabilit# probabilit# that 7 or 11 )omes &p in a single toss o' a pair o' 'air di)e. )e. Ans. # 5 =
E"ut!orHs Aint3 7o of blue balls is more t!an t!e no of red balls
. ; pair o' di)e di)e is thro(n. 9'9' it is Fno(n Fno(n that one die die sho(s a 4$ (hat is the probabilit# that a. the other dies sho(s sho(s a 5 b. the total o' both di)e is greater than 7 Ans2 a2/ $ G44 &2/ J G44
!. ind the probabi probabilit# lit# o' gettin gettingg 1$ 2 or 3 in 'o&r tosses o' a 'air 'air die. Ans. - 5 -2 -2
4. irso and
10. here are three shooters ;$8$?. he probabilit# probabilit# that probabilities that ;$ 8 and ? )an hit the target are 12$ 13 and 14 respe)tivel#. 9' the# are sim<aneo&sl# shooting the target$ 'ind the probabilit# that t(o o' them (ill hit the target. Ans. ).#'
5. P&ring P&ring a )ertain (ar$ (ar$ an o''ensive ve troop released released 4 bombs bombs ea)h o' (hi)h has has probabilit# probabilit# o' hitting the enem# o' 0.25. 9' 4 bombs bombs are released sim<ane sim<aneo&sl# o&sl#$$ (hat is the probabilit# o' hitting the enem#T Ans2 (2SHJ 6. ; bo+ )ontains )ontains an assortment assortment o' red$ red$ bl&e and #ello( balls. 9' three balls are dra(n dra(n 'rom it at random$ the probabilit# that 2 red balls and 1 bl&e ball are dra(n is 30[ o' the
11. he probabilit# that ;lvin )an (in in a )hess game (henever he pla#s is 20[. 9' he pla#s 5 games$ 'ind his probabilit# o' losing e+a)tl# 2 games. Ans2 5$ GS$J 12. ;n &rn )ontai )ontains ns 5 (hite (hite and 6 green green balls. balls. 9' 4 balls balls are dra(n dra(n together together$$ 'ind 'ind the probabilit# that all are o' the same )olor. Ans. # 5 %% %%
probabilit# that 2 bl&e balls and 1 #ello( ball are dra(n. &rthermore$ the probabilit# that 3 bl&e balls are dra(n is 54 times the probabilit# that one ball o' ea)h )olor is dra(n. Go(ever$ i' onl# t(o balls are dra(n$ the probabilit# that one is bl&e and the other is a
13. &t o' 00 'amilies (ith 5 )hildren )hildren ea)h$ ho( man# (o&ld #o& e+pe)t to have a. 3 bo#s b. 5 girls and ). either 2 or 3 bo#sT Ans2 a2/ $J( &2/$J c2/ J((
#ello( ball is times the probabilit# that both are red. Go( man# are red balls$ bl&e balls$ and #ello( balls are there in the bo+. E"ut!orHs Aint3 !e number of balls are in arit!metic progression2 progression2
14. ; bo+ )ontains a ver# large n&mber o' red$ (hite$ bl&e and #ello( marbles marbles in the ratio 4:3:2:1 respe)tivel#. ind the probabilit# in 10 dra(ings. a. 4 red$ 3 (hite$ 2 bl&e and 1 #ello( o( marble (ill be dra(n and ).)%16 4. ).)))#=' b. red and 2 #ello( #ello( marbles marbles (ill be dra(n dra(n Ans. a. ).)%16
22. ; )ontra)tor (ishes to b&ild 'ive 'ive ho&ses$ ea)h di''erent di''erent in design. 9n ho( man# (a#s (a#s )an 16. a Go( man# (a#s )an be 'ive people be lined &p to pa# their ele)tri) billsT
he pla)e these homes on a street i' t(o lots are on one side and three lots on the
22. ; )ontra)tor (ishes to b&ild 'ive 'ive ho&ses$ ea)h di''erent di''erent in design. 9n ho( man# (a#s (a#s )an 16. a Go( man# (a#s )an be 'ive people be lined &p to pa# their ele)tri) billsT b 9' t(o parti)&lar parti)&lar persons re'&se to 'ollo( ea)h other$ ho( man# (a#s are possibleT possibleT Ans. a -#) a&s a&s
4 /# a&s
17. ;n engineering 'reshman m&st taFe a )hemistr# )o&rse$ a h&manities )o&rse$ and a mathemati)s )o&rse. 9' the# ma# be sele)t an# o' 2 )hemistr# )o&rses$ ho( man# (a#s )an he arranged his programT Ans. #1 a&s 1. 9n ho( man# di''erent (a#s )an a ten- %&estion tr&e- 'alse e+amination be ans(eredT Ans. -)#1 a&s
he pla)e these homes on a street i' t(o lots are on one side and three lots on the opposite sideT Ans. -#) a&s
22. 9n ho( man# (a#s )an 'o&r bo#s and three girls sit in a ro( i' the bo#s and girls m&st alternatel# be seatedT Ans. #66 a&s
23. 9n ho( man# (a#s )an seven trees be planted in a )ir)leT Ans. /#) a&s 24. 9n ho( man# (a#s )an t(o mango trees$ three )hi)o trees$ and t(o avo)ado trees be arranged in a straight line i' one does no disting&ish bet(een trees o' the same FindT
1!. Go( man# distin)t perm&tations )an be made 'rom the letters o' the (ord mathemati)sT
Ans. #-) a&s
Ans. 1,=6=,2)) 1,=6=,2))
25. ; )ollege team pla#s eight basFetball games d&ring an intram&ral. 9n ho( man# (a#s )an 20. Go( man# (a#s )an the 'irst 'ive pla#ers in a basFetball team be 'illed (ith t(elve men
the team end and games (ith 'o&r (in$ three losses and one tieT Ans. #6) a&s
(ho )an pla# an# o' the positionsT 26. Nine people (ill be shooting the rapids o' agsanCan in three ban)as that (ill hold 2$ 4 and 5 passengers$ respe)tivel#. Go( man# (a#s is it possible to transport the nine
Ans. =',)1) a&s a&s
21. Go( man# three- digit n&mbers )an be 'ormed 'rom the digits 0$1$2$3$4 and 5 a i' ea)h digit is &sed onl# on)e in a given n&mberT b i' digits ma# be repeated in a given n&mber ) Go( man# in a are odd n&mbersT d Go( man# in a are even n&mbersT e Go( man# in b are even n&mbersT ' Go( man# are less than 330T g Go( man# are greater than 330T Ans. a -)) 4-6)
people to the 'allsT Ans. 1,'%2 a&s a&s
27. rom a gro&p o' three men and seven (omen$ ho( man# )ommittees o' 'ive people are possibleT a (ith no restri)tionsT b (ith t(o men and three (omenT ) (ith one man and 'o&r (omen i' a )ertain (oman m&st be on the )ommitteeT Ans. a #'#
4 -)'
c2)
c16 d'#e=) 9=) 6=
2. rom three red$ 'o&r green$ green$ and 'ive #ello( b&bbleg&ms$ b&bbleg&ms$ ho( man# sele)tions )onsisting o' 'ive b&bbleg&ms are possiblei' t(o o' ea)h )olor are to be sele)tedT Ans. %=)
2!. ; shipment o' 10 on# 8etama+ video re)orders )ontains 3 de'e)tive sets. 9n ho( man#
37. Zoe# prepares 3 )ards 'or his 3 girl'riends. Ge addresses 3 )orresponding envelopes. ;
(a#s )an a hotel p&r)hase 4 o' these sets and re)eive at least 2 o' the de'e)tive setsT
bro(n-o&t s&ddenl# o))&rred and he h&rriedl# pla)ed the )ards in the envelope at
2!. ; shipment o' 10 on# 8etama+ video re)orders )ontains 3 de'e)tive sets. 9n ho( man#
37. Zoe# prepares 3 )ards 'or his 3 girl'riends. Ge addresses 3 )orresponding envelopes. ;
(a#s )an a hotel p&r)hase 4 o' these sets and re)eive at least 2 o' the de'e)tive setsT
bro(n-o&t s&ddenl# o))&rred and he h&rriedl# pla)ed the )ards in the envelope at random. hat is the probabilit# that
Ans. /)
a ea)h )ard is send to its proper addresseeT 30. ; bag )ontains 'o&r bl&e$ 'ive red$ and si+ #ello( plasti) )hips.
b no )ard is sent to the proper addressT Ans. a '5-2
4=52
a 9' t(o )hips are dra(n$ 'ind the probabilit# that both are #ello( b 9' si+ )hips are dra(n$ 'ind the probabilit# probabilit# that there (ill be t(o )hips o' ea)h )olor ) 9' nine )hips are dra(n$ 'ind the probabilit# that t(o (ill be red$ 'ive #ello( and t(o bl&e Ans. a -5/ 4-6)5-,))-
3. ; bo+ )ontains 15 red eggs and 20 (hite eggs. 9' 12 eggs are taFen on random$ (hat is the probabilit# that these (ill have an e%&al n&mber o' red and (hite eggsT Ans. /5-#
c /#5-,))-
31. 9n a single thro( o' t(o di)e$ (hat is the probabilit# o' thro(ing not more than 'iveT Ans. '5-6
3!. P&ring a '&nd raising lotter#$ 250 ti)Fets (ill sold to the 'reshman$ o' (hi)h 3 are (inners. Marissa$ a 'reshman has 2 ti)Fets. hat is her probabilit# o' (inning somethingT Ans. #165 -)%/'
32. ind the probabilit# that all 'ive )ards dra(n 'rom a de)F are all heartsT Ans. 1.=' -) +1
40. 9' the probabilit# that Nini (ill go to O 'or a )ertain seminar is 13 and the probabilit# that she (ill go to O that seminar is 14$ 'ind the probabilit# that she (ill go to )ollage in
33. a team o' 5 st&dents is to be )hosen 'or a math )ontest. 9' there (ere ten male and eight 'emale st&dents to )hoose 'rom$ (hat is the probabilit# that three team members (ill be
one o' the t(o s)hools. Ans. /5-#
male and t(o (ill be 'emaleT Ans. #)5'41. 9' the probabilit# that "inebra$ ;lasFa$ and hell (ill (in the 8; open )on'eren)e 34. ; bag )ontains 'ive pairs o' so)Fs. 9' 'o&r so)Fs are )hosen$ (hat is the probabilit# that there is no )omplete pair taFenT Ans. 65#-
)hampionship are 15$ 16$ and 110$ respe)tivel#$ 'ind the probabilit# that one o' theme (ill (in the title. Ans. /5-'
35. 9n the game spin-a-(in the rim o' a (heel is divided in to 30 e%&al parts (ith ea)h
42. he probabilit# that Zoseph strada (ill be nominated to r&n 'ir president is $ and the
marFed 10$ 20$...$300. 20$...$300. he (in is indi)ated b# a 'i+ed pointer at the top. 9' the (heel
probabilit# o' his ele)tion i' nominated is 13. ind the probabilit# t# a o' his being ele)ted
is sp&n$ (hat is the probabilit# that a three-digit n&mber (ill be the pla#er@s taFe home
president$ b o' his being nominated and not being ele)ted. Ans. a -5-# 4-52
(inningT Ans. /5-) 43. ind the probabilit# o' obtaining a 4 in ea)h o' t(o s&))essive tosses o' a pair o' di)e. 36. 9' eight di''erent booFs are arranged at random in a shel'$ (hat is the probabilit# that a
Ans. -5-#=2
)ertain pair o' booFs a (ill be beside ea)h otherT b(ill not be togetherT Ans. a 44. ne bo+ )ontains 'ive bla)F and three (hite handFer)hie's and another seven bla)F and
4
'ive (hite handFer)hie's. 9' one handFer)hie' is dra(n 'rom ea)h bo+$ 'ind the probabilit#
that both (ill be a bla)F$ b (hite$ ) the same )olor. Ans. a %'5=2 c#'516
4
'5%#
52. the probabilit# o' an event happening e+a)tl# t(i)e in 'o&r trials is 1 times the probabilit#
that both (ill be a bla)F$ b (hite$ ) the same )olor. Ans. a %'5=2
4
'5%#
52. the probabilit# o' an event happening e+a)tl# t(i)e in 'o&r trials is 1 times the probabilit#
c#'516
o' it happening e+a)tl# 'ive times in si+ trials$ 'ind the probabilit# that it (ill happen in one 45. he probabilities that Marita (ill (in the preliminar#$ semi'inal$ and 'inal )ontest in singing$
trial. Ans. -5%
are 3$ 16$ and X$ respe)tivel#. ail&re in an# )ontest prohibits parti)ipation in the 'ollo(ing one. ind the probabilit# that she (ill a rea)h the 'inal )ontest b (in the 'inal )ontest. Ans. a -5-2
4 -5-=#
53. 9' the probabilit# that an event (ill happen e+a)tl# three times in 'ive trials is e%&al to the probabilit# that it (ill happen e+a)tl# t(o times in si+ trials$ 'ind the probabilit# that it (ill happen in one trial. Ans. ).1'-
46. hree h#si)s booFs$ 'ive ;lgebra booFs$ and t(o ?hemistr# booFs are on the shel'. Z&dd de)ides to taFe t(o booFs and sele)ts them at random. ind the probabilit# that the 'irst booF dra(n (ill be h#si)s and the se)ond ?hemistr#. Ans. -5-' 47. ind the probabilit# o' thro(ing in three tosses o' a die$ a e+a)tl# t(o 4@s b at least t(o 4@s.
54. Go( man# perm&tations )an be 'ormed 'rom the letters o' the (ord )onstit&tionT Ans. =,=/=,)))
55. Go( man# 'o&r- pla)e n&mbers )an be (ritten &sing the digits 'rom 1 to !T Ans. %,)#1 56. ind in i' n$3= 6 ?n$5 Ans. n(6
Ans. a '5./# 4 #5#/
57. (o di)e are rolled. ind the probabilit# that the s&m o' the t(o di)e is greater than 10. 4. ; bag )ontains three (hite$ 'o&r red$ and 'ive green )andies. ive (ithdra(als o' one )and# ea)h are made$ and the )and# is repla)ed a'ter ea)h. ind the probabilit# that all 'ive (ill be red. Ans.-5#1% 4!. 9' the probabilit# that ;lasFa basFetball team (ill (in the 8; ?on'eren)e ?hampionship is 23$ 'ind the probabilit# that it (ill (in e+a)tl# three )hampionships in 5 #ears.
Ans. -5-#
5. ; pair o' di)e is thro(n. ind the probabilit# o' having a 7 or 11. Ans. #5= 5!. ; pair o' di)e is thro(n. 9' it is Fno(n that one die sho(s a 4$ (hat is the probabilit# that the other die sho(s a 5T Ans. #5--
Ans. 6)5#1%
60. Nine ti)Fets$ n&mbered to 1 to !$ are in the bo+. 9' t(o ti)Fets are dra(n at random$ 50. i+ ;lgebra booFs$ 'o&r h#si)s booFs$ and t(o ?hemistr# booFs are on the table. 9' a
determine the probabilit# that both are odd. Ans. '5-6
booF is removed and repla)ed$ then another is removed and repla)ed$ and so on &ntil si+ removals and repla)ements have been made$ 'ind the probabilit# that an ;lgebra booF (as removed and repla)ed a three times b at least three three times. Ans. -25#1% 51. 9' the probabilit# that 9melda (ill be ele)ted to o''i)e is 23$ 'ind the probabilit# that she (ill be ele)ted 'or s&))essive terms and then de'eated on the 'i'th term. Ans. -25#1%
63. ; bag )ontains ! balls n&mbered 1 to !. (o balls are dra(n at random. ind the
61. ; )ommittee o' three is to be )hosen 'rom a gro&p o' 5 men and 4 (omen. 9' the sele)tions is made at random$ 'ind the probabilit# that t(o are men. Ans. -)5#62. hree balls are dra(n 'rom bo+ )ontaining 5 red$ bla)F$ and 4 (hite balls. Petermine the probabilit# that all are (hite. Ans. -5-/)
63. ; bag )ontains ! balls n&mbered 1 to !. (o balls are dra(n at random. ind the probabilit# that one is even and the other is odd. Ans.'5= 64. Go( man# )ars )an be given li)ense plates having 5 digits n&mbers &sing the digits 1$2$3$4 Y 5 (ith no digit repeated in an# li)ense plateT Ans. -#) 65. ; )ommittee o' 4 is sele)ted b# lot 'rom a gro&p o' 6 men and 4 (omen. hat (ill be the probabilit# that (ill )onsist o' e+a)tl# 2 men and 2 (omenT Ans. %5/ 66. here are 52 ti)Fets ti)Fets in lotter# lotter# in (hi)h there is a 'irst and a se)ond priQe. priQe. hat is the probabilit# o' a man dra(ing a priQe i' he o(ns 5 ti)FetsT Ans. ).-6%2/ 67. here are three )andidates 'or ;$ 8$ and ? 'or ma#or o' a )ertain to(n. 9' the odds are 7:5 that )andidate ; (ill (in and those o' 8 are 1:3$ (hat is the probabilit# that )andidate ? (ill (inT Ans. -52 6. ; )oin is biased so that the head is t(i)e as liFel# to o))&r as tail. 9' the )oin is tossed 3 times$ (hat is the probabilit# o' getting a 2 tails and 1 headT Ans. #5=
b at least 2 headsT Ans. #)5#/
6!. hree man are r&nning 'or p&bli) o''i)e. ? )andidates ; Y 8 are given abo&t the same )han)e o' (inning. 8&t )andidate ? is given t(i)e the )han)e o' either ; and 8. ind the probabilit# that: a ? (ins Ans. a
b ; does not (in 4%51
70. ; pla#er sinFs 50[ o' all his shots. hat is the probabilit# that he (ill maFe e+a)tl# 3 o' his ne+t 4 shotsT Ans. #'
