Problem Sets Ce Review 2014

ENGINEERING ECONOMICS SET 1

simple interest of 6 % per ann"m Ans. P30,000.00

SIMPLE INTEREST 

1. Find Find the the intere interest st on P6800.00 for 3 years at 11 % simple interest. Ans. P2,244.00 2. A man man borr borrow owed ed P10,000.00 from his friend and arees to pay at the end of !0 days "nder 8 % simple interest rate. #hat is the re$"ired amo"nt Ans. P10,200.00 3. Annie b"ys a tele&is tele&ision ion set from a mer'hant who o(ers P2),000.00 at the end of 60 days. Annie wishes to pay immediately and the mer'hant o(ers to 'omp"te the re$"ired amo"nt on the ass"mption that money is worth 14 % simple interest. #hat is the re$"ired amo"nt Ans. P24,42!.!* 4. #hat is the prin'ipal prin'ipal amo"nt if the amo"nt of interest at the end of  2 + year is P4)00 for a

). ow lon lon m"st m"st a P40, P40,000 000 note bearin 4 % simple interest r"n to amo"nt to P41,3)0.00 Ans. 304 days 6. -f P16,000 P16,000 earns earns P480 P480 in ! months, what is the ann"al rate of interest Ans. 4% *. A time time depo deposit sit of of P110,000 for 31 days earns P8!0.3! on mat"rity date after ded"'tin the 20 % withholdin ta on interest in'ome. Find the rate of interest per ann"m. Ans. 11.*) % 8. A ban/ ban/ 'har 'hares es 12 % simple interest on a P300.00 loan. ow m"'h will be repaid if the load is paid ba'/ in one l"mp s"m after three years. Ans. P408.00 !. A pri'e pri'e ta ta of P1200 P1200 is is spe'ied if paid within 60 days b"t o(ers a 3 % dis'o"nt for 'ash in 30

days. Find the rate of interest. Ans. 3*.11 % 10.-t is the pra'ti'e of almost all ban/ in the Philippines that when they rant a loan, the interest for one year is a"tomati'ally ded"'ted from the prin'ipal amo"nt "pon release on money to a borrower. et "s therefore ass"me that yo" applied for a loan with a ban/ and the P80000 was appro&ed at an interest rate of 14 % of whi'h P11200 was ded"'ted and yo" were i&en a 'he'/ of P68800. in'e yo" ha&e to pay the amo"nt of P80000 one year after, what then will be the e(e'ti&e interest rate Ans. 16.28 % 11.r. . de la 5r" borrowed money from a ban/. e re'ei&ed from the ban/ P1,340.00 and promised to pay P1,)00.00 at the end of ! months. 7etermine the simple interest rate and the 'orrespondin dis'o"nt rate or often

referred to as the 9an/ers dis'o"nt:. Ans. 1).!2 % ; 13.*3 % 12.A man borrowed borrowed from a ban/ "nder a promissory note that he sined in the amo"nt pf P2),000.00 for a period of one year. e re'ei&ed only the amo"nt of P21,!1).00 after the ban/ 'olle'ted the ad&an'e interest and an additional amo"nt of P8).00 for notarial and inspe'tion fees. #hat was the rate of interest that the ban/ 'olle'ted in ad&an'e Ans. 13.64 % 13.Anes Abanilla was ranted a loan of P20,000 by her employer 5P -nd"strial Fabri'ator and 5onstr"'tion 5orporation with an interest at 6 % for 180 days on the prin'ipal 'olle'ted in ad&an'e.
the pro'eeds on the note. Ans. P18,800 14.P4000 is borrowed borrowed for *) days at 16 % per ann"m simple interest. ow m"'h will be d"e at the end of *) days Ans. P4133.33 1).r. Almaro made a money mar/et pla'ement of P1,000,000 for 30 days at *.) % per year. -f the withholdin ta is 20 %, what is the net interest that r. Almaro will re'ei&e at the end of the month Ans. P)000

ENGINEERING ECONOMICS SET 2

16.A bill for motorboat spe'ies the 'ost as P1200 d"e at the end of 100 days b"t o(ers a 4 % dis'o"nt for 'ash in 30 days. #hat is the hihest rate, simple interest at whi'h the

b"yer 'an a(ord to borrow money in order to ta/e ad&antae of the dis'o"nt Ans. 21.4 % 1*.-n b"yin a 'omp"ter dis/, the b"yer was o(ered the options of payin P2)0 'ash at the end of 30 days or P2*0 at the end of 120 days. At what rate is the b"yer payin simple interest if he aree to pay at the end of 120 days. Ans. 32 % 18.Find the dis'o"nt if P2000 is dis'o"nted for 6 months at 8 % simple dis'o"nts. Ans. P80.00 1!.5omp"te the dis'o"nt if P2000 is dis'o"nted for 6 months at 8 % simple interest. Ans. P*6.!2 20.>n ar'h 1, 1!!6 r. Almaro obtains a loan of  P1)00 from r. Abella and sins a note promisin to pay the prin'ipal and a''"m"lated simple interest at the rate of ) % at the end of 120 days.

>n ay 1), 1!!6, r. Abella dis'o"nts the note at the ban/ whose dis'o"nt rate is 6 %. #hat does he re'ei&ed Ans. P1)13.)6 21.A deposit of P110,000 was made for 31 days.  
was the rate of dis'o"nt and the rate of interest that the ban/ 'olle'ted in ad&an'e. Ans. 4 % ; 4.1* % COMPOUND INTEREST 

2).A''"m"late P),000.00 for 10 years at 8 % 'ompo"nded $"arterly. Ans. P11,040.20 26.A''"m"late P),000.00 for 10 years at 8 % 'ompo"nded semi = ann"ally. Ans. P10,!)).61 2*.A''"m"late P),000.00 for 10 years at 8 % 'ompo"nded monthly. Ans. P11,0!8.20 28.A''"m"lates P),000.00 for 10 years at 8 % 'ompo"nded ann"ally. Ans. P10,*!4.62 2!.ow lon will it ta/e P1,000 to amo"nt to P1,346 if in&ested at 6 % 'ompo"nded $"arterly Ans. ) years 30.ow lon will it ta/e for an in&estment to do"ble its amo"nt if in&ested at an interest rate of 6 % 'ompo"nded bi = monthly Ans. 12 years 2

the pro'eeds on the note. Ans. P18,800 14.P4000 is borrowed borrowed for *) days at 16 % per ann"m simple interest. ow m"'h will be d"e at the end of *) days Ans. P4133.33 1).r. Almaro made a money mar/et pla'ement of P1,000,000 for 30 days at *.) % per year. -f the withholdin ta is 20 %, what is the net interest that r. Almaro will re'ei&e at the end of the month Ans. P)000

ENGINEERING ECONOMICS SET 2

16.A bill for motorboat spe'ies the 'ost as P1200 d"e at the end of 100 days b"t o(ers a 4 % dis'o"nt for 'ash in 30 days. #hat is the hihest rate, simple interest at whi'h the

b"yer 'an a(ord to borrow money in order to ta/e ad&antae of the dis'o"nt Ans. 21.4 % 1*.-n b"yin a 'omp"ter dis/, the b"yer was o(ered the options of payin P2)0 'ash at the end of 30 days or P2*0 at the end of 120 days. At what rate is the b"yer payin simple interest if he aree to pay at the end of 120 days. Ans. 32 % 18.Find the dis'o"nt if P2000 is dis'o"nted for 6 months at 8 % simple dis'o"nts. Ans. P80.00 1!.5omp"te the dis'o"nt if P2000 is dis'o"nted for 6 months at 8 % simple interest. Ans. P*6.!2 20.>n ar'h 1, 1!!6 r. Almaro obtains a loan of  P1)00 from r. Abella and sins a note promisin to pay the prin'ipal and a''"m"lated simple interest at the rate of ) % at the end of 120 days.

>n ay 1), 1!!6, r. Abella dis'o"nts the note at the ban/ whose dis'o"nt rate is 6 %. #hat does he re'ei&ed Ans. P1)13.)6 21.A deposit of P110,000 was made for 31 days.  
was the rate of dis'o"nt and the rate of interest that the ban/ 'olle'ted in ad&an'e. Ans. 4 % ; 4.1* % COMPOUND INTEREST 

2).A''"m"late P),000.00 for 10 years at 8 % 'ompo"nded $"arterly. Ans. P11,040.20 26.A''"m"late P),000.00 for 10 years at 8 % 'ompo"nded semi = ann"ally. Ans. P10,!)).61 2*.A''"m"late P),000.00 for 10 years at 8 % 'ompo"nded monthly. Ans. P11,0!8.20 28.A''"m"lates P),000.00 for 10 years at 8 % 'ompo"nded ann"ally. Ans. P10,*!4.62 2!.ow lon will it ta/e P1,000 to amo"nt to P1,346 if in&ested at 6 % 'ompo"nded $"arterly Ans. ) years 30.ow lon will it ta/e for an in&estment to do"ble its amo"nt if in&ested at an interest rate of 6 % 'ompo"nded bi = monthly Ans. 12 years 2

31.-f yo" borrowed borrowed P10,000 from a ban/ with 18 % interest per ann"m, what is the total amo"nt to be repaid at the end of one year Ans. P11,800.00 32.#hat is the e(e'ti&e rate for an interest rate of 12 % 'ompo"nded 'ontin"o"sly Ans. 12.*) % 33.ow lon will it ta/e for an in&estment to &efold its amo"nt if money is worth 14 % 'ompo"nded semi = ann"ally Ans. 12 34.An interest rate of 8 % 'ompo"nded semi = ann"ally is how many per'ent if 'ompo"nded $"arterly Ans. *.!2 % 3).A man is epe'tin to re'ei&e P4)0,000.00 at the end of * years. -f money is worth 14 % 'ompo"nded $"arterly, how m"'h is it worth at present Ans. P1*1,*44.4)

ann"al interest rate is 8 %. Ans. P1)2.8* ENGINEERING ECONOMICS SET 3

36.A man has a will of P6)0,000.00 from his father. -f his father deposited an amo"nt of P4)0,000.00 in a tr"st f"nd earnin 8 % 'ompo"nded ann"ally, after how many years will the man re'ei&e his will Ans. 4.*8 years 3*.r. Adam deposited P120,000.00 in a ban/ who o(ers 8 % interest 'ompo"nded $"arterly. -f the interest is s"b@e't to a 14 % ta, how m"'h will he re'ei&e after ) years Ans. P1*0,14!.** 38.#hat interest 'ompo"nded monthly is e$"i&alent to an interest rate of 14 % 'ompo"nded $"arterly Ans. 13.84 % 3!.#hat is the present worth of two P100.00 payments at the end of the third and the fo"rth year
40.A rm borrows borrows P2000.00 for 6 years at 8 %. At the end of 6 years, it renews the loan for the amo"nt d"e pl"s P2000 more for 2 years at 8 %. #hat is the l"mp s"m d"e. Ans. P6,034.66 41.At an ann"al rate of ret"rn of 8 %, what is the f"t"re worth of P1000 at the end of 4 years Ans. P1360.)0 42.A st"dent has money i&en by his randfather in the amo"nt of P20,000.00. ow m"'h money in the form of interest will he et if the money is p"t in a ban/ that o(ers 8 % rate 'ompo"nded ann"ally, at the end of * years Ans. P34,2*6.48 43.-f the interest rate on an a''o"nt is 11.) % 'ompo"nded yearly, approimately approimately how many years will it ta/e to triple the amo"nt

Ans. 10 years 44.
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years in a 'ontin"o"sly 'ompo"nded a''o"nt Ans. P6,*)0.00 4!.A mer'hant p"ts in his P2,000.00 to a small b"siness for a period of si years. #ith a i&en interest rate on the in&estment of 1) % per year, 'ompo"nded ann"ally, how m"'h will he 'olle't at the end of the sith year Ans. P4,626.00 )0.A person in&ests P4,)00.00 to be 'olle'ted in 8 years. i&en that the interest rate on the in&estment is 14.) % per year 'ompo"nded ann"ally, what s"m will be 'olle'ted in 8 years Ans. P13,2!4.02 )1.-n year ero, yo" in&est P10,000.00 in a 1) % se'"rity for ) years. 7"rin that time, the a&erae ann"al inBation is 6 %. ow m"'h, in terms of year ero pesos, will be in the a''o"nt at the mat"rity Ans. P1),030.00

)2.A 'ompany in&ests P10,000.00 today to be repaid in &e years in one l"mp s"m at 12 % 'ompo"nded ann"ally. -f the rate of inBation is 3 % 'ompo"nded ann"ally, how m"'h prot in present day pesos is realied o&er &e years Ans. P),202.00 )3.5omp"te the e(e'ti&e rate for an interest rate of  16 % 'ompo"nded ann"ally. Ans. 16 % )4.5omp"te the e(e'ti&e rate for an interest rate of  16 % 'ompo"nded $"arterly. Ans. 16.!8 %

ENGINEERING ECONOMICS SET 4

)).5on&ert 12 % 'ompo"nded semi = ann"ally to  % 'ompo"nded $"arterly. Ans. 11.83 %

)6.5on&ert 12 % 'ompo"nded semi = ann"ally to  % 'ompo"nded monthly. Ans. 11.*1 % )*.A ban/ is ad&ertisin !.) % a''o"nts that yield !.84 % ann"ally. ow often is the interest 'ompo"nded aC 7aily bC onthly 'C 9i = monthly dC D"arterly Ans. $"arterly )8.9y the 'onditions of a will, the s"m of P2),000.00 is left to a irl to be held in a tr"st f"nd by her "ardian "ntil it amo"nt to P4), 000.00. #hen will the irl re'ei&e the money if the f"nd is in&ested at 8 % 'ompo"nded $"arterly Ans. *.42 years )!.#hat is the e(e'ti&e rate 'orrespondin to 16 % 'ompo"nded daily  
'onsiderin interest at 6 % 'ompo"nded $"arterly Ans. P11,02).2) 61.#hat amo"nt will be a''"m"lated by a present in&estment of P1*,200 in 6 years at 2 % 'ompo"nded $"arterly Ans. P1!,38*.1) 62.#hat rate of interest 'ompo"nded ann"ally m"st be re'ei&ed if an in&estment of P)4,000.00 made now will res"lt in a re'eipt of P*2,000.00 ) years hen'e Ans. ).!2 % 63.#ith interest at 6 % 'ompo"nded ann"ally, how m"'h is re$"ired * years hen'e to repay an P8  loan made today Ans. P12,02!,042 64.-f money is worth 6 % 'ompo"nded ann"ally, what payment 12 years from now is e$"i&alent to a payment of P*,000.00 ! years from now Ans. P8,33*.10

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6).-f money is worth 6 % 'ompo"nded ann"ally, how m"'h 'an be loaned now if P6,000.00 will be repaid at the end of 8 years. Ans. P3,*64.)0 66.A person in&est P4,)00 to be 'olle'ted in 8 years. i&en that the interest rate on the in&estment is 14.) % per year, 'ompo"nded ann"ally, what s"m, in pesos, will be 'olle'ted eiht years hen'e Ans. P13,2!4.00 6*.-f P)00,000 is deposited at a rate of 11.2) % 'ompo"nded monthly, determine the 'ompo"nded interest after * years and ! months. Ans. 6!0,84! 68.P200,000 was deposited on an"ary 1, 1!88 at an interest rate of 24 % 'ompo"nded semi = ann"ally. ow m"'h wo"ld the s"m on  an"ary 1, 1!!3 Ans. P621,1*0.00 6!.Find the nominal rate that if 'on&erted $"arterly 'o"ld be "sed

instead for 1) % 'ompo"nded semi = ann"ally Ans. 14.*3 % *0.#hi'h of the followin has the least e(e'ti&e ann"al interest rate aC 12 % 'ompo"nded $"arterly bC 11.) % 'ompo"nded monthly 'C 11.* % 'ompo"nded semi = ann"ally dC 12.2 % 'ompo"nded ann"ally Ans. 11.* % 'ompo"nded semi  ann"ally *1.>ne h"ndred tho"sand pesos was pla'ed in a time deposit that earns ! % 'ompo"nded $"arterly, ta free. After how many years wo"ld it be able to earn a total interest of fty tho"sand pesos Ans. 4.)6 *2.
Ans. P4,214.40 *4.
*).#hat rate G%C 'ompo"nded $"arterly is e$"i&alent to 6 % 'ompo"nded semi = ann"ally Ans. ).!6 % *6.Fifteen per'ent G1) %C when 'ompo"nded semi = ann"ally will ha&e e(e'ti&e rate of ? Ans. 1).)6 % **.#hat rate of interest 'ompo"nded ann"ally is the same as the rate of interest of 8 % 'ompo"nded $"arterly Ans. 8.24 % *8.ow lon will it ta/e the money to triple itself if in&ested at 10 % 'ompo"nded semi = ann"ally Ans. 11.3 years *!.#hat is the a''"m"lated amo"nt after three G3C

years of P6,)00.00 in&ested at the rate of 12 % per year 'ompo"nded semi = ann"ally Ans. P!,221.00 80.#hat interest rate, 'ompo"nded monthly is e$"i&alent to 10 % e(e'ti&e rate Ans. !.)* % 81.A man wishes his son to re'ei&e P)00,000.00 ten years from now. #hat amo"nt sho"ld he in&est now if it will earn interest of 12 % 'ompo"nded ann"ally d"rin the rst ) years and 1) % 'ompo"nded $"arterly d"rin the net ) years Ans. P12!,8)3.48 82.A sa&ins asso'iation pays 4 % interest $"arterly. #hat is the e(e'ti&e ann"al interest rate Ans. 16.!8) % 83.A ban/ o(ers 1.2 % e(e'ti&e monthly interest. #hat is the e(e'ti&e ann"al rate with monthly 'ompo"ndin Ans. 1).4 % 5

84.#hat is the present worth of P2*,000.00 d"e in 6 years if money is worth 13 % and is 'ompo"nded semi = ann"ally Ans. P12,681.00 8).
held in tr"st f"nd by her "ardian "ntil it amo"nts to P)0,000.00. #hen will the irl re'ei&e the money if the f"nd is in&ested at 8 % 'ompo"nded $"arterly Ans. 11.)* years 8!.A man epe'ts to re'ei&e P2),000 in 8 years. ow m"'h is that worth now 'onsiderin interest at 8 % 'ompo"nded $"arterly Ans. P13,26).83 !0.P)00,000 was deposited at an interest of 6 % 'ompo"nded $"arterly. 5omp"te the 'ompo"nd interest after 4 years and ! months. Ans. P163,4*).3* !1.-f the nominal interest rate is 3 %, how m"'h is P),000 in 10 years in a 'ontin"o"sly 'ompo"nded a''o"nt Ans. P6,*)0 !2.P200,000 was deposited for a period of 4 years and 6 months and bears on interest of P8),64!.2). what is the rate of interest if it is 'ompo"nded $"arterly

Ans. 8% !3.ow many years will P100,00 earned a 'ompo"nd interest rate is ! % 'ompo"nded $"arterly Ans. 4.)) years !4.A 'ertain amo"nt was deposited ) years and ! months ao at an interest of 8 % 'ompo"nded $"arterly. -f  the s"m mow is P31),3*!.8), how m"'h was the amo"nt deposited Ans. P200,000 ENGINEERING ECONOMICS SET 6

!).-f the rate of interest is 12 % 'ompo"nded ann"ally , nd the e$"i&alent rate of interest if it is 'ompo"nded $"arterly. Ans. 11.4! % !6.ow lon will it ta/e money to do"ble itself if in&ested at ) % 'ompo"nded ann"ally Ans. 14 years !*.5omp"te the e(e'ti&e ann"al interest rate

whi'h is e$"i&alent to ) % nominal ann"al interest 'ompo"nded 'ontin"o"sly. Ans. ).13 % !8.5omp"te the nominal interest rate of a 'ontin"o"sly 'ompo"nded loan if the e(e'ti&e interest rate is 2) % Ans. loge 1.2) !!.A man wishes to ha&e P40,000 in a 'ertain f"nd at the end of 8 years. ow m"'h sho"ld be in&est in a f"nd that will pay 6 % 'ompo"nded 'ontin"o"sly Ans. P24.*)1.34 100. -f the e(e'ti&e ann"al interest rate in 4 %, 'omp"te the e$"i&alent nominal ann"al interest 'ompo"nded 'ontin"o"sly. Ans. 3.!2 % 101. #hat is the nominal rate of interest 'ompo"nded 'ontin"o"sly for 10 years if the 'ompo"nd

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amo"nt fa'tor is e$"al to 1.34!86 Ans. 3% 102. Ameri'an Hpress 5orp. 'hares 1 + % interest per month, 'ompo"nded 'ontin"o"sly on the "npaid balan'e p"r'hases made on this 'redit 'ard. 5omp"te the e(e'ti&e rate of interest. Ans. 1!.*2 % 103. -f the nominal interest is 12 % 'ompo"nded 'ontin"o"sly, 'omp"te the e(e'ti&e ann"al rate of interest. Ans. 12.*) % 104. 5omp"te the di(eren'e in the f"t"re amo"nt of P)00 'ompo"nded ann"ally at nominal rate of ) % and if it is 'ompo"nded 'ontin"o"sly for ) years at the same rate. Ans. P3.8* 10). -f the e(e'ti&e interest rate is 24 %, what nominal rate of interest is 'hared for a

'ontin"o"sly 'ompo"nded loan Ans. 21.)1 % 106. #hat is the nominal rate of interest 'ompo"nded 'ontin"o"sly for 8 years if  the present worth fa'tor is e$"al to 0.618*83) Ans. 6% 10*. #hat is the di(eren'e of the amo"nt 3 years from now for a 10 % simple interest and 10 % 'ompo"nd interest per year Ans. P1)) 108. Find the dis'o"nt if P2,000 is dis'o"nted for 6 months at 8 % 'ompo"nd $"arterly. Ans. P**.66 10!. -f a s"m of money triples in a 'ertain period of time at a i&en rate of interest, 'omp"te the &al"e of the sinle payment present worth fa'tor. Ans. 0.333 110. -f the sinle payment amo"nt fa'tor for a period of ) years is 1.33822. #hat is the nearest &al"e of the interest rate

Ans.

6%

111. -f the sinle payment present worth fa'tor for a period of 8 years is e$"al to 0.)8201, 'omp"te the nearest &al"e of the rate of interest for that period. Ans. *% 112. -f money is worth 8 % 'ompo"nded $"arterly, 'omp"te the sinle payment amo"nt fa'tor for a period of 6 years. Ans. 1.60844 113. A s"m of P1,000.00 is in&ested now and left for 8 years, at whi'h time the prin'ipal is withdrawn.
11). ow m"'h sho"ld yo" p"t into a 10 % sa&ins a''o"nt in order to ha&e P10,000.00 in &e years Ans. P6,210.00

FLUID MECHANICS and HYDRAULICS SET 1 PROPERTIES OF FLUIDS

1.

A reser&oir of ly'erin has a mass of 120 / and a &ol"me of 0.!)2 m3. Find its weiht, mass density, spe'i' weiht and spe'i' ra&ity. Ans. 11.** /I, 1261 /Jm3, 12.36 /IJm 3, 1.26 2.
at a rate of 0.20 mJs 2. 7etermine the mass of the body in / and in sl"s. Ans. )00 /, 34.3 sl"s 4.
-f an ob@e't has a mass of 22 / at sea le&el. aC #hat will be its

weiht at a point where the a''eleration d"e to ra&ity  E !.*) mJs 2 bC #hat will be its mass at that point Ans. 214.) I, 22 / !. >ne '"bi' foot of ly'erine has a mass of 2.44 sl"s. Find its spe'i' weiht in lbJft 3 and /IJm 3. Ans. *8.6 lbJft3, 12.3) /IJm3 10.
14.An ob@e't at a 'ertain lo'ation has a mass of 2.0 / and weihs 1!.0 I on a sprin balan'e. #hat is the a''eleration d"e to ra&ity at this lo'ation Ans. !.)0 mJs 2 1). -f an ob@e't has a mass of 2.0 sl"s at sea le&el, what wo"ld be its mass be at a lo'ation where the a''eleration d"e to ra&ity is 30.00 ftJs 2 Ans. 2.0 sl"s 16. #hat wo"ld wo"ld be the weiht of a 3 / mass on a planet where the a''eleration d"e to ra&ity is 10.00 mJs 2 Ans. 30.00 I 1*. 7etermine the weiht weiht of a )sl" bo"lder at a pla'e where the a''eleration d"e to ra&ity is 31.* ftJs 2. Ans. 1)8 lb 18. -f 200 ft3 of oil weihs 10)20 lb, 'al'"late its spe'i' weiht, density and spe'i' ra&ity. Ans. )2.6 lbJft3, 1.63 sl"sJft3, 0.843 1!. Find the heiht of the free s"rfa'e if 0.8 ft 3 of

water is po"red into a 'oni'al tan/ 20 in hih with a base radi"s of 10 in. ow m"'h additional water is re$"ired to ll the tan/ Ans. *13 in 3 20. A 'oni'al tan/ 20 in hih and a base radi"s 10 in., holds 30.) / of salad oil, what is the density of the oil Ans. 88! /Jm3 UNIT WEIGHT OF AIR AND GASES

21. #hat is the spe'i' weiht of air at 480 /Pa absol"te and 21o5 Ans. )).81 IJm3 22. A as at 20o5 and 0.21 pa abs has a &ol"me of  41  and a as 'onstant K of 210 ImJ/L. 7etermine the density and mass of the as. Ans. 3.41 /Jm3, 0.140 / 23. Find the mass density of  heli"m at a temperat"re temperat"re of 4o5 and a press"re of 184 /Pa ae, if atmospheri' press"re is

8

101.!2 /Pa. K E 20*!  J/ L. Ans. 0.4!6) /Jm3 24. 5al'"late the density of water &apor at 3)0 /Pa abs and 20o5 if its as 'onstant K is 0.462 /Pa m 3J / L. Ans. 2.)! /Jm3 2). At 32o5 and 20) /Pa ae, the spe'i' weiht of a 'ertain as was 13.* IJm 3. 7etermine the as 'onstant of this as. Ans. *18.8* J/ L  FLUID MECHANICS and  HYDRAULICS SET 2

26.

#hat is the spe'i' weiht if air at a temperat"re of 30 o5 and a press"re of 4*0 /Pa abs. Ans. 0.0)2! /IJm3 2*. Air is /ept at a press"re of 200 /Pa abs and a temperat"re of 30o5 in a )00 liter 'ontainer. #hat is the mass of air Ans. 1.1) / 28.A as at 40 o5 "nder a press"re press"re of 21.868 bar

abs has a "nit weiht of 362 IJm3. #hat is the &al"e of K for this as Ans. 1!.3 mJL   ABSOLUTE AND KINEMATIC KINEMATIC VISCOSITY 

2!. -f the &is'osity of water at 68oF is 0.01008 poise, 'omp"te its absol"te &is'osity in po"ndse'ond per s$"are foot. -f the spe'i' ra&ity at 68 oF is 0.!!8, 'omp"te its /inemati' &is'osity in s$"are feet per se'ond. Iote? 1 lb.sJft 2 E 4*8.8 poises. Ans. 2.1110 ) lb.sJft 2, 1.0!10) ft2Js. 30.-f the &is'osity of water at *0o5 is 0.00402 poise and its spe'i' ra&ity is 0.!*8, determine its absol"te &is'osity in Pas and its /inemati' &is'osity in m 2Js and in sto/es. Ans. 0.000402 Pas, 4.1110* m2Js, 4.1110 3 sto/e 31.5on&ert 1).14 poises to /inemati' &is'osity in ft 2Js if the li$"id has a spe'i' ra&ity of 0.!64. Ans. 0.016! ft 2Js

32.
33. A li$"id 'ompressed in a 'ylinder has a &ol"me of  1000 'm 3 at 1 IJm 2 and a &ol"me of !!) 'm 3 at 2 IJm2. #hat is its b"l/ mod"l"s of elasti'ity, L Ans. 200 Pa 34. #hat press"re is re$"ired to red"'e the &ol"me of water by 0.6% 9"l/ mod"l"s of elasti'ity of water, H E 2.2 Pa. Ans. 13.2 Pa 3). Find the b"l/ mod"l"s of elasti'ity of a li$"id if a press"re of 1)0 psi applied to 10 ft 3 of the

li$"id 'a"ses a &ol"me red"'tion of 0.02 ft 3. Ans. *)000 psi 36. Find the 'hane in &ol"me of 1 ft 3 of water at 80oF when s"b@e'ted to a press"re in'rease of  300 psi. #aterMs b"l/ mod"l"s of elasti'ity at this temperat"re is 32) 000 psi Ans. 0.000!2 ft3 3*. #ater in a hydra"li' press, initially at 13* /Pa abs, is s"b@e'ted to a press"re press"re of 116,280 /Pa abs. Nsin H E 2.) Pa, determine the per'entae de'rease in the &ol"me of water. Ans. 4.6)% de'rease COMPRESSION OF GASSES

38.

-f a &essel that 'ontains 3,)00 ft 3 of water at )0 oF G62.4 p'fC atmospheri' press"re is heated to 160 oF G61 p'fC, what will be the per'entae 'hane in its &ol"me #hat weiht of water m"st be remo&ed to maintain the oriinal &ol"me Ans. 2.*% in'rease, 4.!0 lb 9

3!. A &erti'al, 'ylindri'al tan/ with a diameter of 12.00 m and a depth of  4.00 m is lled to the top with water at 20 o5 G!.*! /IJm3C. -f the water is heated )0 o5 G!.6! /IJm3C, how m"'h water will spill o"t Ans. 4.* m3 40. -f ! m 3 of an ideal as at 24 o5 and 1)0 /Paa is 'ompressed to 2 m3, GaC what is the res"ltin press"re ass"min isothermal 'onditions, GbC what wo"ld ha&e been the press"re and temperat"re if the pro'ess is isentropi'. isentropi'. Nse / E 1.3. Ans. 6*) /Paa, 1060 /Paa, 466.4 L or 1!3.4o5 41.eli"m at 14! /Pa abs and 10 o5 is isentropi'ally 'ompressed to one forth of its oriinal &ol"me. #hat is its nal &ol"me / E 1.66 Ans. 1488 /Pa abs 42. -f 12 m 3 of nitroen at 30o5 and 12) /Pa abs is

permitted to epand isothermally to 30 m 3, what is the res"ltin press"re press"re #hat wo"ld ha&e been if the pro'ess had been isentropi' / E 1.40 Ans. )0 /Paa, 34.* /Paa SURFACE SURFACE TENSION AND CAPILLARITY 

43. #hat is the &al"e of the s"rfa'e tension of a small drop of water 0.) mm in diameter whi'h is in 'onta't with air if the press"re within the droplet is )61 Pa Ans. 0.042 IJm FLUID MECHANICS and  HYDRAULICS SET 3

44.

A small drop of water at 80oF is in 'onta't with the air and has a diameter of 0.02 in. -f the press"re within the droplet is 0.082 psi reater than the atmosphere, atmosphere, what is the &al"e of the s"rfa'e tension Ans. 0.004!2 lbJft

4). An atomier forms water droplets 4) µm in diameter. 7etermine the e'ess press"re within these droplets "sin σ E 0.0*12 IJm. Ans. 632! Pa 46.
in a lass t"be 2 mm diameter. Nse σ E 0.)14 IJm and θ E 140 o. Ans. ).! mm depression )0. Hstimate the depression h for mer'"ry in the lass 'apillary t"be. Anle θ E 40o, "nit weiht of water is !*!0 IJm3. Ans. ).!1 mm )1. Find the 'apillary rise in the t"be for a mer'"ry airlass airlass interfa'e with θ E 130o if the t"be radi"s is 1 mm and the temperat"re is 20 o5. "rfa'e tension is 0.)14 IJm. Ans. =).0 mm )2. Find the 'apillary rise in the t"be for a water airlass interfa'e θ E 0o if the t"be is 1 mm and the temperat"re is 20 o5. "rfa'e tension is 0.0*28 IJm. Ans. 14.8 mm )3. -f a b"bble is e$"i&alent to an air water interfa'e with s"rfa'e tension 0.00) lbJft, what is the press"re di(eren'e 10

between the inside and o"tside of a b"bble of diameter 0.003 in Ans. 80.0 lbJft2 o )4.At 30 5, what diameter lass t"be is ne'essary to /eep the 'apillary heiht 'hane of water less than 2 mm "rfa'e tension is 0.0*12 IJm. Ans. 14.6 mm )).#hat for'e is re$"ired to lift a thin wire rin 6 'm in diameter from a water s"rfa'e at 20 o5 "rfa'e tension is 0.0*28 IJm. Ans. 0.02*4 I )6.#hat is the press"re di(eren'e between the inside and o"tside of a 'ylindri'al water @et when the diameter is 2.2 mm and the temperat"re is 10 o5 "rfa'e tension is 0.0*42 IJm. Ans. 6*.) Pa )*.7istilled water stands in a lass t"be of ! mm diameter at a heiht of 24 mm. #hat is the tr"e stati' heiht Nse σ E 0.0*42 IJm. Ans. 20.64 mm

)8.Find the anle the s"rfa'e tension lm lea&es the lass for a &erti'al t"be immersed in water if the diameter is 0.2) in and the 'apillary rise is 0.08 in. Nse σ E 0.00) lbJft )!.7istilled water at 10 o5 stands in a lass t"be of !.0 mm diameter at a heiht of 24.0 mm. #hat is the tr"e stati' heiht Nse σ E 0.0*42 IJm. FLUID S TATICS

60.A depth of li$"id of 1 m 'a"ses a press"re of * /Pa, what is the spe'i' ra&ity of li$"id Ans. 0.*14 61.#hat is the press"re 12.) m below the o'ean Nse sp. r. E 1.03 for salt water. Ans. 126.3 /Pa 62. -f the press"re 23 m below a li$"id is 338.44) /Pa, determine its "nit weiht, mass density, and spe'i' ra&ity. Ans. 14.*1) /IJm 3, 1)00 /Jm 3, 1.) 63.-f the press"re at a point in the o'ean is 60 /Pa, what is the press"re 2* m below this point

Ans. 332.82 /Pa 64.-f the press"re in the air spa'e abo&e an oil Gs E 0.*)C s"rrfa'e in a 'losed tan/ is 11) /Pa absol"te, what is the press"re 2 m below the s"rfa'e Ans. 28.41) /Pa 6).Find the absol"te press"re in /Pa at a depth of 10 m below the free s"rfa'e of oil of sp. r. E 0.*) of the barometri' readin is *)2 mm. Ans. 1*3.! /Pa FLUID MECHANICS and  HYDRAULICS SET 4

66.A press"re ae 6 m abo&e the bottom of the tan/ 'ontainin a li$"id reads !0 /Pa.; another ae heiht 4 m reads 103 /Pa. 7etermine the spe'i' weiht of the li$"id. Ans. 6.) /IJm 3 6*.An open tan/ 'ontains ).8 m water 'o&ered with 3.2 m of /erosene G γ  E 8 /IJm 3C. Find the

press"re at the interfa'e and the bottom of the tan/. Ans. 2).6 /Pa, 82.4!8 /Pa 68.-f atmospheri' press"re is !).* /Pa and the ae atta'hed to the tan/ reads 188 mm &a'""m, nd the absol"te press"re within the tan/. Ans. *0.62 /Pa  6!. A weather report indi'ates the barometri' press"re is 2!.*) in of mer'"ry. #hat is the atmospheri' press"re in psi Ans. 14.61 psi *0.A press"re ae *.0 m abo&e the bottom of a tan/ 'ontainin a li$"id reads 64.!4 /Pa, another ae at heiht 4.0 m reads 8*.)3 /Pa. 5omp"te the spe'i' weiht and mass density of the B"id. Ans. *)60 IJm3, *86 /Jm3 *1.An open tan/ 'ontains ).* m of water 'o&ered with 2.8 m of /erosene G γ  E 8.0 /IJm 3C. Find the press"re at the interfa'e 11

and at the bottom of the tan/. Nnit weiht of  water E !.*! /IJm 3. Ans. 22.4 /Pa, *8.2 /Pa *2.-f air had a 'onstant spe'i' weiht of 0.0*6 lbJft3 and were in'ompressible, what wo"ld be the heiht of the atmosphere if sea le&el press"re were 14.!2 psia Ans. 282*0 ft *3.
the absol"te press"re within the tan/ Nnit weiht of water is !.*! /IJm3. Ans. 38.1 /Pa *6.-f air had a 'onstant spe'i' weiht of 12.2 IJm3 and were in'ompressible, what wo"ld be the heiht of the atmospheri' press"re Gsea le&elC is 102 /Pa Ans. 8360.66 m **.Ass"min spe'i' weiht of air to be 'onstant at 12 IJm3, what is the approimate heiht of t. 9anahaw if a mer'"ry barometer at the base of the mo"ntain reads 6)4 mm and at the same time another barometer at the top of the mo"ntain reads 480 mm. Ans. 1!34.)3 m *8.5on&ert *60 mm of mer'"ry to aC oil of sp. r. 0.82 and bC water. Ans. 12.60) m, 10.34 m *!.A barometer reads *60 mm and a press"re ae atta'hed to a tan/ reads 8)0 'm of oil Gsp. r. E 0.80C. #hat is the

absol"te press"re in the tan/ in /Pa Ans. 168.1 /Pa 80.Hpress an absol"te press"re of ) atm in meters of water ae when the barometer reads *60 mm. Nnit weiht of water is !.*! /IJm3. Ans. 41.4 m of water 81.Ass"min sea water to ha&e a 'onstant spe'i' weiht of 10.0) /IJm 3, what is the absol"te press"re at a depth of 10 /m Ans. !!3 atm 82.A hydra"li' press is "sed to raise an 80/I 'aro tr"'/. -f oil of sp. r. 0.82 a'ts on the piston "nder a press"re of 10 Pa, what diameter of piston is re$"ired GEOTECHNICAL ENGINEERING SET 1 SOIL COMPOSITION

1.
weiht, bC &oid ratio, 'C porosity, dC deree of sat"ration. Ans. aC 1*.4!; bC 0.)08; 'C 0.33*; dC )1.*! % 2. For a i&en soil, the followin are /nown.  s E 2.*4, moist "nit weiht E 20.6 /IJm 3, and moist"re 'ontent is 16.6 %. 7etermine the followin? aC dry "nit weiht, bC &oid ratio, 'C porosity, dC deree of sat"ration, eC weiht of water in /I to be added per '"bi' meter of soil for a 100 % deree of sat"ration. Ans. aC 1*.6* ; bC 0.)2 ;  'C 0.34; dC 8*. ) %; eC 0.44 3.
'ontent and the spe'i' ra&ity of soil solids are determined in the laboratory to be 11 % and 2.* respe'ti&ely. 5al'"late the followin? aC moist "nit weiht, bC dry "nit weiht, 'C &oid ratio, dC porosity, eC deree of sat"ration. Ans. aC 18; bC 16.22; 'C 0.63 ; dC 0.3! ; eC 46.!2 % ). -n its nat"ral state, a moist soil has a &ol"me of 0.00!3) m 3 and weihs 1*8 I.
sat"rated "nit weiht of soil in /IJm 3. Ans. aC 0.631; bC 2.61; 'C 1!.4! *.
Final &ol"me of soil in a dry state E 1).! 'm 3 -nitial mass in a sat"rated state E 44  Final mass in a dry state E 30.1  7etermine the followin? aC shrin/ae limit of soil, bC &oid ratio if  s E 2.*0, 'C dry "nit weiht Ans. aC 1*.28 %; bC 0.46*; 'C 18.06 10.
dC porosity, eC deree of sat"ration, fC &ol. o''"pied by water. Ans. aC 1!.2 ; bC 1*.68 ; 'C 0.)04 ; dC 0.33) ; eC 46.2 % ; fC 0.186 GEOTECHNICAL ENGINEERING SET 2

12.For a sat"rated soil, the followin are i&en? water 'ontent E 23 %, s E 2.6*. 7etermine the followin? aC sat"rated "nit weiht, bC dry "nit weiht, 'C moist "nit weiht when the deree of sat"ration be'omes *0 %. Ans. aC 1!.!6; bC 16.23; 'C 18.84 /IJm 3 13.
7etermine the followin? aC sat. "nit weiht, bC moist"re 'ontent when the moist "nit weiht is 1*.), 'C shrin/ae limit. Ans. aC 20.)8; bC 2.1 %; 'C 20.0* % 1).
bC #hat is the moist "nit weiht of soil when water 'ontent is * %. Ans. 1*.24 /IJm 3 'C #hat is the deree of sat"ration Ans. 30 % 1*.A soil sample has the followin properties? Porosity E 0.38! pe'i' ra&ity E 2.*) aC 5omp"te the &oid ratio of the soil sample. Ans. 0.66 bC 5omp"te the sat"rated "nit weiht of soil sample. Ans. 20.1) 'C 5omp"te the e(e'ti&e "nit weiht of the soil sample. Ans. 18.34 18.A soil sample ta/en from the ro"nd has a moist"re 'ontent of 12 % and deree of sat"ration of )2 %. -f its spe'i' ra&ity is 2.*. 7etermine the followin? aC &oid ratio, bC sat"rated "nit weiht, 'C "nit weiht. Ans. aC 0.62; bC 20.10; 'C 10.2! 1!.A soil sample has a dry "nit weiht of 1*.1 /IJm 3 and a &oid ratio of 0.)2.

7etermine the followin? aC sp. r. of soil, bC "nit weiht of soil, 'C shrin/ae limit. Ans. aC 2.6); bC 10.6); 'C 1*.62 % 20.
'C 5omp"te the e(e'ti&e "nit weiht of the silty sand. Ans. 10.11 /IJm 3 22.A sat"rated soil sample has the followin data? #ater 'ontent E 34 % oids ratio E 0.!2 7etermine the followin? aC sp. r. of soil, bC e(e'ti&e "nit weiht, 'C moist "nit weiht. Ans. aC 2.*1; bC 8.*4; 'C 18.)) 23.
14

SET 3

24.
aC 5omp"te the water 'ontent of the 'lay. Ans. 31 % bC 5omp"te the deree of sat"ration if the sp. r. of the 'lay is 2.*0 and a &oid ratio of 0.!2 Ans. !0.!8 % 'C 5omp"te the moist "nit weiht of the 'lay if the &oid ratio is 0.!2,  s E 2.*0 Ans. 18.0* /IJm3 2*.
bC 5omp"te the shrin/ae limit of soil. Ans. 1!.26 % 'C 5omp"te the e(e'ti&e "nit weiht of  soil. Ans. 10.** /IJm3 2!.A soil has a moist"re 'ontent of 18.2 % and a deree of sat"ration of 80 %. -f the &oid ratio of soil is 0.60*. 7etermine the followin? aC sp. r. of the soil, bC moist "nit weiht of the soil, 'C sat"rated "nit weiht of soil. Ans. aC 2.6*; bC 1!.2* /IJm3; 'C 20 /IJm 3 30.For a i&en sandy soil with ma. and min. &oid ratios of 0.*) and 0.46 respe'ti&ely has a sp. r. of 2.68. -f the density inde is *8 % and moist"re 'ontent of ! %. aC #hat wo"ld be the in sit" &oid ratio Ans. 0.)238 bC 7etermine the deree of sat"ration of the soil. Ans. 46.0) % 'C #hat will be the moist "nit wt. of 'ompa'tion in the eld

Ans. 18.81 /IJm 3 31.A sample of soft sat"rated 'lay has a &ol"me of 100 '". m. and weihs 1*) . -f the o&en dry weiht is 120 . aC 5omp"te the water 'ontent of 'lay. Ans. 4).8 % bC 5omp"te the &oid ratio of the 'lay. Ans. 1.22 'C 5omp"te the sp. r. of the 'lay. Ans. 2.66 32.
of 18 % and a sp. r. of 2.*0. 7etermine the followin? aC &oid ratio, bC deree of sat"ration, 'C air &oid ratio. Ans. aC 0.6*; bC *2.)4 %; 'C 0.11 GEOTECHNICAL ENGINEERING SET 4

34.
followin? aC density inde, bC deree of sat"ration, 'C air &oid ratio. Ans. aC 20 %; bC 42.!* %; 'C 0.24) 36.
the soil is 1)0 I. -f the soil has a sp. r. of 2.*0, 'omp"te the followin? aC moist"re 'ontent, bC &oid ratio, 'C deree of sat"ration. Ans. aC 20 %; bC 0.)!; 'C !1.)3 % 3!.
'ontent, bC &oid ratio, 'C deree of sat"ration. Ans. aC 2) %; bC 0.!); 'C 68.4 % 42.A soil sample has a weiht of 120  before it was pla'ed on the o&en. After dryin for 20 min. it weihs only 100 . pe'i' ra&ity of soil sample is 2.6. ol"me of  soil spe'imen before dryin is 66 'm 3. 5omp"te for the followin? aC water 'ontent, bC &oid ratio, 'C deree of sat"ration. Ans. aC 20 %; bC 0.*2; 'C *2.* % 43.ilty sand has a &oid ratio in loosest 'ondition e$"al to 0.86 and a &oid ratio of 0.30 in the densest 'ondition. -f the relati&e density of sand is )4 %, 'omp"te the followin? aC in sit" &oid ratio, bC porosity, 'C if the dry "nit wt. of the silty sand is 16.8) /IJm3, 'omp"te the spe'i' ra&ity of sand. Ans. aC 0.)6; bC 0.36; 'C 2.68 44.
/IJm3. -t has a spe'i' ra&ity of 2.6).
46.A loose, "n'ompa'ted sand ll 1.8 m depth has a relati&e density of 40 %. aboratory test indi'ated that the minim"m and maim"m &oid ratios of the sand are 0.46 and 0.!0 respe'ti&ely. pe'i' ra&ity of the solids of the sand is 2.6). aC 7etermine the &oid ratio of the sand ha&in a relati&e density of 40 %. Ans. 0.*24 bC #hat is the dry "nit weiht of the sand Ans. 1).08 /IJm3 'C -f the sand is 'ompa'ted to a relati&e density of *) %, what is the de'rease in the thi'/ness of the 1.8 m ll Ans. 160.*! mm 4*.
b"l/ sp. r., bC dry density, 'C &oid ratio, dC porosity, eC deree of sat"ration. Ans. aC 2.63; bC 2).8; 'C 1.0 ; dC 0.)0 ; eC 0.*4 48.
porosity, 'C water 'ontent, dC sp. r. of the soil parti'le, eC sat"rated "nit weiht of the soil, fC e(e'ti&e "nit weiht of the soil, C dry "nit weiht of soil. Ans. aC 0.62 ; bC 0.38 ; 'C 24.3 % ; dC 2.)) ; eC 1!.20 /IJm 3 ; fC !.3! /IJm3 ; C 1).44 /IJm 3 )0.An "ndist"rbed soil sample in sat"rated 'ondition has 6 &oid ratio of 0.81 and sp. r. of 2.6!. Find the followin? aC water 'ontent, bC sat"rated "nit weiht, 'C dry "nit weiht. Ans. aC 30.1 %; bC 18.!*; 'C 14.)8 )1.7ry sand is pla'ed in a 'ontainer ha&in a &ol"me of 0.30 ft 3.
17

sat"ration, 'C b"l/ sp. r.. Ans. aC 0.64 ; bC 0.436 ; 'C 1.66 )2.aboratory test data on a sample of sat"rated soil show that the &oid ratio is 0.4) and the spe'i' ra&ity of solids is 2.6). For these 'onditions, determine the followin? aC water 'ontent, bC sat"rated "nit weiht, 'C dry "nit weiht. Ans. aC 1* %; bC 20.!*; 'C 1*.!3 )3.A soil sample has a b"l/ "nit weiht of 1!.6 /IJm3 at a water 'ontent of 10 %. 7etermine the followin if it has a sp. r.  s E 2.*? aC &oid ratio, bC per'entae air in &oids Gair &oidsC, 'C deree of sat"ration. Ans. aC 0.486; bC 14.) %; 'C )).6 % )4.For a 'ompa'ted soil, spe'i' ra&ity  s E 2.*2, water 'ontent E 18 %, γ d E 0.!0 γ  zav . 5omp"te the followin? aC ero air &oid "nit weiht, bC dry "nit

weiht, 'C deree of sat"ration. Ans. aC 1*.!1 /IJm 3; bC 16.12 /IJm 3; 'C *4.18 % )).For a sandy soil, ema E 0.*2, emin E 0.46 and  s E 2.68. aC #hat is the moist "nit weiht of 'ompa'tion in the eld if 7 r E *8 % and water 'ontent of ! %. Ans. 18.8! /IJm3 bC 5omp"te the sat"rated "nit weiht. Ans. 20.6* /IJm3 'C 5omp"te the deree of sat"ration. Ans. 46.64 % GEOTECHNICAL ENGINEERING SET 6

)6.A soil at a 'onstant moist"re 'ontent shows the followin when 'ompa'ted 7eree of 7ry "nit at"ration #eiht 40 % 14. )0 /IJm3 *0 % 1*.8! /IJm3

aC 7etermine the spe'i' ra&ity of soil. Ans. 2.64 bC 7etermine the moist"re 'ontent of soil. Ans. 11.86 % 'C 7etermine &oid ratio of the 1*.8! /IJm 3 soil. Ans. 0.44* )*.A soil sample has a nat"ral water 'ontent of 22.) % and it is /nown to ha&e a sp. r. of 2.6. -n order to determine the moist density of the soil, a portion of soil weihin 224  is p"t in a )00 'm3 'ontainer. -t is lled with 382 'm 3 of water to ll the 'ontainer. 7etermine the followin? aC moist "nit weiht of soil, bC dry "nit weiht of soil, 'C &oid ratio. Ans. aC 18.62 /IJm 3; bC 1).2 /IJm 3; 'C 0.6*8 )8.Followin are the res"lts of a eld "nit weiht determination test "sin the sand 'one method. 7ry "nit weiht of sand E 16.36 /IJm 3 #t. of sand to ll the 'one

E 1.1) I #t. of @ar O 'one O sand Gbefore "seC E )8.! I #t. of @ar O 'one O sand Gafter "seC E 2*.6) I #t. of moist soil from hole E 32.)) I oist"re 'ontent of moist soil E 11.6 % 7etermine the dry "nit weiht of 'ompa'tion in the eld. Ans. 1).8)4 /IJm 3 )!.A 'lay sample was pla'e in a lass dish.
60.
'ompa'tion mo"ld ha&in mass of )32)  and a &ol"me of !48 'm 3. ass of mo"ld lled with soil po"red in loosely E 6!64  ass of mo"ld lled with soil dynami'ally 'ompa'ted E *368  -f the dry density of the soil in = sit" is 1.6) J'm 3 and sp. r. of the soil is 2.*0, 'al'"late the relati&e density of the in = sit" soil. Ans. 80.! % 63.#hen sand = po"rin 'ylinder was "sed in a eld density test, the mass of sand r"n into the hole was fo"nd to be 1)68 .
mm and a lenth of 12) mm was rammed into an in = sit" soil mass. After remo&in it and trimmin the ends Bat, its mass was fo"nd to be 3482 .
6).A eld 'ompa'ted sample of sandy loam was fo"nd to ha&e a wet density of 136 lbJ'". ft. at a water 'ontent of 10 %.
dC Porosity GnC eC Per'ent 'ompa'tion of the eld sample Ans. 123.6 lbJ'". ft.; 130.! p'f; **.!) %; 2).4 %; !8.88 % 66.A sample of moist $"art sand was obtained by 'aref"lly pressin a sharpened hollow 'ylinder with a &ol"me of 1)0 '". m. into the bottom of an e'a&ation.  
6*.
weiht to whi'h this soil 'an be 'ompa'ted witho"t 'hane in its moist"re 'ontent LIQUID LIMIT TEST, PLASTIC LIMIT TEST  6!.A soil sample was determined in the laboratory to ha&e a li$"id limit of 41 % and a plasti' limit of 21.1 %. -f the water 'ontent is 30 %, determine the followin? aC plasti'ity inde, bC li$"idity inde, 'C what is the 'hara'teristi' of soil. Ans. aC 1!.! %; bC 0.44*; 'C plasti' *0.For a sandy soil, ema E 0.*), emin E 0.)2 and  s E 2.*0. 7etermine the followin ? aC &oid ratio of  the soil at 7 r E 6) %, bC dry "nit weiht of soil, 'C b"l/ "nit weiht if the moist"re 'ontent is 28 %, dC ero air &oid "nit weiht. Ans. aC 0.60; bC 16.)) /IJm3 ; 'C 21.18 /IJm 3 ; dC 1).08 /IJm 3 *1.
Io. of 9lows oist"re 'ontent GIC G%C 1) 42 20 40.8 28 3!.1  
dC #hat is the 'onsisten'y inde Ans. 0.81 *2.Followin are the res"lts from the li$"id and plasti' limit test for a soil. Io. of 9lows oist"re 'ontent GIC G%C 16 36.)

20 33.1 28 2*.0 Plasti' limit E 12.2 % -n sit" moist"re 'ontent E 31 %. aC 5omp"te the li$"id limit. Ans. 28.) % bC 5omp"te the plasti'ity inde. Ans. 16.3 % 'C 5omp"te the li$"idity inde if the in sit" moist"re 'ontent is 26.) %. Ans. 0.8** dC 5omp"te the 'onsisten'y inde. Ans. 0.164 *3.-n a li$"id limit, "sin a 'one penetrometer, the followin readins were re'orded and tab"lated as shown. Plasti' imit
20

) 113.4

136.0

aC 7etermine the li$"id limit of the soil. Ans. )2% bC 7etermine the plasti'ity inde of the soil. Ans. 31% 'C -f the nat"ral water 'ontent of the soil is 38%, determine the li$"idity inde. Ans. 0.)) *4.-n a li$"id limit test, "sin a 'one penetrometer, the followin readins were re'orded. oist"re 5ontent 5one Penetration   G%C GmmC   30.! 14.4   42.0 16.4   )1.8 18.2   68.2 21.1   **.6 22.3 Plasti' imit
#eiht of wet soil O 'ontainer E 22.12  #eiht of dry soil O 'ontainer E 20.42  #eiht of 'ontainer E 13.0*  Iat"ral #ater 5ontent
*6. A 'lay has the followin Atterber limits? li$"id limit E 60%, plasti' limit E 40%, shrin/ae limit E 2)%.
3J" = 2J& O 1Jw E 11 1J" O )J& = 2Jw E ! 2J" O 1J& = 3Jw E 6 Ans. 1J2, 1, 1J3 2. Find the &al"es of a, b, and '; the s"m of a, b and '; the prod"'t of a, b, and '? 4a O 3b O 4' E ! 2a O b = 3' E 3 3a O 2b O ' E 4 Ans. 4, ), 2; 1 ; 40 3. Find the &al"e of , y, and  in the followin e$"ations

MATHEMATICS  ALGEBRA SET 1

SYSTEMS OF LINEAR EQUATIONS

1. ol&e for ", &, and w in the followin e$"ations?

 E 21 y E 3) y E 1) Ans.

3, ) ,*

NUMBER PROBLEMS

4. A rope 2)m lon is '"t into two pie'es where one is )m loner than the other. Find their lenths. Ans. 10 and 1)m

21

). Find two 'onse'"ti&e positi&e odd inteers whose s$"are di(er by 24. Ans. ) and * 6. ne n"mber is &e less than the other n"mber. -f the s"m is 13), what are the n"mbers Ans. 6) and *0 10.
the re'ipro'al of the other n"mber. Find the n"mbers. Ans. 1J) and 1J6 11.
14.
the $"otient is 3 and the remainder is 14. #hat are these parts Ans. 32 and 110 1!.
A. 20 9. 21

5. 7.

22 23

MATHEMATICS SET 2 DIGIT PROBLEMS

24.-n a twodiit n"mber, the "nitMs diit is 3 reater than the tenMs diit. Find the n"mber if  it is 4 times as lare as the s"m of its diits. Ans. 36 2).
Ans. 842 2*.
30.ow many min"tes after 2 oM'lo'/ will the hands of  the 'lo'/ are perpendi'"lar for the rst time

Ans. 2*.2* min 31.ow many min"tes after 3?00 P will the min"te hand of the 'lo'/ o&erta/es the ho"r hand Ans. 16.36 min 32.ow many min"tes after 10?00 oM'lo'/ will the hands of the 'lo'/ are dire'tly opposite ea'h other for the rst time Ans. 21.82 min 33.#hat time between the ho"rs of 12?00 noon and 1?00 pm wo"ld the ho"r hand and the min"te hand of a 'ontin"o"sly dri&en 'lo'/ is in straiht line Ans. 12?33 P 34.ow many min"tes after 12?00 noon will the ho"r hand and the min"te hand of a 'lo'/ rst form an anle of 120 o Ans. 21.82 min 3).From the time 6?1) P to the time *?4) P of the same day, the min"te hand of a standard 'lo'/ des'ribe an ar' of how m"'h Ans. )40o 36.-t is now between 3 and 4 oM'lo'/ and in 20

min"tes the min"te hand will be as m"'h as the ho"r hand as it is now behind it. #hat is the time now Ans. 3?06.36 P 3*.A man left his home at past 3?00 P as indi'ated in his wall 'lo'/. 9etween two to three ho"rs after, he ret"rned home and he noti'ed that the hands of the 'lo'/ inter'haned. At what time did he left his home Ans. 3?31.4* P 38.
Ans.

12.)* AGE PROBLEMS

41.
be three times as old as her da"hter Ans. 1) yrs MATHEMATICS SET 3

46.
4!.9 is as old as A was fo"r years ao.
WORK PROBLEMS

)2.A p"mp 'an p"mp o"t a tan/ in 11 ho"rs. Another p"mp 'an p"mp o"t the same tan/ in 20 ho"rs. ow lon will it ta/e both p"mps toether to p"mp o"t the tan/ Ans. *.10 hrs

)3.r. 9rown 'an wash his 'ar in 1) min"tes, while his son ohn ta/es twi'e as lon to do the @ob. -f they wor/ toether, how many min"tes 'an they do the washin Ans. 10 min )4.>ne pipe 'an ll a tan/ in ) ho"rs and another pipe 'an ll the same tan/ in 4 ho"rs. A drainpipe 'an empty the f"ll 'ontent of the tan/ in 20 ho"rs. #ith all the three pipes open, how lon will it ta/e to ll the tan/ Ans. 2.)0 hrs )).A swimmin pool is lled thro"h its inlet pipe and then emptied thro"h its o"tlet pipe in 8 ho"rs. -f water enters thro"h its inlet and sim"ltaneo"sly allowed to lea&e thro"h its o"tlet, the pool is lled in *.) ho"rs. Find how lon will it ta/e to ll the pool with the o"tlet 'losed. Ans. 3 hrs )6.
hrs, 4 hrs, and 6 hrs, respe'ti&ely. #hat fra'tion of the @ob 'an they nish in one ho"r wor/in toether Ans. 3J4 )*.A father and his son 'an di a well if the father wor/s 6 hrs and his son wor/s 12 hrs or they 'an do it if the father wor/s ! hrs and the son wor/s 8 hrs. ow lon will it ta/e for the son to di the well alone Ans. 20 hrs )8.Peter and Pa"l 'an do a 'ertain @ob in 3 hrs. >n a i&en day, they wor/ toether in 1 hr then Pa"l left and Peter nishes the rest of the wor/ in 8 hrs. ow lon will it ta/e Peter to do the @ob alone Ans. 12 hrs )!.7elia 'an nish a @ob in 8 hrs. 7aisy 'an do it in ) hrs. -f 7elia wor/ed for 3 hrs and then 7aisy was as/ed to help her nish it, how lon will 7aisy ha&e to wor/ with 7elia to nish the @ob Ans. 1.!2 hrs

60.Hle&en men 'an nish the  @ob in 1) days. Fi&e men were wor/in at the start and after 6 days fo"r men were added. ow many days will it ta/e to nish the @ob Ans. 21 days 61.
63.A and 9 'an do a pie'e of wor/ in 42 days, 9 and 5 in 31days, and A and 5 in 20 days. #or/in toether, how many days 'an all of them nish the wor/ Ans. 18.86 days 64.A tan/ is lled with two pipes.
days. A wor/in alone 'an nish it in &e less than 9. ow lon will it ta/e ea'h of them to nish the wor/ alone Ans. 10 and 1) days 6*.-t ta/es 9"t'h twi'e as lon as it ta/es 7an to do a 'ertain pie'e of wor/. #or/in toether they 'an do the wor/ in 6 days. ow lon wo"ld it ta/e 7an to do it alone Ans. ! days 68.A and 9 wor/in toether 'an nish a pie'e of wor/ in 20 days. After wor/in toether for 4 days, A $"its and 9 nishes the wor/ in 24 days more. Find the n"mber of days that 9 'o"ld nish the wor/ alone. Ans. 30 days 6!.*2 men 'an do a 'ertain  @ob in 100 days.
25

remainin wor/for'e to 'omplete the @ob Ans. 80 days *0.n the 61 st day ten more men were hired. Find the total n"mber of days to nish the @ob. Ans. 10*.14 days *2.A team of 28 wor/ers were wor/in on the pro@e't that they 'o"ld had to nish in 60 days. At the start of the 16 th day &e wor/ers resined. After 4) th days ten more wor/ers were hired. 9y how many days wo"ld the

'ompletion of the pro@e't be delayed Ans. 2.3 days *3.An enineer estimated that 40 wor/ers 'an nish a pro@e't in 120 days. -f he hired 20 wor/ers at the start, 20 more on the 21st day and 20 more on the 41st day, how many days earlier 'o"ld the pro@e't be done Ans. 20 days *4.A 'ontra'tor ot a pro@e't that had to be 'ompleted in 120 days. e was s"re 40 wor/ers 'o"ld nish the pro@e't on time. e started the wor/ with )0 wor/ers b"t red"'ed them to 30 after 60 days. ow many days wo"ld it ta/e him to nish the @ob Ans. 120 days *).Peter 'an do the whole  @ob in half the time it ta/es enry to do it.  
*6.-t ta/es i'hael 60 se'onds to r"n aro"nd a 440yard tra'/. ow lon does it ta/e ordan to r"n aro"nd the tra'/ if they meet in 32 se'onds after they start toether in a ra'e aro"nd the tra'/ in opposite dire'tions Ans. 68.)* se' **.A man tra&els in a motoried ban'a at the rate of 12 /ph from his barrio to the pobla'ion and 'ome ba'/ to his barrio at the rate of 10 /ph. -f his total time of tra&el ba'/ in forth in 3 ho"rs and 10 min"tes, what is the distan'e from the barrio to the pobla'ion Ans. 1*.2* /m *8.Lim and Len tra&eled at the same time at the rate of 20 mJmin from the same point on a 'ir'"lar tra'/ of radi"s 600m. -f Lim wal/s alon the 'ir'"mferen'e and Len towards the 'enter, nd their distan'e after 10 min. Ans. 2)*.*2 m

MATHEMATICS SET 5

*!.A boat tra&els downstream in 2J3 the time as it does oin "pstream. -f the &elo'ity of the ri&er '"rrent is 8 /ph, determine the &elo'ity of the boat in still water. Ans. 40 /ph 80.A man rows downstream at the rate of ) mph and "pstream at the rate of 2 mph. ow far downstream sho"ld he o if he ret"rn in *J4 ho"rs after lea&in Ans. 2.) miles 81.A @oer starts a 'o"rse at a steady rate of 8 /ph. Fi&e min"tes later, a se'ond @oer with the same 'o"rse at 10 /ph. ow lon will it ta/e for the se'ond @oer to 'at'h the rst Ans. 20 min 82.An airplane 'o"ld tra&el a distan'e of 1000 miles with the wind in the same time it 'o"ld tra&el a distan'e of 800 miles is 40 mph, what is the speed of the plane 26

Ans. 360 mph 83.An airplane Byin with the wind too/ 2 hrs to tra&el 1000 /m and 2.) hrs in Byin ba'/. #hat was the wind &elo'ity in /ph Ans. )0 /ph 84.
40 /ph, what is the speed of the '"rrent Ans. 20 mph 8*.
-f he starts from point A at 8?00 A and wal/s at 2 /ph, he will be at 5 3 min"tes earlier. -f he starts at 8?30 A and wal/s at 3 /ph, he will arri&e at 5 6 min"tes late. At what time is he s"pposed to be at 5 and what is the distan'e from A to 5 in /m Ans. !?06 A; 2.1 !0.An enineer is re$"ired to report at ta. 9 e&ery mornin at a ed time. #hen he left at ta. A at 8?00 A and wal/ed at 1.8 /ph, he arri&ed at 9 ) min"tes earlier. #hen he left at 8?30 A and wal/ed at 2.* /ph, he arri&ed at 9 ) min"tes late. ow far in /m is ta. 9 from ta. A. Ans. 1.8 /m !1.A salesman started wal/in from oQ'e A at 8?30 A at the rate of 2.) /ph. e arri&ed at oQ'e 9 12 se'onds late. ad he started at A at 8?00 A and wal/ed at 1.) /ph, he wo"ld ha&e arri&ed at 9 one min"te the appointment time.

At what time was he s"pposed to be at 9 Ans. !?13 A !2.A boat oin a'ross a la/e 8 /m wide pro'eed 2 /m at a 'ertain speed and then 'ompletes the trip at a speed + /ph faster. 9y doin this, the boat arri&es 10 min"tes earlier than if the oriinal speed had been maintained. Find the oriinal speed of the boat Ans. 12 /ph MIXTURE PROBLEMS !3.ow many rams of old m"st be added with )00 rams of an alloy 'ontainin 30% old and *0% sil&er in order to prod"'e another alloy analyin 40% old and 60% sil&er Ans. 83.33 rams !4.2000 / of steel 'ontainin 8 % ni'/el is to be made by miin steel 'ontainin 14 % ni'/el with another steel 'ontainin 6% ni'/el. ow m"'h of the steel 'ontainin 14% ni'/el is needed

27

Ans.

)00 /

MATHEMATICS SET 6

!).A 40ram alloy 'ontainin 3)% old to be melted with a 20 ram alloy 'ontainin )0% old. ow m"'h per'entae of old is the res"ltin alloy Ans. 40% !6.
&ol"me of 100 liters is mied by *.)% al'ohol and 2)% asoline. -t was fo"nd o"t that a )0)0 per'ent mit"re is appropriate in the sol"tion. ow m"'h asoline sho"ld be added to the hih 'on'entrated sol"tion in order to ha&e a )0)0% mit"re Ans. )0  asoline !!.ow many po"nds of 'ream 'ontainin 12% b"tterfat m"st be added to 1800 lb of mil/ 'ontainin 2% b"tterfat to obtain mil/ ha&in 3% b"tterfat Ans. 200 lb 100.
prod"'e a mit"re of )0 millimeters that 'ontains 42% of the 'hemi'al sol"tions Ans. 20 mm 102. A 'hemi'al enineer mied 40 mm of 3)% hydro'hlori' a'id sol"tion with 20 mm of )0% hydro'hlori' a'id sol"tion. #hat is the per'entae of the hydro'hlori' a'id of the new sol"tion Ans. 40%

22.)% al'ohol, what is the &al"e of  Ans. 30% 106. A 'ontainer holds 3 liters of a 2)% al'ohol sol"tion. >nehalf liter of  the sol"tion is withdrawn and repla'ed with one half liter of p"re al'ohol. #hat is the per'entae of the al'ohol in the new mit"re Ans. 3*.)%

103. >ne liter of a 2)% s"ar sol"tion is mied with 3 liters of a 40% s"ar sol"tion. #hat is the per'entae of s"ar in the new mit"re Ans. 36.2)% 104.
ARIATION 0ROLEMS

0RO0ORTION AND

10*. Find the mean proportion of 4 and 36. Ans. 12 108.
28

110. Find the third proportional to the s$"are of 2 and the '"be root of 8 Ans. 1 111.
MATHEMATICS SET 7 113.
114. y &aries in&ersely as the s$"are of  and dire'tly as t. #hen t E 4 and  E 6, y E 1J3. Find y when t E 6 and  E 2. Ans. 4.) 11). w &aries dire'tly as the s$"are root of  and in&ersely as y. #hen  E ! and y E 3, w E ). Find the &al"e of w when y E 4 and  E 12. Ans. 4.33 116. i&en that w &aries dire'tly as the prod"'t of

 and y and in&ersely as the s$"are of  and that w E 4 when  E 2, y E 6, and  E 3. Find w when  E 1, y E 4, and  E 2. Ans. 3 11*.
11!.
ENN DIAGRAM

122. An enineerin professor 'ond"'ted a s"r&ey reardin the fa&orite s"b@e'ts of the st"dents. nly 1) st"dents li/e all the three s"b@e'ts. ow many st"dents were s"r&eyed Ans. !) 123. -n a 'lass of 40 st"dents, 2* li/es 5al'"l"s and 2) li/e 5hemistry. ow many li/e both 5al'"l"s and 5hemistry Ans. 12 124.
eams abo"t Pro@e'tile and H'onomi's. ow many were able to sol&e both problems if only ))% of the eaminees sol&ed the pro@e'tile problem and *0% sol&ed the e'onomi's problems.  ne who drin/s whis/y does not drin/ 'o/e. ow many are in the ro"p Ans. !00 126. A s"r&ey of 100 st"dents reported that the n"mber of those who enrolled in &ario"s athemati's s"b@e'ts were Alebra, eometry, and 5al'"l"s, 20; Alebra and eometry, 30; Alebra and 5al'"l"s, 3);

eometry and 5al'"l"s, 3); Alebra, *0; 5al'"l"s, 60. ow many enrolled in eometry Ans. )0 MATHEMATICS SET /

12*.
128. >n one h"ndred parti'ipants in a seminar? 80 are 5H, 30 are H, 20 are HH, 1! are both 5H and H, 2 are both H and HH, 10 are both HH and 5H, and 1 is a 5H, H, and HH. aC ow many parti'ipants are 5H only Ans. )2 bC ow many parti'ipants are H only Ans. 10 'C ow many parti'ipants are HH only Ans. ! 12!. -n a s"r&ey 'ontainin the smo/in habits of 'ons"mer, it was fo"nd that ))% smo/e 'iarette A, )0% smo/e 'iarette 9, 40% smo/e 'iarette 5, 30% smo/e 'iarette A and 9, 20% smo/e 'iarette A and 5, 12% smo/e 'iarette 9 and 5, only 10% smo/e all three 30

'iarettes. #hat per'entae of the pop"lation did not smo/e Ans. *% 130. f these 14 ha&e already ser&ed in the war of Lorea, 12 in the war of ietnam, 10 in the war of apan.
football and ho'/ey ames, )0 wat'h ho'/ey and bas/etball ames, and )0 do not wat'h any of the three ames. ow many wat'h ho'/ey ames only Ans. !) 0ERMUTATION 132. ow many perm"tations 'an be made o"t of the letters in the word 7-H> ta/en 3 at a time Ans. 60 ways 133. ow many perm"tations 'an be made o"t of the letters in the word -HAI-A Ans. 4)3,600 ways 134. ow many perm"tations 'an be made o"t of the letters in the word -AI7 ta/in fo"r letters at a time Ans. 360 ways 13). ow many 4 diit n"mbers 'an be formed witho"t repeatin any diit, from the followin diits 1, 2, 3, 4 and 6. Ans. 120 136. ow many perm"tations 'an be

made o"t of the letters of the word HI-IHHK-I Ans. 2**,200 ways 13*. ow many ways 'an 3 men and 4 women be seated on a ben'h if the women are to be toether Ans. )*6 ways 138. -n how many ways 'an ) people line "p to pay their ele'tri' bills Ans. 120 ways 13!. -n how many ways 'an ) people line "p to pay their ele'tri' bills if two parti'"lar persons ref"se to follow ea'h other Ans. *2 ways 140. ow many ways 'an * people be seated at a ro"nd table Ans. *20 ways 141. -n how many relati&e orders 'an we seat * people at a ro"nd table with a 'ertain 3 people side by side. Ans. 144 142. -n how many ways 'an we seat * people in a ro"nd table with a 'ertain 3 people not in 'onse'"ti&e order

Ans. )*6 ways 143.
144. ow many two = diit n"mbers 'an be formed by the "se of diits 1, 2, 3, 4 and 6 if one diit is "sed only on'e in one n"mber Ans. 20 14). ow many three = diit n"mbers 'an be formed by the "se of diits 1, 2, 3, 4 and 6 if one diit is "sed only on'e in one n"mber Ans. 60 146. -n how many ways 'an 11 women be seated in a ro"nd table with 11 seats 31

Ans. 3,628,800 14*. -n how many ways 'an 4 persons be seated in a ro"nd table with 4 seats Ans. 6 148. A lare horiontal 'ir'"lar rin has 8 hoo/s aro"nd its 'ir'"mferen'e. -n how many ways 'an 8 di(erent lanterns be h"n with one lantern on ea'h hoo/ Ans. ),040 14!.
be formed by the "se of diits 1, 2, 3, 4, 6 and * if one diit is "sed only on'e in one n"mber Ans.

360

1)2.ow many 3 diit n"mbers 'an be formed from the diits 2, 4, 6, 8 and ! if repetitions are allowed Ans.

12) ways

1)3.-n how many ways 'an 4 persons be seated in a room where there are ! seats Ans.

3,024 ways

1)4.7etermine the n"mber of perm"tations of a 8 distin't ob@e't ta/en 3 at a time.

Ans.

336

1)).P ta/en fo"r at a time i&es fo"r perm"tations say I>P, I>P, >PI and PI>. ow many perm"tations are there for ! letters ta/en ! at a time Ans.

!

1)6.ow many perm"tations are their if the letters PIK5H are ta/en si at a time Ans.

*20

1)*.Fo"r di(erent 'olored Bas 'an be h"n in a row to ma/e a 'oded sinal. ow many sinals 'an be made if a sinal 'onsists of the display of  one or more Bas Ans.

64

1)8.A perm"tation is an ordered arranement of any n"mber of thins.
$po, 24 perm"tation s of the letters r, s, t, & ta/en fo"r at a time. ow many perm"tation s are there of the si letters r, s, t, ", &, w ta/en si at a time Ans.

*20

1)!.A fa'tory b"ildin has 8 entran'e doors. -n how many ways 'o"ld a person enter and lea&e.

the letters in the word 5>PN
1,680

#hat is the n"mber of perm"tations of the letters in the word 9AIAIA Ans.

60

Find the n"mber of perm"tations that 'an be formed from the letter P--PP-IH. Ans.

1,108,800

by di(erent door

ow many perm"tations 'an be made o"t of the letters in the word 9HA-A.

Ans.

Ans.

by any door Ans.

64 ways

)6 ways

160.ow many perm"tation s 'an be made o"t of

&erti'al line 'an be formed from 4 identi'al red Bas and 2 identi'al bl"e Bas

!0,*20

ow many di(erent sinals ea'h 'onsistin of 6 Bas in a

Ans.

1)

16).
2*,*20 ways

166.-n how many ways 'an ! boo/s, 6 athemati's and 3 7esin

be arraned on a shelf if 7esin boo/s are not to be separated Ans.

30,240 ways

16*.-n how many ways 'an 6 people be seated in a ro"nd table with 6 seats Ans.

120

MATHEMATICS SET 1

168.Hiht 'hildren @oin hands. -n how many ways 'an they form a 'ir'le Ans.

),040 ways

16!.Fo"r 'o"ples are to eat at a ro"nd table with the men and women alternatin. -f  the hostess reser&es a pla'e for 33

herself, in how many ways 'an she assin seats to the other Ans.

144 ways

1*0.ow many di(erent ways 'an ) people line "p to pay their telephone bills at the eral'o oQ'e in any order Ans.

120 ways

1*1.i boys @oin hands. -n how many ways 'an they form straiht line Ans.

*20 ways

from a de'/ of )2 'ards Ans.

24 ways

1*3.ow many ways 'an yo" in&ite any one or more of yo"r &e friends to yo"r birthday party

Ans.

140 ways

ow many ways 'an a 'ommittee of &e be sele'ted from an oraniation with 3) members

A semi'ond"'tor 'ompany will hire * men and 4 women. -n how many ways 'an the 'ompany 'hoose from ! men and 6 women who $"alied for the position

Ans.

Ans.

Ans.

31 ways

324,632 ways

ow many line sements 'an be formed by 13 distin't point

5>9-IA<->I Ans. 1*2.ow many ways 'an yo" draw 3 DNHHIs and 2 L-Is

women if she is to a&oid ha&in a parti'"lar two of them toether at the l"n'heon

*8

-n how many ways 'an a hostess sele't si l"n'heon "ests from 10

)40 ways

178.How many two – digit combinations can be o!med "sing digits 0 to 9# $ns. 45 179.How many t%!ee – digit combinations can be o!med o"t o digits 0 to 9# $ns. 120

180.How many o"! – digit combinations can be o!med "sing digits 0 to 9# $ns. 210

181.ow many &e = diit 'ombinations 'an be formed o"t of diits 0 to ! Ans. 2)2 182.ow many si = diit 'ombinations 'an be formed o"t of diits 0 to ! Ans. 210 183.ow many three = letter 'ombination 'an be formed "sin all the letters in the alphabet Ans. 2,600 184.ow many fo"r = letter 'ombinations 'an be formed "sin all the letters in the alphabet Ans. 14,!)0 18).ow many eiht = letter 'ombination 'an be formed "sin all the letters in the alphabet Ans. 1,)62,2*) 186.ow many ten = letter 'ombinations 'an be formed "sin all the letters in the alphabet Ans. ),311,*3) 34

18*.ow many twel&e = letter 'ombinations 'an be formed "sin all the letters in the alphabet Ans. !,6)*,*00 188.ow many 'ombinations 'an be made o"t of the letters A957 and H ta/en two at a time. Ans. 10 18!.ow many trianles are determined by 8 points, no three of whi'h are 'ollinear Ans. )6 1!0.
1!3.ow many 'ommittees 'an be formed by 'hoosin 4 men from an oraniation of a membership of 1)men Ans. 1,36) 1!4.9A9--
-f two 'hips are drawn at random, what is the probability that one is bl"e and the other is red Ans. ) J ! 1!*.
Ans. 1 J 18 201.ne 'ard is drawn, followed by another. #hat is the probability of ettin an A'e then a Lin Ans. 4 J 663 20).A sinle de'/ of playin 'ards is sh"Sed well. >ne 'ard is drawn, followed by another. #hat is the probability of ettin D"een then another D"een Ans. 1 J 221 206.A sinle de'/ of playin 'ards is sh"Sed well. >ne 'ard is drawn, followed by 35

another. #hat is the probability of ettin two A'es Ans. 1 J 221 20*.A ba 'ontains 3 white and ) red balls. -f two balls are drawn in s"''ession witho"t repla'ement, what is the probability that both balls are white Ans. 3 J 28 208.A ba 'ontains 3 white and ) bla'/ balls. -f two balls are drawn in s"''ession witho"t repla'ement, what is the probability that both balls are bla'/ Ans. ) J 14

20!.A ba 'ontains 3 white and ) bla'/ balls. -f two balls are drawn in s"''ession witho"t repla'ement, what is the probability that the two balls are of the same 'olor Ans. 13 J 28 210.A ba 'ontains 3 white and ) bla'/ balls. -f two balls are drawn in

s"''ession witho"t repla'ement, what is the probability that two balls are di(erent 'olors Ans. 1) J 28 211.-n lotto there are pin pon balls n"mbered from 1 to 4). si balls are drawn, one at a time, to determine the winnin 'ombination in any order. 7etermine the probability of winnin this lotto. Ans. 1 J 8,14),*86 212.A ba 'ontains 4 red balls, 3 reen balls, and ) bl"e balls. #hat is the probability of not ettin the red ball in rst draw Ans. 2 J 3 213.
A<HA<-5 SET 12 214.Ki'/y and eore ea'h throw two di'e. -f Ki'/y ets

a s"m of 4, what is the probability that eore will et less Ans. 1 J 12 21).
the probability of drawin a white ball. Ans. + 21!.-f se&en 'oins are tossed sim"ltaneo"sly, nd the probability that they will  @"st ha&e three heads. Ans. 3) J 128 220.-f se&en 'oins are tossed sim"ltaneo"sly, nd the probability that there will be at least si tails. Ans. 1 J 6 221.A fa'e of a 'oin is either head or tail. -f three 'oins are tossed, what is the probability of ettin three tails Ans. 1 J 8 222.
mar/ for a wron answer. -f  the probability that ary 'orre'tly answers a $"estion in a test is 2J3, determine the probability that ary ets two mar/s. Ans. 4 J ! 22).A mar/sman hits *)% of all his tarets. #hat is the probability that he will hit ea'tly 4 of his net 10 shots Ans. 0.01622 226.A two = diit n"mber is 'hosen randomly. #hat is the probability that is di&isible by * Ans. 13 J !0 22*.>ne bo 'ontains fo"r 'ards n"mbered 1, 3, ), and 6. Another bo 'ontains three 'ards n"mbered 2, 4, and *. >ne 'ard is drawn from ea'h ba. Find the probability that the s"m is e&en. Ans. * J 12 228.ne letter is ta/en from ea'h of the words PAKAH and HH at

random. #hat is the probability of ettin the same letter Ans. 1 J ) 230.-n a shootin ame, the probability that 9otoy and  
23).
Ans. P)0 23!.
repla'ement, what is the probability that both balls are bla'/ Ans. ) J 14 246.A ba 'ontains 3 yellow and ) bla'/ balls. -f 2 balls are drawn in s"''ession witho"t repla'ement, nd the probability that the to balls drawn are both yellow. Ans. 3 J 28 24*.A ba 'ontains 3 yellow and ) bla'/ balls. -f 2 balls are drawn in s"''ession witho"t repla'ement, nd the probability that the balls drawn are one yellow and one bla'/ ball. Ans. 1) J 28 248.From a ba 'ontainin 4 bla'/ balls and ) white balls, two balls are drawn one at a time. Find the probability that both balls are bla'/. Ass"me that the rst ball is ret"rned before the se'ond ball is drawn. Ans. 16 J 81 24!.-n the H5H 9oard Haminations, the probability that an eaminee will pass in ea'h s"b@e't is 0.80. #hat is the probability that an

eaminee will pass at least 2 s"b@e'ts Ans. 0.64 AK-<H<-5 PK>KH->I 2)0.#hat is the )0 th term of the proression 3, ), *, ! VV Ans. 101 2)1.#hat is the s"m of the rst )0 terms of the proression 3, ), *, ! VVV Ans. 2,600 2)2.#hat is the 30 th term of the proression =1), 13, 11, ! V. Ans. 43 2)3.
2)*.
264.
38

Arth"r. ow m"'h is Arth"rMs hare Ans. P4, 000 26!.P36, 000 is to be di&ided amon Arth"r, 9ernard and 5arlos s"'h that their shares in same order form an arithmeti' proression. 9ernardMs share is 3 times that of Arth"r. ow m"'h is 9ernardMs share Ans. P12, 000 2*0.ow many terms of the proression 3, ), * V m"st be in order that their s"m will be 2600 Ans. )0 2*1.#hat is the s"m of the proression 4, !, 14, 1! V "p to the 20 th term Ans. 1030 2*2.A eodeti' Hnineerin st"dent ot a s'ore of 30% on n the last n"mber he ot !0% in whi'h a 'onstant di(eren'e more on ea'h n"mber than he had on the immediately pre'edin one. #hat was his a&erae s'ore in "r&eyin

Ans. 60 2*3.-n a pile of los, ea'h layer 'ontains one more lo than layer abo&e and the top 'ontains @"st one lo. -f there are 10) los in the pile, how many layers are there Ans. 14 layers 2*4.Find the 30 th term of the A.P. 4, *, 10 V.. Ans. !1 2*).-n a ra'in 'ontest, there are 240 'ars that will ha&e pro&isions that will last for 1) ho"rs. Ass"min 'onstant ho"rly 'ons"mption for ea'h 'ar, how lon will the f"el pro&isions last if 8 'ars withdraw from the ra'e e&ery ho"r after the rst Ans. 2) 2*6.
Ans. )8 'enta&os 2**.A new 5i&il enineer failin to 'omplete his rst b"ildin 'ontra't worth P*0, 000 in an spe'ied time is 'ompelled to pay a penalty of + of 1% per day for the rst 6 days of etra time re$"ired and for ea'h additional day thereafter, the stip"lated penalty is in'reased by 10% or P3) ea'h day. -f he pays a total penalty of P6, 160.00, how many days did he o&err"n his 'ontra't time Ans. 14 days 2*8.An arithmeti' proression starts with 1, has ! terms and the middle term is 21. 7etermine the s"m of the rst ! terms. Ans. 18!

2*!.A small line tr"'/ ha"ls poles from a s"bstation sto'/yard to pole sites alon a proposed distrib"tion line.
distan'e tra&eled by the line tr"'/ ba'/ and forth, after ret"rnin from deli&erin the 30 th pole Ans. )2.) /m 280.Find the s"m of all the odd inteers between 100 and 1000. Ans. 24*)00 281.Find a positi&e &al"e of  x  so that 4 x , ) x  O 4 and 3 x 2 = 1 will be in A.P. Ans. 3 282.Find the s"m of the rst 40 e&en n"mbers. Ans. 1640 283.-f the s"m is 220 and the rst term is 10, nd the 'ommon di(eren'e if the last term is 30. Ans. 2 284.
28).
n"mbers namely 8)0 and 2)0 are remo&ed, what is the arithmeti' mean of he remainin n"mbers Ans. 42.31 286.Find the s"m of the se$"en'e 2), 30, 3) V. Ans. ) J 2 Gn2 O !nC 28*.#hat is the s"m of the followin nite se$"en'e of terms 18, 2), 32V.6* Ans. 340 288.Find the $"otient of the s"m of all odd inteers between 100 and 1000 when it is di&ided by !. Ans. 2*)00 28!.Find the $"otient of the s"m of all e&en inteers between !!! and !!!! when it is di&ided by !. Ans. 2*4!)00 2!0.7etermine the s"m of the odd n"mbers from 1 to 61 of the followin inteers. Ans. !61 2!1.7etermine the s"m  of the followin series  E 2 O ) O 8 O 11 V.. with 100 terms. Ans. 1)0)0 2!2.Find the s"m of the rst n e&en positi&e inteers.

 Ans. n Gn O 1C 2!3.Find the s"m of the rst n positi&e interal m"ltiplies of 4. Ans. 2n Gn O 1C 2!4.Find the s"m of the n"mbers di&isible by 6 that lie between *) and 1!0. Ans. 2)08 2!).A sta'/ of bri'/s has 61 bri'/s at the bottom layer, )8 bri'/s in the 2 nd layer, )) bri'/s in the 3 rd layer and so on, "ntil the last layer whi'h has 10 bri'/s. 7etermine the total n"mber of bri'/s "sed "p. Ans. 63! 2!6.#hi'h of the followin fra'tion is wron to form an A.P. 2*.)J14, 33J28, 4)J28, and 1*.)J14. Ans. 33 J 28 2!*.-f !x , !" , !z  are in A.P. then y is e$"al to Ans. 2 xz  J G x  O z C 2!8.
300.ow many terms of the se$"en'e =!, 6, 3 V. "st be ta/en so that the s"m is 66 Ans. 11 AK>I-5 PK>KH->I 301.Find the &al"e of  x  if it forms a harmoni' proression. 1, 1J3, 1J x , 1J*. Ans. ) 302.Find the &al"e of  x  if the followin forms a harmoni' proression. =1J2, 1J x , 1J16 Ans. * 328. 303. Find the positi&e &al"e of x  so that  x , x 2 = ), 2 x  will be in harmoni' proression. Ans. 3 304.
Ans. ) 30*.Find the fo"rth term of the proression +, 0.2, 0.12) V. Ans. 1 J 11 308.Find the 12 th term of the proression +, 1J6, 1J10 V Ans. 1 J 46 30!.7etermine the harmoni' mean between a: and b:. Ans. 2ab J Ga O bC

GEOMETRIC 0ROGRESSION

310.Find the &al"e of  if T, 1J6, 1J and 2J2* are 'onse'"ti&e terms of a eometri' proression. Ans. ! 311.Find the &al"e of x if +, 1J x , 1J2) and 2J12) are 'onse'"ti&e terms of a eometri' proression. Ans. ) 312.1J x ,1J), 3J2) and !J12) are 'onse'"ti&e terms of a eometri' proression. #hat is the &al"e of x . 40

Ans. 3 313.1J), 1J8, )J and 2)J)12 form a eometri' proression. proression. #hat is the &al"e of  Ans. 64 314.2J*, 3J14, !J)6 and 2*J form a eometri' proression. proression. #hat is the &al"e of  Ans. 224 31).#hat is the &al"e of  in the eometri' proression 1J3, 4J, and 16J2* Ans. ! 316.
MATHEMATICS SET 16

31!.
*2!. Find the fo"rth term of the eometri' proression. Ans. 81 32).A 'o"ntry has a billion dollars in paper '"rren'y.
Ans. 20 32*.
the net se'ond? 144 m on the third se'ond and so on. Find how far the body falls d"rin the *th se'ond. Ans . 11664 333.Find the &al"e of  from the i&en eometri' Proression Proression 1J), 2J, 4J4)V. Ans. 1) 334.
33*.
Ans. 32 342.-f the 3rd term of a .P. is 20, and the 6th term is 160, what is the rst term Ans. ) MATHEMATICS SET 17

343.
346.
348.-f a stro/e of a &a'""m p"mp remo&es 1)% of the air from the 'ontainer, how many stro/es are re$"ired to remo&e !)% of the air Ans. 1! stro/es 34!.A 'ertain ball dropped from a heiht rebo"nds 2J3 of the distan'e from whi'h it last fell. Find the total distan'e tra&eled by the ball from the time it is dropped from a heiht of 60 meters "ntil it stri/es the ro"nd the )th time. Ans. 2)2.)! m

42

3)0.A ball is thrown &erti'ally "pward to a heiht of 100 m. Ha'h time the ball stri/es the ro"nd the rebo"nd is + of the pre&io"s heiht. ow far has the ball tra&eled "p and down when it stri/es the ro"nd to the ) th time Ans. 38*.) m 3)1.
3)3.A 10$"art 'ontainer is lled with water. >ne $"art of water is drained o"t and repla'ed with al'ohol. After miin, a $"art of the sol"tion is drained o"t and repla'ed with al'ohol.
Find the s"m of all the terms. Ans. 131320 3)8.Find 3)8.Find the s"m of the rst 10 terms of the eometri' Proression 2, 4, 8, 16 V Ans. 2046 3)!.7etermine  so that , 2 O *, 10 =* will form a eometri' proression. proression. Ans. * 360.
363.Find the s"m of the innite eometri' proression proression 6, 2, 2J3V. Ans. ! J 2

MATHEMATICS SET 1/

364.Find the ratio of an innite eometri' series if the s"m is 2 and the rst term is +. Ans. 3 J 4 36).Find 36).Find the s"m of 1, 1J), 1J2)V. Ans. ) J 6 366.7etermine the s"m of the innite series  E + O T O 1J8 O V..G1J2C n Ans. 1 36*.7etermine the s"m of the eometri' series with an innite n"mbers of terms? 3 O 3J2 O U O 3J8 O V.. Ans. 6 368.A eometri' proression is 1 O  O  2 O 3 O V.  n where  W 1. 7etermine the s"m of the series as n approa'hes innity. Ans. 1 J G1 = C

43

36!.A r"bber ball is made to fall from a heiht of )0 ft. and is obser&ed to rebo"nd 2J3 of the distan'e it falls. ow far will the ball tra&el before 'omin to rest if the ball 'ontin"es to fall in this manner Ans. 2)0 3*0.
innite n"mber of ins'ribed s$"ares th"s formed. Ans. *2 3*3.A hamster re'ei&es a dose of 3 m of a 'ompo"nd and then 2J3 as m"'h as the pre&io"s dose at the end of e&ery 3 hr. #hat is the maim"m amo"nt of the 'ompo"nd it will re'ei&e Ans. ! m. 3*4.A man who is on diet losses 24 lb. -n 3 months, 16 lb. in the net 3 months and so on for a lon time. #hat is the maim"m total weiht loss Ans. *2 3*).Find the s"m of the eometri' series 1J3 O 2J! O 4J2* O 8J81 O V. Ans. 1 3*6.Hpress 3.2181818 as a fra'tion. Ans. 1** J )) 3**.Hpress 0.444 as a rational n"mber in lowest terms. Ans. 4 J ! 3*8."ppose a ball rebo"nds one half the distan'e if falls. -f it is dropped from a heiht of 40 feet, how far does it tra&el before 'omin to stop Ans. 120 feet

3*!."ppose that ea'h swin of a pend"l"m bob is 80% as lon as the pre'edin swins. -f the rst swin is 20 in'hes lon, how far does the bob tra&el before 'omin to rest Ans. 100 in'hes 380.7etermine the s"m of the innite eometri' series of 1, 1J), O1J2), VV. Ans. ) J 6 AK-<H<-5, AK>I-5 X H>H
7etermine the harmoni' mean. Ans. ! 386.
A<HA<-5 H< 1! 44

3!0.
mean is 4 and the harmoni' mean is 16. Ans. 12

3!).
QUADRATIC EQUATIONS 2

400.-n the e$"ation A  O 14 O 12 E 0, one root is 6 times the other. #hat is the &al"e of A Ans. 2 401.-n the e$"ation  2 O 11 O 28 E 0, one root is 3 more than the other. #hat are the roots of the e$"ation Ans. 4, * 402.7etermine the &al"e of / so that the s"m and prod"'t of the roots are e$"al from the i&en e$"ation. * 2 O G2/ = 1C  = 3 O 2 E 0. Ans. 1 403.
406.Find the &al"e of 5 from the i&en $"adrati' e$"ation  2 = 4 O 5 E 0 if the prod"'t of the roots is =). Ans. ) 40*.Find the &al"e of / from the i&en $"adrati' e$"ation 2 2 = / O 6 E 0 if the s"m of the roots is e$"al to 4. Ans. 8 408.Find the s"m of the roots of ) 2 = 10 O 2 E 0. Ans. 2 40!.Find the &al"e of h in the e$"ation 2 2 = h2 O 4 O )h E 0 so that the s"m of the roots is 2. Ans. 4 410.Find h in the $"adrati' e$"ation 2h 2 O 32 E 4 = )h E 0 so that the prod"'t of the roots is = 4. Ans. 4 411.-f T and =*J2 are the roots of the $"adrati' e$"ation, what is the &al"e of 9 Ans. 26 412.-n the e$"ation 3 2 O 4 O G2h = )C E 0, nd h 45

if the prod"'t of the roots is 4 Ans. 1*J2 413.-f the roots in the $"adrati' e$"ation a 2 O b O ' E 0 are 3 and 2, nd a O b O '. Ans. 2 414.
41!.ne st"dent made a mista/e in the 'oeQ'ient of the rstderee term, ot roots of 2 and =3.
420.Find the 3 rd term in the epansion of G 2 O yC). Ans. 106y2 421.Hpand G = yC 8. 422.5omp"te the n"meri'al 'oeQ'ient of the ) th term of the epansion G O 4yC12   Ans. 126*20 423.Find the 'onstant term free of  in the epansion of G O 1J 3J2C1). Ans. )00)

424.Find the &al"e of  if the 8th term of the epansion of   G3 O 1C12 is e$"al to 2)!)22)6. Ans. 2 42).
431.Find the 'oeQ'ient of the epansion of G2 O yC12 'ontainin the term 3y!. Ans. 1*60 432.#hat is the ) th term of the epansion of Ge  O e  8 C Ans. *0 433.Find the middle term of  the epansion of G 2 = )C8. Ans. 43*)0  8 434.Find the term in&ol&in 8 in the epansion of G = 2yC10. Ans. 180 8y2 43).Find the middle term of  the epansion of G 1J2 O y1J2C12. Ans. !24 3y3 436.Find the 6 th term of the epansion of G2 = C 14. Ans.  102)024 ) 43*.Find the ) th term of the epansion of G1J2 O C 12. Ans. G4!)J2)6C  4 438.Find the 'oeQ'ient of the 8th term of the epansion of  G4 O C14. Ans. )622!888 43!.Find the 'oeQ'ient of the 4th term of the epansion of G1J2 = C10. 46

Ans.  1)J16 440.-n the epansion of  G O 4yC12, what is the n"meri'al 'oeQ'ient of  the )th term Ans. 126,*20 441.#hat is the middle term in the epansion of G2 = 3C8 Ans. )6*0 8 442.
448.Find the s"m of the 'oeQ'ients in the epansion of G O 2y = C8. Ans. 2)6 44!.
4)2.Find the remainder when G4 = 10 2 = ! = 20C is di&ided by G = 4C. Ans. 40 4)3.Find / s"'h that G = 3C is a fa'tor of G/3 = 62 O 2/ =12C. Ans. 2 4)4.7i&ide G 4 = 10 2 = ! = 20C by G = 4C. Ans.

4)).Find the remainder when G 12 O 2C is di&ided by G = 3 1J2C. Ans. *31 4)6.-f G3 3 = 42y O )y2 O 6 y3C is di&ided by G2 = 2y O 3y 2C, what is the remainder Ans. 0 MATHEMATICS SET 21

4)*.-f G4y 3 O 18y 2 O 8y = 4C is di&ided by G2y O 3C, what is the remainder Ans. 11 4)8.i&en? fGC E G O 3C G = 4C O 4 when di&ided by G = /C, the remainder is /. Find /. Ans. 4 4)!.
G3) = 43 O 22 O 36 O 48C di&ided by G 3 = 22 O 6C. Ans. 32 O 6 O 8 4 461.-f G  = 23 = 32 = 4 = 8C is di&ided by G =2C, what is the remainder Ans.  28 462.Nsin syntheti' di&ision, 'omp"te the remainder if we di&ide G23 O 2 = 18 O *C by G = 2C. Ans. ! 463.-f G 2 O 3 = 40C is di&ided by G = )C, what is the remainder Ans. 0 464.-f G4 3 = 82 = ! O *C is di&ided by G2 = 3C, what is the remainder Ans.  11 46).-f G8 O 11a O 8a 3 O 18a2C is di&ided by G4a 2 O 1 O 3aC, what is the remainder Ans. ) 466.#hen Ga 3 O 22 = 18 O *C is di&ided by G O 1C the remainder is = 1). Find the &al"e of a. Ans. 42 46*.-f G8y 3 = 18y 2 = 6 O 11yC is di&ided by G4y 2 = 3y O

47

2C, what is the remainder Ans.  2y 468.-f G)3 = 14 O 3C is di&ided by G = 2C, what is the remainder Ans. 1) 46!.
Ans.

2 O 6y2

S0ECIAL 0RODUCTS  FACTORING

4*3. G4y = 4 2 = y2C is e$"al to? Ans.  G2 = yC2 4*4.Fa'tor G 4 = y2 O y =  2C as 'ompletely as possible. Ans. G2 = yC G 2 O y = 1C 4*).Fa'tor the epression G2 O 6 O8C as 'ompletely as possible. Ans. G O 4C G O2C 4*6.Fa'tor the epression G4 = y4C as 'ompletely as possible. Ans. G2 O y2C G O yC G = yC 4**.Fa'tor the epression G33 O 32 = 18C as 'ompletely as possible. Ans. 3 G O 2C G = 3C 4*8.Fa'tor the epression G16 = 10 O  2C. Ans. G = 8C G = 2C 4*!.Fa'tor the epression G6 = 1C as 'ompletely as possible. Ans.G O 1C G = 1C

G4 O 2 O 1C 480.#hat are the roots of the e$"ation G = 4C 2 G O 2C E G O 2C2 G = 4C Ans. 4 and =2 only 481.Ked"'e to lowest terms YGb2 = 4b O 16C Gb 2 = 16CZ di&ided by Gb 3 O 64C Ans. b=4 482. Ga = bC 3 E  Ans. a3 = 3a2b O 3ab 2 = b3 483.i&e the fa'tors of Ga 2 = 2C Ans. Ga = C G a O C 484.Fa'tor the epression G3 = 2*C Ans. G =3C G 2 O 3 O !C 48).Fa'tor the epression G4 O !2 O 81C Ans. G2 = 3 O !C G 2 O 3 O !C 486.Find the prod"'t of the followin alebrai'

G33 O 22 O  = 2C G O 4C Ans.34 O 14 3 O 10 2 O 2 = 8 488.Fa'tor the epression G2 = 2 = 8C Ans. G =4C G O 2C 48!.Fa'tor the epression G1)2 O 2 = 8C Ans. G3 =2CG)O4C 4!0.Fa'tor the epression G4a3 =12a2 = a O 3C Ans. Ga =3C G2aO1C G2a1C 4!1.Fa'tor the epression G!4 = 16 2y2 O 4y4C Ans. G3 2 = 2y2 O 2yC G32 = 2y2 = 2yC PRIME NUMBERS

4!2.#hi'h of the followin is not a prime n"mber aC 2* bC 13

'C 1* dC *

epression G2 O 4C G32 = 2 O 4C Ans. 63 O 82 O 16 48*.Find the prod"'t of the followin alebrai' epression

MATHEMATICS SET 22

48

4!3.#hi'h of the

bC 0.003 dC 0.002*86 4!*.Ko"nd o( 102.48886

followin n"mber is not

)01.
to fo"r sini'ant "res. a prime n"mber

36.140 is? aC 102.48! 'C 102.48886 bC 102.) dC 102.48! 4!8.
aC 3 'C ) bC 4 dC 6 )02.
followin n"mber is not

sini'ant "res in

sini'ant "res in

a prime n"mber

0.02)48 is?

0.006423 is?

aC 31 bC 1!

aC 3 'C 4 bC ) dC 6 4!!.
aC 33 'C * bC 13 dC 1! 4!4.#hi'h of the

'C ) dC **

aC 3 'C 6 bC 4 dC * PK>PHK<-H >F >AK-<

SIGNIFICANT FIGURES

4!).Ko"nd o( 12.048*!

sini'ant "res in

to fo"r sini'ant

16.433 is?

"res.

aC 2 'C 3 bC 4 dC ) )00.
aC 12.0) 'C 12.0488 bC 12.04!dC 12.048*! 4!6.Ko"nd o( 0.002*86

sini'ant "res in

to three sini'ant

0.00018 is?

"res.

aC 2 bC )

aC 0.002*! 0.0028

'C

'C 4 dC 6

Ans. 4 )0!.ol&e for  if ln G 2 O  =2C E ln 2 O ln G1C Ans. 2,1 )10.ol&e for  if lo 3 81 E 16 Ans. 4 )11.-f lo y E 6 and lo Jy E 2. ol&e for  and y. Ans.  E 10000, y E 100 )12.
)03.#hat is the &al"e of  x  in lo x  62) E 4 Ans. ) )04.implify the epression lo 8 412 E  x  Ans. 2.!0 )0).H&al"ate the lo 6 84) E Ans. 3.*6 )06.-f a 6J8 E 0.001, sol&e for a. Ans. 10000 )0*.Find the &al"e of y if y E ln G e Je2C Ans. 2 )08.ol&e for y if y E 4 ln Ge2Je21C

Find the &al"e of . Ans. ! )12.
49

Ans. * )13.#hat is the &al"e of Glo ) to the base 2C O Glo ) to the base 3C Ans. 3.*! )14.-f lo of 2 to the base 2 pl"s lo of  to the base 2 is e$"al to 2, nd the &al"e of . Ans. 2 )1). -f G2 lo  to the base 4C = Glo ! to the base 4C E 2, nd . Ans. 12 )16.ol&e for the simplied &al"e of  "  E lo 20 e. Ans. y E lo e J Glo 2 O 1C )1*.ol&e for the simplied &al"e of y E lo 20 e3. Ans. y E G3 lo eC

Glo 2 O 1C )18.Find the &al"e of ? lo 6 O  lo 4 E lo 4 O lo G32 O 4 C Ans. 3 )1!.
)28.#hat is the &al"e of lo to the base 10 of 1000 3.3 Ans. !.! )2!.-f lo  to the base 3

Ans. + ln 3 )3).-f ln  2 E Gln C 2 sol&e for . Ans. e2 )36.-f ln Glo C E 1

min"s lo of ) to the sol&e for . base 3 is e$"al to 2, nd the &al"e of . Ans. 40 )30.-n *.18 y is e$"al to? Ans. 1.!* y )31.-f lo G![C E ).))!8, what is the lo of 10[ Ans. 6.))!8 )32.ol&e for ? lo G2 O *C = lo G = 1C E lo ) Ans. 4 )33.Hpress tan h 1G4J)C in terms of nat"ral loarithm. Ans.  ln G1J3C MATHEMATICS SET 23

)34.Find the &al"e of  if  Ge O eC J Ge = eC E 2

Ans. 10e )3*.-f lo  2 O lo 2  E2 sol&e for . Ans. 2 )38.An earth$"a/e is "s"ally meas"red by the manit"de  on the Ki'hter s'ale.
 
Ans. Gy lo C J G lo yC )4).-f 10 a = b E P, what is the &al"e of  Ans. G1JaC Glo P = bC )46.Find the &al"e of loGa aCa Ans. a2 lo a )4*.ol&e for ? Glo b aC Glo ' dC Glo d 'C Ans. lo b a )48.Find the positi&e &al"e of  if lo  36 E 2 Ans. 6 )4!.Find  if lo  2* O lo  3 E 2. Ans. ! ))0.Find a if lo 2 Ga O2C O lo 2 Ga = 2C E ). Ans. 6 ))1.ol&e for  if lo )  E 3 Ans. 12) ))2.Find lo P if -n P E 8. Ans. 3.4* ))3.-f lo 8  E  n, then  is e$"al to? Ans. 1 J 8n ))4.-f 3 lo 10  = lo 10 y E 0, nd y in terms of . Ans. y E 3 ))).#hi'h of the followin is 'orre't

a. =2 lo * E 1J4! b. lo * G2C E 1J4! '. lo * G1J4!C E 2 d. lo * G1J4!C E 2 Ans. lo * G1J4!C E 2 ))6.o of the nth root of  e$"als lo of  to the 1Jn power and also e$"al to?

prod"'t of two n"mbers are 0.3)2182)18 and 1.))6302)01, respe'ti&ely. Find the rst n"mber Ans. ! )61.
Ans. lo GC J n ))*.#hat is the nat"ral

and the di(eren'e of

loarithm of e to the y

their loarithms is =

power

0.0)*!!1!. one of the

Ans. y ))8.#hat epression is

n"mbers is? Ans.

8

e$"i&alent to lo  = EXPONENTS & RADICALS

lo Gy O C Ans. loY  J Gy O CZ ))!.lo 6 84) E  Ans. 3.*61 )60.
)62.

H&al"ate

4G)2nO1C = 10G) 2n1C  y E  2G)2nC 51

Ans.

!

)63.ol&e for  if   4n O 2 = 24G4n = 1C E 10G4n O 1C Ans. T )64.ol&e for  if 8  E 2y O 2 and 163 = y E 4y Ans. 2 )6).-f 8  E 3, nd the &al"e of 2 6. Ans. ! )66.ol&e for ? 16  = O 1 E 8 Ans. T )6*.-f !  E ), what is the &al"e of 3 4 Ans. 2) )68.ol&e for y if y E Ge  O eC2 = Ge = eC2 Ans. 4 )6!.-f  to the U power e$"als 8, nd the &al"e of .

)*3.-f 8  E 3, what is the &al"e of 8 3 O 1 Ans. 216 )*4.ol&e for ?  E  G1J2*C  2J3

Ans. ! )*).ol&e for a in the e$"ation?

√ = √1 =  E 1  √ Ans. 16J2) X 0 )80.implify 3 √24  3√164 O 2 3√)44 Ans. ) 3√24 )81.ol&e for ? 3  ) O 1 E 6 O2

a E 644y. Ans. 43 O y MATHEMATICS SET 24

)*6.implify 3  = 3 = 1 = 3 Ans. ) G3 =2C )**.#hi'h of the

 =2

.

followin is tr"e a. b.

Ans. 16 )*0.-f 4 2 O 1 E 1024, nd the &al"e of 4  Ans. 16 )*1.-f 3  O 1 E 81, nd 3 2 = 1 Ans. 243 )*2.#hat is the &al"e of 102O1 if 10  E ) Ans. 2)0

Ans. 2 √ 2 )*!.ol&e for ?

'.

√ 2  √ 2 E 2 24 E 4 √ 6 √ 10 E √ ) O √ 2

)) O )) O )) O )) O )) E 6 ) An-. (!!" d. )*8.ol&e for ?  E √ 18 

Ans. 2.1)44 )82. Ga2b3C2 implify  a2 b1 6 Ans. a  b* )83. G3C is e$"al to? Ans. 3\2 )84.ol&e for ? 3 * O 1 E 6)61 Ans. 1 )8).-f 3a E *b, then 3a 2 J *b2 E  Ans. *J3 )86.ol&e for N if N E √1 = √1 =√1  V. Ans. 0.618

d.

√ *2 O √ )0.

)8*.-f  to the U power e$"als 8, then  e$"als? Ans. 16 )88.-f 33 y E 1, what is the &al"e of yJ33

Ans. 0 )8!.Find the &al"e of  that will satisfy the followin epression? √ = 2 E  √ O 2 Ans. E!J4 )!0.e3 is e$"al to? Ans. 0.04!*8* )!1.b to the mJn th power is e$"al to? Ans. nth root of b to the m power )!2.Find  from the followin e$"ations? 2* E !y 81y3 E 243 Ans. 1 )!3.ol&e for a if Ga mCGanC E 100,000, a nJam E 10 and amn E 100,000. Ans. 31.06 )!4. G 2y32C3G 3 3 1J2 y C implify  Gy3C)J2 1 Ans.  2y*3 )!).implify the followin?

52

 *a O 2 = 8 G*a O 1C O ) G* aC O 4! G*a = 2C. Ans. *a )!6. implify? G2y2C3 (y1C4  ÷  G2y3C4 G3y3C3 1 Ans.  3y )!*.implify the followin? √)  √3  √) O √3 Ans. 4  √1) )!8.#hi'h of the followin is e$"i&alent to m√ n√a mn Ans. √a 5.S'(8 &'" 9 &"'# !% *$8n :a!$'n+

Ans.

G1 O √bC J G1 =

bC 602.
603.
4

√8 3√2 √8 E 2 the denominator

  Ans. 2 600.ol&e for  if 

√a  √a O 1  √a O √a O 1 Ans. 2 √a√a O 1 = 2a

√8 3√4 4√ E 4 Ans. 16 601.Kationalie the denominator and determine the transformed fra'tion  a1J2  a1J2  a1J2 b1J2

1 COMPLEX NUMBERS

604.implify i 30 = 2i2) O 3i1* Ans. 1 O i 60).implify G3 O 2iC G4 = 3iC

Ans. 18 = i 606.implify G3 O iC 2 Ans. 8 O 6i 60*.implify G2 O 3iC G2 = 3iC Ans. 13 608.implify 3 O 4i  2 = i Ans. 2 O 11 i ) ) 60!.ol&e for  X y if 3 = 2yi E 6 O 8i Ans. 2 X 4 610.implify ) ∠300  6∠4)0 Ans. 30∠*)0 611.implify 4)∠6*0 1)∠1*0 Ans. 3∠)00 612.implify G) ∠1)0C3 Ans. 12)∠4)0 613.Find the roots of √) O 12i Ans. 3 O 2i X 3 = 2i 614.Find ln G3 O 4iC Ans. 1.60! O 0.!2*3i 61).-n the 'omple n"mber 3 O 4i, what is the absol"te &al"e Ans. ) MATHEMATICS SET 25

616.-n the 'omple n"mber 8 = 2i, what is the amplit"de Ans. 34).!60 61*.G6 'is 120 0C G4 'is 30 0C is e$"al to? Ans. 24 'is 1)0 0 618. 30 'is 80 0 is e$"al to?  10 'is )0 0 Ans. 3 'is 30 0 61!. #hat is the &al"e of  O y in the 'omple e$"ation 3 O i E y O 2i. Ans. ) 620. "ltiply G3 = 2iC G4 O 3iC Ans. 18 O i 621. 7i&ide 4 O 3i  2 = i Ans. 1 O 2i 622.Find the &al"e of i !. Ans. i 623.implify i 1!!*  O i1!!!, where i is an imainary n"mber. Ans. 0 624.Hpand G2 O √ !C3 Ans. 46 O !i 62).#rite =4 O 3i in polar form. Ans. )∠143.130 626.H&al"ate the &al"e of √ 10  √ *

53

Ans.  √ *0 62*.Perform the indi'ated operation? √ !  3√ 343 Ans. 21i 628.#hat is the $"otient when 4 O 8i is di&ided by i3 Ans. 8 O 4i 62!.#hat is the eponential form of the 'omple n"mber 4 O 3i Ans. ) e i 36.8*° 630.#hat is the alebrai' form of the 'omple n"mber 13 e i 6*.38°  Ans. ) O 12i 631.ol&e for  that satisfy the e$"ation  2 O 36 E ! = 2 2. Ans. ± 3i 632.H&al"ate ln G) O 12iC Ans. 2.)6) O 1.1*6i 633.Find the &al"e of  and y in the e$"ation? G3 O yC O G = yCi E G18 = 2iC Ans.  E 4, y E 6 634.#hat is the &al"e of  and y in the e$"ation? 2 O )y O 3yi = 1) =3i E0

Ans.  E ), y E 1 63).#hat is the &al"e of  and y in the e$"ation? G3 O 2yC = G3 O 3yCi E ) = !i Ans.  E 1, y E 4 636.#hat is the &al"e of  and y in the e$"ation?  O 2y O G3 = 2yCi E G2 O 3iC = G1 O 2iC Ans.  E 0.), y E 2 63*.Find the &al"e of  and y in the e$"ation?  = 4yi O ) = 8i E 0 Ans.  E ), y E 2 638.Find the &al"e of  and y in the e$"ation? G O yiC G1 = 2iC E * = 4i Ans.  E 3, y E 2 63!.Hpress 3 O 4i in trionometri' form. Ans.  ) G'os )3.13 0 O i sin )3.130C 640.Hpress 2 G'os 30 0 O sin 300C in alebrai' form. Ans. √ 3 O i

641.Hpress 4 'is 60 0 in alebrai' form a 'omple n"mber. Ans. 2 O 2√ 3i 642.
6)0.Hpress the eponential form of the 'omple n"mber 6 O 8i. Ans. 10 e i )3.13° 6)1.Hpressed G2 O 3iC 4 in the polar form. Ans. 16! e 3.!312i 6)2.Hpressed ) e 0.!2*30i to re'tan"lar form. Ans. 3 O 4i 6)3.Find the &al"e of sin θ if  e 0.!2*30i . Ans. 0.80 6)4.Nsin H"lers e$"ation, nd the &al"e of 'os 30 0 O i sin 30 0. Ans. e 0.)236i PARTIAL FRACTIONS

6)).Find the &al"e of A, 9, 5, 7, and H in the e$"ation? 24 O 3 3 O *2 O 20 G =1C G2 O 3C2 A 9O5 7OH  =1 2 O 3 G 2 2 O 3C Ans. 2, 0, 3, 2, 11 6)6.Find the &al"e of A in the e$"ation G2 O 4 O 10C A G3 O 22 O )C  9 G2 O 2C 5 G2 O 2 O )C G 2 O 2 O )C Ans. 2 54

6)*.Kesol&e into partial fra'tions? O2 2 = * O 12 Ans. 6 ) = 4 = 3 MATHEMATICS SET 26

6)8.Kesol&e into partial fra'tions? 32 = 8 O ! G = 2C3 Ans. 3 4 )  = 2 G  = 2C 2 G = 2C3 6)!.Kesol&e the followin into partial fra'tions and 'omp"te the s"m of A, 9, 5, 7 and H. 3 O 42 O ) O 3 A  G O 1C G 2 O  O 1C 2 O1   9 O 5 7 O H 2 O  O 1 G 2 O  O 1C2 Ans. 3 660.Nsin distin't linear fa'tors, resol&e its fra'tions into partial

fra'tions and sol&e for A, 9 and 5. 42 = 2* O 20 A G = 3C G = 4C G O 2C = 3 9 5 =4 O2 Ans. A E ), 9 E 4, 5 E3 661.Nsin repeated linear fa'tors, resol&e its fra'tions into partial fra'tions and sol&e for A, 9 and 5. 62 = 11 = 32 A G O 6C G O 1C 2  O6 9 5  O 1 G O 1C2 Ans. A E 10, 9 E 4, 5 E 3 662.Nsin distin't $"adrati' fa'tors, resol&e its fra'tions into partial fra'tions and sol&e for A, 9 and 5. 2

1 = 13 = 10 A G = 2C G2 2 O 1C = 2 9 O 5 22 O 2 O 1 Ans. A E ), 9 E 0, 5 E 3

663.Nsin repeated $"adrati' fa'tors, resol&e its fra'tions into partial fra'tions and sol&e fro A, 9, 5, 7 and H. 4 = 63 O !2 = 10 O ) A G = 3C G2 = 2 O 2C 2  =3 9 O 5 7 O H 2 = 2 O 2 G 2 = 2 O 2 2C Ans. A E 1, 9 E 2, 5 E 0, 7 E 1, H E 3 MATRICES

664.i&en the (. matri'es X &e'tors? 1 2 3 aE 4 , 9E 0 2 3 0 1 4 5E 6 3 0 2 HE

6 0

2 3

, dE 4

aC

6 G 9 = 3H 
Ans.

11 8 20 ) 666.Find A O 9 if A E 4 2 ) 3 0 3 9E

3 1

6 0 1 0 )

3

5al'"late the (. epressions or i&e reasons why they are "ndened.

2 2

4 8

1 6

Ans.

6 2 4 ) 8 3 66*.Find A  9 * 3 4 1 AE 3 19E 2 3 6 0 4 2

55

Ans.

3 4 1 4 10 2 668.Find 59 if 5 E 2 2 9E

6*2.ol&e for  and y from the i&en matri epansion? 0 1

3 4

2 3 1 1 4 2

Ans.

8 0 13 13 66!.-f Y a b ' Z E Y 2 1 3 Z, nd the &al"e of a, b and '. Ans. 2, 1, 3 6*0.i&en the matri e$"ation, sol&e for  X y? 1 3

1 2

 y

2 0

Ans.  E 4, y E 6 6*1.From the i&en matri e$"ation, sol&e for , y and . 1 2 1

1 1 1 1y 2 1

2 E1 0

 y

1 2

2 6

Ans. ), ) 6*3.ol&e for , y, , and w from the i&en matri e$"ation. 2O1 y ) 3 O2 w1 4 2

4 12 16 13 22 4 6*6.Find the matri : that satises the e$"ation? ) 1 2 6 2 6 0  4 E 4 4 2 ) 0 2

Ans. 2, 3, 2, 1 6*4.ol&e for , y and  from the i&en matri e$"ation. Y G2C y G23C Z E Y GyOC 2 G1C Z Ans.

10, 2, 6

MATHEMATICS SET 27

6*).Find the &al"e of AG95C? 2 0 A E 3 1 1 =2

1, 2, 3

1 3 2 1

1 1 2 0

Ans.

Ans.

2 1 4 1 1 2 6**.Find A O 9? 2 6 2 if A E 1 1 3 ) * 2 and 9 E

* 6 4 8 0 2

) 8

0 6 1 2

2 0 3 4

2) 2 1 13

6*!.Find 57? if A E 7E

Ans.

9E Ans.

1 1 2 2

2 =1

 2 1 1 Y 1

2

3

Z Ans.

2 1 1

4 2 2

6 3 3

680.Find A9 ?

) 3 3

3 -f A E 2

1

1

0

2

0

3

0

4

)

2 Ans.

6*8.Find A9? -f 1

) ) )

0 ! )

1 2

4 0

3 0 1

0 3

2 0 1  2 Ans.

4 ) 1

6 1 3 3 4 ! )

2 5E

4

3 56

681.Find the &al"es of a, b, ' and d from the i&en e$"ation. a  1 b 3

4 ' 1 d

! 8 1 3

Ans.

2, 1, 3, 2

DETERMINANTS

682.
684.
5onsiderin the elements of the rst

6

3

*

4

8

1

'ol"mn, determine the Ans.

4

minor of 3. Ans.

MATHEMATICS SET 2/

10

68*.5onsiderin the 68).
68!.
8

2

3

4

)

686.
1

Ans.

2

 1

2

4

2

1

1

4

3

2

1

13 5onsiderin the

follows?

688.5onsiderin the 2

3

4

elements of the se'ond

3

2

1

row, nd the minor of 3.

1

1

2

4

1

2

elements of the rst 'ol"mn, nd the minor of 4. Ans.

6

57

6!0.7etermine the

determinant whose

&al"e of the

elements are as follows?

determinant whose

4

3

0

0

0

)

2

0

0

2

1

6

0

1

4

*

8

1

8

2

the determinant whose Ans.

elements are shown

16

elements are? 2

6!4.#hat is the &al"e of

below 1

)

6

6!3.7etermine the &al"e of the determinant whose elements are as

0

3

3

follows? Ans.

2 0

0

1

6!2.7etermine the &al"e

8

3

1

0

0

1

2

3

2

1

3

3

2

1

0

Ans.

46

6!).#hat is the &al"e of

0 0

3

2

4

elements are? 0

)

288

of the determinant whose

1

1

the determinant whose elements are shown

0 6

)

0

below

2 Ans.

18

0 0

0

!

)

2

6

1

3

11

2

1

1

6!1.7etermine the

2

0

0

6 &al"e of the

0 0

2

3 Ans.

)

120 Ans.

!2

58

6!6.#hat is the &al"e of the determinant

6!8.#hat is the &al"e of

whose elements are

elements are shown 2

3

4

1

2

 6

1

1

2

1

1

2

1

1

10

2

2

)

determinant whose elements are?

Ans.

*0

*00.Find the &al"e of the

elements are? 2

2

3

0

3

)

3

1

1

6

4

1

1)

3

determinant whose

28

3)

3 3

0 2

6!!.Find the &al"e of the

whose elements are

3

determinant whose

shown below 2

3

1

1

2

8

1

1

4

3

1 3

1

0

1

2

2

3 6 3

)

3

elements are?

!

*0

1

3

of the determinant

Ans.

Ans.

Ans.

*01.Find the &al"e of the 8

below

6!*.#hat is the &al"e

2

3

1

3)

1

2

the determinant whose

shown below

Ans.

2

)

3

10)

*02.Find the &al"e of the 4

1

Ans.

1

)

determinant whose elements are?

59

2

6

0

3

2

)

1

1 3

10

0

1

2

2

'os  sin 

2

2

3

3

2

 sin  'os 

3 4

2

)

3

Ans.

3)

*04.H&al"ate the Ans.

*0

determinant?

*03.Find the &al"e of the determinant whose

Ans.



ln 

1

1J

1 = ln 

elements are? *0).H&al"ate the 2 4

followin determinant

) 1

e)

*06.H&al"ate the

1

3

Ans.

3

1

followin determinant e2

e3

2e2

3e3

Ans. 1 *0*.H&al"ate the followin determinants se'  tan  se'  tan  se'2  Ans. se'  *08.H&al"ate the followin determinants  ln  1 1 O ln  Ans.  O ln  G = 1C *0!.-f the determinant of the matri 5 E 1, sol&e for the &al"e of ? 1  2 ) Ans. 3

*10.Find the determinants of ? 1 0 4 E 2 2 6   3 1 12 Ans. 46 *11.Find the determinant of  ? 4 1 2 3 2 0 2 1 10 3 0 1 14 2 4 ) Ans. 28  ADJOINT OF A MATRIX 

*12.Find the ad@oint of the matri, when it eist, nd the in&erse

AE Ans.

 1 1 3

2 0 1 3

A1 E 1J* MATHEMATICS SET 2

4 3 2 8

6

* 14

* 1 ) 2 *13.Find the ad@oint matri of  60

1 0

4 2

 

S0HERICAL TRIGONOMETRY 

Ans. 2 4 0 1 *14.Find the in&erse matri of  2 1

PLANE TRIGONOMETRY  TRIGONOMETRIC IDENTITIES

1. -f tan a E 4J), determine 4 sin a = 'os a 3 'os a O sin a Ans. 11J1! 2.
3 1

Ans. 1 3  1 2 *1).7etermine the in&erse matri of  'os θ  sin θ sin θ   'os θ

*16.Find the ad@oint of the matri

Ans.

  0LANE and

0 2 2

2 0 1 2 1 4

4 3 2

4 2 2

3. #hat &al"e of F satisfy the e$"ation tan G8F O 1C E 'ot 1*0 where all anles are in derees Ans. ! 4. -f tan G27 = 3C E 1 tan G)7 =!C determine 7 in derees. Ans. 14.)*0 ). -f 'os G!5C E 1 's' G35 O 6C nd 5 in derees. Ans. *

12. -f se' 2 A is )J2, the $"antity 1 = sin 2 A is e$"i&alent to ]]]]]]. Ans.

0.40

sin2 θ G1 O 'ot 2 θC Ans.

of the same anle. Find

300

20.!00, 6!.10

e$"al to ]]]]]]].

the s$"are of the 'osine

Ans.

Ans.

13. 'os4 A = sin 4 A is

e$"ation

Ans.

the anle.

Ans.  'os θ  sin θ  sin θ  'os θ

1 1  1

a 'ertain anle is twi'e of 

6. -f tan 4A E 'ot 6A, nd the &al"e of A. Ans. ! * -f sin 3A E 'os 69 then GA O 29C is? Ans. A O 29 E 300 8. ol&e for θ in the followin e$"ation? sin 2θ E 'os θ Ans. 300 !. implify the

'os 2A

1 14. sin G9 = AC is e$"al to

10. Find the &al"e of A between 2*0 0 and 360 0

 ]]]]]], when 9 E 2*0 0 and A is an a'"te anle.

if Ans.

 'os A

2

2 sin  A = sin A E 1 1). ol&e the e$"ation Ans.

330

0

11. -f sin  'os  O sin 2

'os2 A E 1 = 'os 2 A Ans.

4)0, 31)0

E 1, what are the &al"es of  61

16. -f sin A E 4J), A in

Ans.

2nd $"adrant, sin 9 E

200

sin Gar' 'os 1)J1*C

20. H&al"ate 'os Gar' tan

Ans.

8J1*

Ans.

1

28. -f tan A E + and tan

st

*J2), 9 in 1  $"adrant.

1)J8 = ar' sin *J2)C

Find sin GA O 9C. Ans. Ans.

MATHEMATICS SET 3

2!*J42)

1*. ow many di(erent

Ans.

2!. -f A O 9 O 5 E 180 G tan ar' 'ot 3 CZ

Ans.

πJ4

0

0  and 180  for the

Ans.

1

22. H&al"ate

e$"ation G2 sin  =1C

26. H&al"ate ar' sin 'os

G'os  O 1C E 0

ar' 'ot Y2 'os Gar' sin

ar' tan √3.

and tan A O tan 9 O tan 5 E ).6*, nd the & al"e of tan A tan 9 tan 5. Ans.

).6*

0.)CZ

3

Ans.

18. -f Gsin  O 1Jsin C E

Ans.

300

√ 2, then sin   O 1Jsin

2

, is e$"al to? 0

1!. -f ** 0 O 0.40 E ar' tan G'ot 0.2)C, nd ?

30. ol&e for  in the

0

30

2*. -n the i&en 2

Ans.

4)0

tan ar' sin Y 3J√2

31J2J2CZ is e$"al to?

&al"es of  lie between

Ans.

GA O 9C.

2). H&al"ate

3J) 21. Ar' tan Y2'os Gar' sin

0

9 E 1J3, nd the anle

23. Find the &al"e in derees of ar' 'os Gtan

e$"ation below sol&e for .

e$"ation? ar' tan G2C O ar' tan GC E πJ4

240C  E Gtan φ O 'ot φC2 Ans.

63.)6

0

Ans.

0.281

sin2 φ  tan 2 φ 24. Find the &al"e of 62

31. -f ar' tan  O ar'

Ans.

0.862

Ans.

0.46!8

tan 1J3 E πJ4, nd the

3). -f sin θ E 3J), 'os β E

3!. ol&e for the &al"e

&al"e of .

)J13, nd Gθ O βC

of  if?

Ans.

+

Ans.

32. ol&e for  if ar' tan

36. ol&e for the &al"e of

= 's'h2  E ese'ant θ Ans.

Ar' sin G3 = 4yC E

0.!6!

42. ol&e for θ if 'oth 2 

πJ2

600

43. Find the &al"e of θ in the e$"ation

0

J2 O ar' tan J3 E 4) . Ans.

1

33.ol&e for  from the

A: when sin A E 3.) and

Ar' 'os G = yC E πJ3

'os A E ).).

Ans.

1

32.4*0

40. -f 'o&ersed sin θ E

3*. -f sin A E 2.)11, 'os

0.134, nd the &al"e of

A E 3.06 and sin 2A E

θ

Ans.

'osh2  = sinh 2  E 2 sin θ. Ans.

300

i&en trionometri' e$"ation. Ar' tan G1 = C O ar' tan G1 O C E ar' tan

3.!3!, nd the &al"e of

se' θ  Gse' θC sin2 θ Ans.

600



Ans.

'os θ

41. Find θ if &ersed sin θ

1J8 Ans. Ans.

44. implify the epression

0.2)6

4 38. -f sin G O yC E 0.*66

34. -f ar' tan 4J3 O ar' tan 12J) E ar' sin G O

E 0.423 Ans.

)4.*60

and sin G = yC E 0.1*36,

4). implify

nd sin  'os y.

4 'os y sin y G1 2 sin 2 yC

yC, nd G O yC 63

Ans.

sin 4y

46. ol&e for θ? sin θ  se' θ O 's' θ  tan 2 θ E

48. i&en three anles A,

)2. ol&e for 'os *2 0 if

)6. ow many di(erent

9, and 5 whose s"m is

the i&en relationship is

&al"e of  from 0 0 to

1800. -f the tan A O tan 9

'os 2A E 2 'os 2 A = 1

O tan 5 E , nd the

0.0866

Ans.

0.30!

&al"e of tan A  tan 9  Ans.

430

4*. #hat are the ea't

tan 5.

of a'"te anle A, i&en that

Ans.

'os A E 2 √

10J* ;

)*. Find the &al"e of  in 4!. #hat is the sine of

10J20

Ans.

Ans.

0.!84

1

)1. -n 'o&ersine θ is

Ans.

Ans.

1*J! the e$"ation 's'  O 'ot

)4. Find the &al"e of 'os

820 

0.134, nd the &al"e of θ. tan A E 3 √

3

sin G ar' 'os 1)J1* C



)0. 's' 2*00 E

sin A E 3J*

E 0 Ans.

Ans.

0

trionometri' f"n'tions

G2 sin  = 1C G'os  O 1C

)3. Find the &al"e of

&al"es of the 'osine and tanent

1800 for the e$"ation

0

60

 E 3.

Y ar' sin G1J3C O ar' tan Ans.

πJ)

G2J√)CZ )8. Find sin  if Ans.

G2J!C G √10 = 1C 2 sin  O 3 'os  = 2 E 0.

)). -f sin 40 0 O sin θ, nd Ans.

1 X )J13

the &al"e of θ. )!. -f 'os θ E √ 3J2, then Ans.

800

nd the &al"e of  if  E 1 = tan 2 θ. 64

Ans.

63. -f 'os θ E J2 then 1 =

2J3

60. -f sin θ  'os θ E 1J3, what is the &al"e if sin 2 θ Ans.

tan2 θ is e$"al to? Ans.

G22 = 4C J  2

6*. implify 'os 4 θ  sin4

θ.

e$"ation? ar' tan G2C O

61. -f  'os θ O y sin θ E 1 and  sin θ  y 'os θ E 3, what is the

ar' tan GC E πJ4 Ans.

2 'os2 θ 1

Ans.

68. ol&e for  ?

0.281



1 = tan2 a

6). ol&e for y if

1 O tan 2 a

y E G1Jsin  = 1Jtan C G1

Ans.

and y O 'os C Ans.

  O y  E 10 sin 

y E G1 O 'os 2 θC tan

0

62. -f  O y E !0 , then 66. ol&e for ? sin  tan y

 E 1 = Gsin θ  'os θC

Ans.

'ot 

θ.

is

e$"al to? sin y tan 

Ans.

'os 2a

6!. Find the &al"e of y?

2

Ans.

sin 2θ

sinh 2

sin G!0 0 O AC Ans.

relationship between 

2

Ans.

*1. Find the &al"e of

64. ol&e for  in the 8J!

MATHEMATICS SET 31

'os A

*2. -f tan θ E 2, whi'h of the

followin is

in'orre't aC sin θ E 1 J √ 1 O 4 bC se' θ E √ 1 O 4 'C 'os θ E 1 J √ 1 O 4 dC 's' θ E √ 1 O 4 J 2 Ans. sin θ E 1 J √ 1 O 4

2

Ans.

sin 2θ

*0.
*3. -n an isos'eles riht trianle, the hypoten"se

is e$"al to?

65

is how m"'h loner

ar' tan G1 = C E ar' tan

than its sides

1J18, what is the &al"e of

Ans.

√ 2 times

*4. -f tan 9 E  J √ 1 = 2, nd 'os 9 E  Ans.

√ 1 = 2



Ans.

)3.130

E 0.422618, nd the

80. -f &ersed sin θ E

Ans.

0.38, what is θ

Ans.

6 Ans.

tan U E ar' sin G O yC, 81. -f 2 'ot  E1, what is

πJ2 and ar' 'os G = yC EπJ3, what is the &al"e of 

nd G O yC. the &al"e of

) sin  = Ans.

3 'os 

 and y 2 sin

O b2C, nd 's' A E  Ans.

1.0 , 0.)

 O 'os 

Ga2 O b2C *8. ol&e for θ if

Ans.

Ga2 = b2C 'os G6θC E

1

's' G3θ O !C Ans. *6. -f ar' tan G1 O C O

0.62

84. -f ar' tan )J3 O ar'

)1.680

**. -f ar' sin G3 = 4yC E

*). -f 'os A E 2ab J Ga 2

Ans.

&al"e of sin  'os y.

!0

*!. 5o&ersed sin θ E 0.2,

GA O 9C E , nd 'os 9. 0.8

83. -f sin G O yC E 0.81!1)2 and sin G = yC

nd the &al"e of θ

8). -f sin A 'os A O sin 2A E 1, what is the &al"e of A in derees Ans. 20.!

1.4

82. -f sin A E 4J) and sin

Ans.

0.!!)

86. -f sin 2A E 2.6*18, sin A E 1.40) and 'os A E 4.323, what is the &al"e of  and A Ans. 0.22, 18 8*. -f ar' 'os Gsin 0.8C E 60 O 0.4), what is the &al"e of  in derees Ans. 24 88. -f ar' 'os Gsin 0.8C E 60 O 0.4), what is the &al"e of tan  66

Ans. 0.44) 8!. -f sin A E 1.*4*, 'os A E 2.83! and sin 2A E 2.!*), what is the &al"e of A in derees Ans. 31.6 !0.
)

!3. A, 9 and 5 are anles of a trianle. -f tan A O tan 9 O tan 5 E 12, nd the ratio tan A O tan 9 O tan 5 tan A  tan 9  tan 5 Ans. 1 !4. -f sin 2A E 'os 49, nd the &al"e if GA O 29C in derees. Ans. 4) !). -f ar' 'os Gsin 0.1)AC E 40 O 0.20A, what is the &al"e of sin A Ans. 0.603 !6. A E 4 sin θ O 6 'os θ and 9 E 4 'os θ  6 sin θ. Find the &al"e of A 2 O 92. Ans. )2 !*.
the opposite ede of the path wal/ the pedestal and Basta( s"btends an e$"al anles. 5omp"te the width of the path wal/. Ans. 4.4* m !!.
TRIANGLES

!8. >n one side of a pa&ed path wal/ is a pedestal with a Basta( on top of it.
MATHEMATICS SET 32

100.
distan'e of 120 m from it. 5omp"te the heiht of the tower. Ans. 6!.28 m 101.A &erti'al pole 'onsists of two parts, ea'h one half of the whole pole. At a point is the horiontal plane whi'h passes thro"h the foot of the pole and 36 m from it, the "pper half of the pole s"btend an anle whose tanent is 1J3. ow hih is the pole Ans. *2 m or 36 m 102.A Bapole 20 m hih stands on top of a tower that is !6 m hih. At what distan'e from the base of the tower will the Bapole s"btend an anle of 4 0. Ans. 46.48 m 103.-f anle θ represents the anle s"btended by a pi't"re ) ft hih when &iewed from a point A that is * ft below the pi't"re and 14 ft away from the wall on whi'h the pi't"re hans, sol&e for θ. Ans. 14.030

67

104.An obser&er ! m horiontally away from the tower obser&es it anle of ele&ation to be only one half as m"'h as the anle of ele&ation of the same tower when he mo&es ) m nearer towards the tower. ow hih is the tower Ans. 3m 10).A man nds the anle of ele&ation of the top of a tower to be 30 derees. e wal/s 8) m nearer the tower and nds its anle of ele&ation to be 60 derees. #hat is the heiht of the tower Ans. *3.61 106.Points A and 9 are 100 m apart and are of the same ele&ation as the foot of the bld.
from the top of tower A is 280 and the anle of ele&ation of the top of tower A from the base of tower 9 is 46 0.
10!.A and 9 are s"mmits of  two mo"ntains rise from a horiontal plain. 9 bein 1200 m abo&e plain. Find the heiht of A, it bein i&en that its anle of ele&ation as seen from a point 5 in the plain Gin the same &erti'al plane with A and 9C is )0 0, while the anle of depression of 5 &iewed from 9 is 28 0)8M and the anle s"btended at 9 by A5 is )0 0. Ans. 3002.33 m

110.A 40 m hih tower stands &erti'ally on a hillside Gslopin ro"ndC, whi'h ma/es an anle of  180 with the horiontal. A tree also stands &erti'ally "p the hill from the tower. An obser&er on top of the tower nds the anle of depression of the top of tree to be 260 and the bottom of the tree to be 38 0. Find the heiht of the tree. Ans. 10.62 m 111.A tr"'/ tra&els from point  northward for 30

min, then eastward for one ho"r, then shifted I 300 #. -f the 'onstant speed is 40 /ph, how far dire'tly from  in /m, will it be after 2 ho"rs Ans. 4*.88 112.A 'ar tra&els northward from a point 9 for one ho"r, then eastward for 30 min then shifted I 300 H. After ea'tly 2 ho"rs, the 'ar will be 64.* /m dire'tly away from 9. #hat is the speed of the 'ar in /ph Ans. 40 /ph 113.A motor'y'le tra&els northward from point  for half an ho"r, then eastward for one ho"r, then shifted I 30 0 #. After ea'tly 2 ho"rs, the motor'y'le will be 4*.88 /m away from . #hat is the speed of the motor'y'le in /ph Ans. 40 /ph MATHEMATICS SET 33 114.A b"s tra&els northward from point 7 for one ho"r, then eastward for 30 min, and then shifted I 300 H.
'onstant speed is 40 /ph. #hen the b"s is 64.* /m dire'tly away from 7, how lon in ho"rs has it tra&eled Ans. 2 11).-n a trianle A95, A9 E 20 m and 95 E 8 m. >ne possible dimension of 5A is? Ans. 2* ^  ^ 12 116.-n trianle 957, 95 E 10 m and 57 E 2) m.  ne possible perimeter of the trianle is? Ans. 3* 118.
a E 1), b E 20. Find '? Ans. 2* 120.i&en anle A E 32 0, anle 9 E *0 0, and side ' E2* "nits. ol&e for side a of the trianle. Ans. 14.63 "nits 121.-n trianle A95, A E 4) derees and anle 5 E *0 derees.
an"lar bise'tor from the &erte 5. Ans. 14.3 12).-n trianle A95, anle 5 E *00, anle A E 4) 0, A9 E 40 m. #hat is the lenth of the median drawn from &erte A to side 95 Ans. 36.3 m 126.
12!.From a point o"tside of an e$"ilateral trianle, the distan'e to the &erti'es are 10 m, 10 m, and 18 m. Find the dimension of the trianle. Ans. 1!.!4 130.Points A and 9 1000 m apart are plotted on a straiht hihway r"nnin east and west. From A, the bearin of a tower 5 is 320 I of # and from 9 the bearin of 5 is 26 0 I of H. Approimate the shortest distan'e of tower 5 to the hihway. Ans. 2*4 m 131.An airplane lea&es an air'raft 'arrier and Bies so"th at 3)0 mph.
'onta't with the 'arrier Ans. after 2.13 hrs 132.A stat"e 2 meters hih stands on a 'ol"mn that is 3 m hih. An obser&er in le&el with the top of the stat"e obser&ed that the 'ol"mn and the stat"e s"btend the same anle. ow far is the obser&er from the stat"e Ans. 2 √ ) m 133.From the top of a b"ildin 100 m hih, the anle of depression of a point A d"e Hast of it is 30 0. From a point 9 d"e o"th of the b"ildin, the anle of ele&ation of the top is 600. Find the distan'e A9. Ans. 100 √ 30J3 134.An obser&er fo"nd the anle of ele&ation of the top of the tree to be 2*0. After mo&in 10 m 'loser Gon the same &erti'al and horiontal plane as the treeC, the anle of ele&ation be'omes )4 0. Find the heiht of the tree.

Ans.

8.0! m

MATHEMATICS SET 34

13).From a point A at the foot of the mo"ntain, the anle of ele&ation of the top 9 is 60 0. After as'endin the mo"ntain one G1C mile at an in'lination of 30 0 to the horion, and rea'hin a point 5, an obser&er nds that the anle A59 is 13)0. #hat is the heiht of the mo"ntain in ft Ans. 124!3 136.A )0 = m &erti'al tower 'asts a 62.3 = m shadow when the anle of ele&ation of the s"n is 41.60. #hat is the in'lination of the ro"nd Ans. 4.*20 13*.A &erti'al pole is 10 m from a b"ildin. #hen the anle of ele&ation of the s"n is 4) 0, the pole 'ast a shadow on the b"ildin 1 m hih. Find the heiht of  the pole. Ans. 11 m

138.A pole 'ast a shadow of  1) m lon when the anle of ele&ation of the s"n is 61 0. -f the pole has leaned 1) 0 from the &erti'al dire'tly toward the s"n, what is the lenth of the pole Ans. )4.23 m

13!.An obser&er wishes to determine the heiht of a tower. e ta/es sihts at the top of the tower from A and 9, whi'h are )0 ft apart, at the same ele&ation on a dire't line with the tower.
120 m, how hih is tower A in m Ans. 40.* m 141.Points A and 9 are 100 m apart and are on the same ele&ation as the foot of a b"ildin.
a 10 m wide road as shown. -f the land 'ost P1, )00.00 per s$"are meter, what is the 'ost of the roadway 40 m 30 m

Ans. P601, 3)0.00 14).A man impro&ises a temporary shield from the s"n "sin a trian"lar pie'e of wood with dimensions of 1.4 m, 1.) m, and 1.3 m. #ith the loner side lyin horiontally on the ro"nd, he props "p the other 'orner of the trianle with a &erti'al pole 0.! m lon. #hat wo"ld be the area of the shadow on the ro"nd when the s"n is &erti'ally o&erhead Ans. 0.) m2

146.At 12?00 noon ship sailed from the port in

the dire'tion of  4) 0 # at a speed of 16 na"ti'al miles per ho"r. At 1?00 P another ship left the same port in the dire'tion I )00 # at a speed of 20 na"ti'al miles per ho"r. At what time will the se'ond ship be ea'tly Iorth of the rst ship Ans. 2?4!.4 P.. 14*.A ship sailed in the dire'tion of  42 0 # at a speed of 1) na"ti'al miles per ho"r.
aC #hat is the heiht of the tower in meter Ans. 32.! m bC ow far is point A from the foot of the tower Ans. )* m

MATHEMATICS SET 35

148.n the side ede of a pa&ed path wal/ is a Basta( restin &erti'ally on top of a pedestal.
the opposite ede a'ross the path wal/, the pedestal and the Basta( s"btend e$"al anles. 7etermine the width of the path wal/ in m. Ans. 2.44 m 1)1.A stat"e stands on top of a pedestal 3 m hih restin on a le&el ro"nd. At a point on the ro"nd, 4 m away from the pedestal, the stat"s and the pedestal s"btend e$"al anles. #hat is the heiht of the stat"e in m Ans. 10.* m

1)2.
71

1)3.From the top of tower A the anle of ele&ation of the top of tower 9 is 460. From the foot of tower 9 the anle of ele&ation of the top of tower A is 28 0. 9oth towers are on le&el ro"nd. -f the heiht of tower 9 is 120 m. aC ow hih is tower A in m Ans. 40.* m bC ow far apart are the towers in m Ans. *6.)* m 1)4.From the top of a b"ildin the anles of depression of points A and 9 are 21 derees and 32 derees, respe'ti&ely. Points A and 9 are 100 m apart and are on the same ele&ation as the foot of the b"ildin. aC ow hih is the b"ildin in m Ans. !!.) m bC ow far is A from the b"ildin in m Ans. 2)!.2 m 1)).A 'ar tra&els from point 5 northwards for

one ho"r, then eastward for half ho"r, and then shifted I 30 0 H. -f the 'onstant speed of the 'ar is 40 /ph, how far dire'tly from point 5 will it be after ea'tly 2 ho"rs Ans. 64.* /m 1)6.A b"s tra&el northward from point 7 for one ho"r, then eastward for 30 min, then shifted I 30 0 H.
what is the heiht of the b"ildin Ans. 26.* m

1)!.
160.A spheri'al trianle A95 has an anle 5 E !00 and sides a E )00 and $ E 800. Find the &al"e of  #: in derees Ans. *4.330 161.-n a spheri'al trianle A95, nd GA O 9C if 5 E !00, # E 4)030M, $ E 1300. Ans. 21*014M 162.From the i&en $"adrantal spheri'al trianle whose parts are

A E )8 010M, 9 E 81 0))M and $ E !00. 5omp"te for the side  a:. Ans. )8o2)M 163.From the i&en isos'eles spheri'al trianle, 'omp"te the &al"e of anle 5. A E 640, 9 E 64 0, #E 810 Ans. 144026M 164.Find the perimeter of a spheri'al trianle in whi'h a E **036M, # E 6301*M and $ E 10*023M if  it has a spe'ial radi"s e$"al to ) m. Ans. 21.66 m MATHEMATICS SET 36

16).From the i&en parts of a spheri'al trianle A95, 'omp"te for anle A. a E 1200, # E *301)M and $ E 6204)M. Ans. 13*0)6M 166.From the i&en parts of s spheri'al trianle, 'omp"te the &al"e of side  a:. A E )2 030M, 9 E *0 01)M and  5 E 120 01)M Ans. )*022M

72

16*.-n the spheri'al trianle shown, the followin parts are i&en? A E 40 018M $ E 100010M 5 E *) 000M # E 6)02)M Find side a and anle 9. Ans. 41013M)1M 63010M2*M 168.i&en the parts of the spheri'al trianle? A E 60 030M # E 3801)M a E 40030M Find the &al"e of side $. Ans. 46043M 16!.An airplane Bew from 7a&ao whose latit"de is 140 I and lonit"de of 121030M H on a 'o"rse  300 # and maintainin a "niform altit"de. At what lonit"de will the plane 'ross the e$"ator Ans. 113033M H 1*0.Iorthwest Airlines Bew from
will be its 'o"rse at the point where it 'rosses the e$"ator Ans.  280)6M # 1*1.A ship sailed from a point A Gat. 20 0 I and lonit"de 120 0 HC. After sailin for 64 ho"rs at a "niform speed alon a reat 'ir'le ro"te it rea'hes point 9 at latit"de 10 0  and lonit"de 1)0 0 H. Find its speed in na"ti'al miles per ho"r. Ans. 3!.) 1*2.Find the distan'e in na"ti'al miles between anila and
na"ti'al miles per ho"r, determine the n"mber of ho"rs it will ta/e to tra&el ydney. Ans. !.0* ho"rs 1*4.5al'"late the area of a spheri'al trianle whose radi"s is ) m and whose anles are 40 0, 6)0 and 1100. Ans. 1).2* s$. m. 1*).A riht spheri'al trianle has an anle 5 E !00, a E )00, and $ E 800. Find the side #. Ans. *4.330 1*6.-f the time is 8?00 a.m. <, what is the time in the Philippines, whi'h is lo'ated at 120 0 Hast lonit"de Ans. 4 p.m. 1**.An airplane Bew from anila G14 036M I, 12100)M HC at a 'o"rse of   300 H maintainin a 'ertain altit"de and followin a reat 'ir'le path. -f its ro"ndspeed is 3)0 /nots, after how many ho"rs will it 'ross the e$"ator Ans. 2.8* ho"rs

1*8.Find the distan'e in na"ti'al miles between anila and an Fran'is'o. anila is lo'ated at 14 036M I latit"de and 121 00)M H lonit"de. an Fran'is'o is sit"ated at 3* 048M I latit"de and 122 024M # lonit"de. Ans. 6046.2 1*!.Find the area of the spheri'al trianle A95, if  A E 12)0, 9 E *3 0, 5 E 840 and r  E 30 m. Ans. 1602.2 m2

180.Find the area of the spheri'al trianle A95, if  a E 3!0, # E 4!0, $ E 620 and r  E )0 m. Ans. 808.04 m2 181.Find the area of the spheri'al trianle A95, if  A E 63.30, 9 E *2.23 0, 5 E 8).2*0 and r  E 10 m. Ans. *1.2 m2 182.7etermine the area of the spheri'al trianle A95, if a % 23.2*0, # E 4).31*0, $ E )0.48 0 and r  E 220. Ans. 8,312 m2

73

183.7etermine the area of the spheri'al trianle A95 if A E 140 0, 9 E *)0, 5 E 86 0 and r  E 40 m. Ans. 3,3*! m2 184.-n the spheri'al trianle A95, A E 11601!M, 9 E )) 030M and 5 E 80 03*M. Find the &al"e of side  #:. Ans. )6.0320 18).-n the spheri'al trianle A95, a E 610, # E 430 and $ E 680. Find the &al"e of A. Ans. *0.)230 186.-n riht spheri'al trianle A95, 5 E !0 0, 9 E 3*0 and # E 310. Find the &al"e of  a: in derees. Ans. )2.880 18!.A riht spheri'al trianle A95, 5 E !0 0, 9 E 3*0 and # E 310. Find the &al"e of A: in derees. Ans. 68.*0 MATHEMATICS SET 37

1!0.-n riht spheri'al trianle A95, 5 E !0 0, A

E *4.*0 and $ E **.20. Find the &al"e of  a: in derees. Ans. *0.1)0 1!1.-n riht spheri'al trianle A95, 5 E !0 0, A E *4.*0 and $ E **.20. Find the &al"e of  #: in derees. Ans. 4!.260  ANALYTIC GEOMETRY  STRAIGHT LINES

1.
4. Find the median thro"h

G2,)C of the

trianle whose &erti'es are G6,2C, G2,2C, and G 2,)C. Ans. ) ). Find the 'entroid of a trianle whose &erti'es are G2,3C, G4,6C and G2, 6C. Ans. G0,1C 6. Find the area of trianle whose &erti'es are A G3,1C, 9 G),3C and

G11, 20C

2.
G2,8C. Ans. 38 *. Find the distan'e between the points G4, 2C and G),1C. Ans. !.48* 8. Find the distan'e between A G4,3C and 9 G2,)C. Ans. 10 !. -f the distan'e between points G8,*C and G3,yC is

13, what is the &al"e of y Ans. 1! or ) 10.
74

1*. Find the slope of the line by y =  E ). Ans. 1 18.
Ans. 3 = 2y O 1 E 0 2). >ne line passes thro"h the points G1,!C and G2,6C, another line passes thro"h G3,3C and G1,)C.  
32. #hat is the  = inter'ept of the line 3 = 4y O 12 E 0. Ans. 4 33. Find the anle, in derees, that the line 2 O )y O 10 E 0 ma/es with the  = ais. Ans. 1)8.2 34. A line passes thro"h G 2,4C and has a slope of 2. Find the y = inter'ept. Ans. 8 3). A line has a y = inter'ept of 12 and a slope of 2. #hat is the abs'issa of a point on the line whose ordinate are 6 Ans. 3 MATHEMATICS SET 3/

36. A line has a slope of 3 and a y = inter'ept of 8. -t passes thro"h point G2,yC. Find the &al"e of y. Ans. 2 3*. Find the area bo"nded by the line 2 = y O 10 E 0 and the 'oordinate aes. Ans. 2)

38. Find the e$"ation of the line whose slope is 2J3 and passes thr" the interse'tion of the lines  = 2y O 4 E 0 and 4 = 2y O 11 E 0. Ans. 3!. Find the slope of the line whi'h passes thr" G4,2C and the interse'tion of the lines  = 2y O 4 E 0 and 4 = 2y O 1 E 0. Ans. 1J6 40. 7etermine the anle between the  = ais and the line passin thr" G0,0C and the interse'tion of the lines 3 O 4y = * E 0 and ) = 2y = 3 E 0. Ans. 4)0 41. Point 5 is U of the way from A G1,4C to 9 G3,8C. 7etermine the abs'issa and ordinate of 5. Ans. 2,) 42. #hat is the distan'e from the oriin to the line  = 2y O 8 E 0  Ans. 3.)8 43. 7etermine the distan'e from the point G1,1C to the line  = 2y O 8 E 0. Ans. 3.13

75

44. Find the distan'e from the point G0,0C to the line 3 = y = 12 E 0. Ans. 3.*! 4). Find the distan'e from the point G4,)C to the line 3 O 2y O 6 E 0. Ans. 4.4 46. Find the anle between the lines y O  = 12 E 0 and 4y =  = 2 E 0. Ans. )!.040 4*. Find the anle between the lines y = 2 O 6 E 0 and 2y O 3 O * E 0. Ans. 11!.*0 48. Find the lenth of the line from point G),3C to the y = ais if its slope is 3J2. Ans. !.01 4!. #hat is the e$"ation of  the line bise'tin the bier anle formed by the interse'tion of the lines 2 = y O 2 E 0 and 2 O y = 4 E 0. Ans. y=3E0 )0.
)1. A line passin thr" G8,2C has a slope of U. #here does it interse't the y = ais Ans. G0,4C )2. #here does the line passin thro"h G),6C ha&in a slope of 3J2 interse't the  = ais Ans. G1,0C )3. A trianle with and area of 2 s$. "nits has two of its &erti'es at G2,4C and G 2,3C. Find the lo'"s of the third &erte. Ans.  = 4y O 10 E 0 )4.
e$"al to 4, nd the &al"e of . Ans. 2 )!. -f G,4C is e$"idistant from G),2C and G3,4C nd  Ans. 13 60.
straiht line, then the e$"ation of the line is? Ans. 2 O y E 1 E 0 66. Find the distan'e of the line 3 O 3y O ) E 0 from the oriin. Ans. 1 6*. Find the distan'e from a point G),3C to the line * = 4y E 28. Ans. 2.36 68. 7etermine the distan'e from G),10C to the line  = y E 0. Ans. 3.)4 6!. 5omp"te the distan'e from the oriin to the line 8 = )y = 121 E 0. Ans. 12.83

MATHEMATICS SET 3

*0.
*2. Find the distan'e between the i&en lines 4 = 3y E 12 and 4 = 3y E  8. Ans. 4 *3. Find the distan'e between the lines, 3 O y = 12 E 0 and 3 O y = 4 E 0. Ans. 8J√ 10 *4. #hat is the lenth of the line with slope 4J3 from a point G6,4C to the y = ais Ans. 10 *). Find the &al"e of y if a line ha&in a slope of 2J3 and passin thro"h the point G6, 2C interse'ts the y = ais. Ans. 2 *6. Find the &al"e of  if a line ha&in a slope of )J2 and passin thro"h point G4,)C interse'ts the  = ais. Ans. 2 **. A line of slope 3 passes thro"h the point G*,10C. -f point on the line has an abs'issa 2, nd its ordinate. Ans. ) *8. -f the slope of the line 4h O 6y O 8 E 0 is

e$"al to =4J3, nd the &al"e of h. Ans. 2 *!. Find the y 0 inter'ept of a line ha&in a slope of 3 and passes thro"h G 1,6C. Ans. ! 80. Find the abs'issa of a point ha&in an ordinate of 4 of a line that has a y = inter'ept of 8 and a slope of 2. Ans. 2 81.
Ans. G16,2C 8). A line sement has its ends on the 'oordinate aes and forms with them a trianle of area 36. -f the sement passes thro"h the point G),2C, what is its slope Ans. 2 86. Find the area bo"nded by the line  = 2y O 10 E 0 the  = ais, the y = ais and  E 10. Ans. *) 8*. -n a 5artesian 'oordinates, the &erti'es of a trianle are dened by the followin points ? G2,0C, G4,0C and G3,3C. #hat is the area Ans. ! 88.
!0. ow far from the y = ais is the 'enter of the '"r&e 22 O 2y2 O 10 = 6y = )) E 0  Ans. 2.) !1. Find the area of the 'ir'le whose 'enter is at G2,)C and tanent to the line 4 O 3y = 8 E 0. Ans. 6π !2. 7etermine the area en'losed by the '"r&e  2 = 10 O 4y O y 2 E 1!6. Ans. 22)π !3. Find the shortest distan'e from the point G1,2C to a point on the 'ir'"mferen'e of the 'ir'le dened by the e$"ation  2 O y2 O 10 O 6y O 30 E 0. Ans. ).81 !4. 7etermine the lenth of  the 'hord 'ommon to the 'ir'les  2 O y2 E 64 and  2 O y2 = 16 E 0. Ans. 13.86 !). -f G3,2C lies on a 'ir'le with 'enter G1,1C, then the area of the 'ir'le is? Ans. 2)π !6.
!*. #hat is the radi"s of a 'ir'le with the followin e$"ation  2 = 6 O y 2 = 4y = 12 E 0. Ans. ) !8.
102.A 'ir'le has a diameter whose ends are at G3,2C and G12,6C. -ts e$"ation is? Ans. 4 2 O 4y2 = 36 O 16y = 1!2 E 0 103.Find the e$"ation of the 'ir'le with 'enter on  O y E 4 and ) O 2y O 1 E 0 and ha&in a radi"s of 3. Ans. 2 O y2 O 6 = 14y O 4! E 0 104.-f G3,2C lies on the 'ir'le with 'enter G1,1C then the e$"ation of the 'ir'le is? Ans. 2 O y2 O 2 = 2y = 23 E 0 10).Find the e$"ation of / for whi'h the e$"ation  2 O y2 O 4 = 2y = / E 0 represents a point 'ir'le. Ans. ) 106.#hat is the area bo"nded by the '"r&e  2 O y2 O 8 O 4y = 61 E 0  Ans. 2)4.) 10*.Find the perimeter of the '"r&e 2 O y2 O 8 O 4y = 61 E 0 Ans. )6.)

108.#hat is the area of the 'ir'le 2 O y2 = 6y E 0 Ans. 28.2* 10!.#hat is the e$"ation of  the 'ir'le whose 'enter is at G),)C with a radi"s of ) Ans.  2 O y2 = 10 = 10y O 2) E 0 110.Find the nearest distan'e from the point G8,10C to the 'ir'le  2 E y2 E 16y. Ans. 0.2) 111.7etermine the distan'e from the point G8,10C to the farthest point on the 'ir'le 2 O y2 E 16y. Ans. 16.2) 112.
Ans. 'ir'le 116.A 'hord of the 'ir'le  2 O y2 = 81 E 0 is tanent to the 'ir'le  2 O y2 = 16 E 0. #hat is the lenth of this 'hord Ans. 16.1 11*.Find the e$"ation of the 'ir'le with 'enter at G 3,8C and tanent to the line  = y O ) E 0. Ans. 2 O y2 O 6 = 16y = )) E 0 118.7etermine the lenth of the tanent to the 'ir'le 2 O y2 = 4 = ) E 0 from the point G8,2C. Ans. ).)* 11!.Find the lenth of the 'hord of the 'ir'le  2 O y2 O 4 O 6y = 32 E 0 if its distan'e from the 'enter of the 'ir'le is )m. Ans. 1*.32 m 120.Find the e$"ation of the radi'al ais of the 'ir'le 2 O y2 = 4 E 0 and  2 O y2 = 18 = 14y O 121 E 0. Ans. 18 O 14y = 12) E0 121.Find the lenth of the 'hord 'ommon to the

78

'ir'les  2 O y2 E 36 and 2 O y2 = 20 E 0. Ans. 11.4) 122.Find the lenth of the 'hord 'ommon to the 'ir'les  2 O y2 E 2) and 2 O y2 = 10y E 0. Ans. 8.66 123.7etermine the lenth of the 'hord 'ommon to the 'ir'le  2 O y2 E 64 and  2 O y2 = 16 E 0 Ans. 13.86 124.Find the lo'"s of the 'enter of the 'ir'le that mo&es s"'h that it is always tanent to the y = ais and another 'ir'le whose 'enter is ed at G4,2C with radi"s of 3. Ans. y2 = 4y =14 O 11 E 0 12).
126.ow lon is the 'hord of the 'ir'le  2 O y2 E 64 if this 'hord is tanent to  2 O y2 O ! E 0. Ans. 14.8 12*.7etermine the area bo"nded by the '"r&e  2 O y2 = 6y E 0. Ans. 28.2* s$. "nits 128.#hat is the radi"s of the 'ir'le 2 O y2 = 6y E 0 Ans. 3 12!.#hat is the radi"s of the 'ir'le 2 O y2 = 10 O 4y = 1!6 E 0 Ans. 1) 130.
Ans. G = 3C2 O Gy O )C 2 E 16 134.#hat is the 'enter of the '"r&e 2 O y2 = 2 = 4y = 31 E 0 Ans. G1,2C MATHEMATICS SET 41

13).ow far the y = ais is the 'enter of the '"r&e 22 O 2y2 O 10 = 6y = )) E 0 Ans. 2.) 136.Find the distan'e of the 'enter of the 'ir'le 2 2 = 8 O 2y2 O 12y E 1 from the  and y aes. Ans. G2,3C 13*.ow far is the 'entroid of the 'ir'le  2 O y2 = 10 = 24y O 2) E 0 from the line y O 2 E 0. Ans. 14 138.5omp"te the ratio of the area of the 'ir'le  2 O y2 = 10 = 24y O 2) and the 'ir'le  2 O y2 = 10 O 4y = * E 0. Ans. 4 13!.
the rst and the se'ond 'ir'le. Ans. ) 140.Find the e$"ation of a 'ir'le that is 'on'entri' with the 'ir'le  2 O y2 = 4 = 6y O 4 E 0 and a radi"s of 4. Ans. 2 O y2 = 4 = 6y = 3 E 0 141.Find the e$"ation of the 'ir'le that is 'on'entri' with the 'ir'le  2 O y2 = 6 O 4y = 23 E 0 and tanent to the line 3 O 4y = 26 E 0. Ans. G =3C2 O Gy O 2C 2 E 2) 142.Find the distan'e between the 'enters of the two 'ir'les  2 O y2 = 4 = 4y O 4 E 0 and  2 O y2 = 4 O 8y O 4 E 0. Ans. 6 143.Find the e$"ation of the 'ir'le tanent to the y = ais and the 'enter is at G),3C. Ans. G =)C2 O Gy = 3C 2 E 2) 144.Find the e$"ation of the 'ir'le whose 'enter is at G1,3C and the 'ir'le passes thro"h G3,)C. Ans. G =1C2 O Gy O 3C 2 E 100 79

14).Find the e$"ation of the 'ir'le tanent to the line 3 O 4y E 1) and the 'enter is at G 3,4C. Ans. G O 3C2 O Gy O 4C 2 E 64 146.Find the area of a 'ir'le with 'enter at G 4,)C and tanent to the line y E  = 4. Ans. 84.)π 14*.Find the e$"ation of a 'ir'le ha&in its 'enter on the line 3 = 2y = 22 E 0 and tanent to the y = ais at G0,1C. Ans. 2 O y2 = 16 = 2y O 1 E 0. 148.Find the e$"ation of the 'ir'le 'ir'"ms'ribin a trianle whose &erti'es are G0,0C, G0,)C and G3,3C. Ans. 2 O y2 =  = )y E 0 14!.Find the e$"ation of a 'ir'le 'ir'"ms'ribin the trianle formed by the lines y E 0, y E  and 2 O 3y E 10. Ans. 2 O y2 = ) O yE0 1)0.A trianle has its sides ha&in e$"ations e$"al

to  O 2y = ) E 0, 2 = y = 10 E 0 and 2 O y O 2 E 0. Find the e$"ation of the 'ir'le ins'ribe in the trianle. Ans. G = 2C2 O Gy O 1C 2 E) 1)1.A 'ir'le has its 'enter at G0,0C and its radi"s is 10 "nits. 7etermine the e$"ations of the lines

1)4.
thro"h G1),1)C and 2 O y2 = 16 O 2y O )2 E 0 1)*.A 'ir'le has its 'enter on the line 2y E 3 and tanent to the  =ais at G4,0C. Find its radi"s. Ans. 6 1)8.Find the e$"ation of a 'ir'le passin thro"h the interse'tion of the lines 4 O y = 4 E 0 and  = y = 6 E 0 with 'enter at G1,3C. Ans. 2 O 2y O y 2 O 6y E0 1)!.
tanent to the 'ir'le. Ans. 10.4)

 = 0,303y E

 = 3.3y E  34.48 1)2.7etermine the lenth of the line @oinin the interse'tion of the e$"ation  2 O y2 E 48 and 2 O 8y E 0. Ans. 11.31 1)3.Find the distan'e between the point of interse'tion of the line  = 2y O 2 E 0 and the 'ir'le 2 O y2 O 6 O 6y O 8 E 0. Ans. 4.4* "nits

160.
164.Find the lenth of the 'ommon 'hord of the 'ir'les 2 O y2 E 64 and 2 = 24 O y 2 O 44 E 0. Ans. 13.22 16).Find the lenth of the 'ommon 'hord of the '"r&es 2 O y2 E 64 and 2 O y2 = 16 E 0. 80

Ans. 13.86 166.
1*0.Find the e$"ation of the line passin thro"h the points of interse'tion of the 'ir'les  2 O y2 O 4 E 0 and 2 O y2 = 4 O 2y = 4 E 0. Ans. 4 = y O 2 E0 1*1.Find the radi'al 'enters of the radi'al ais of the three i&en 'ir'les.  2 O y2 = 12 O 11 E 0,  2 O y2 = 4 = 21 E 0, and  2 O y2 = 4 O 16y O 43 E 0. Ans. G2,4C PARABOLA

1*2.#hat is the lenth of the lat"s re't"m of the parabola  2 = 8 = 20y O )6 E 0. Ans. 20 1*3.#hat is the lenth of the lat"s re't"m of the parabola  2 O 8 = 16y O 32 E 0  Ans. 16 1*4.#hat is the lenth of the lat"s re't"m of the parabola  2 E 4Gy = 2C Ans. 4 1*).#here is the fo'"s of the parabola  2 E 4 Gy = 2C Ans. 0,3

1*6.#hat is the e$"ation of  the dire'tri of the parabola  2 E 4Gy = 2C Ans. y=1E0 1**.#here is the fo'"s of the '"r&e 2 E 12y Ans. 0, 3 1*8.ow far is the dire'tri of the parabola G =4C 2 E  8Gy = 2C from the  = ais. Ans. 6 1*!.Find the distan'e from the 'enter of the parabola G = 3C 2 E 12y to the line 2 = 3y = 12 E 0. Ans. 1.* 180.ow far is the dire'tri of the parabola G = 4C 2 E 8Gy = 3C. Ans. 1 181.Find the distan'e of the &erte of the '"r&e G = 2C2 E 4y to the line 2 = 3y = 8 E 0. Ans. 1.1 182.A parabola has its &erte at G2,3C and its dire'tri is y E *. Find the lenth of its lat"s re't"m. Ans. 16 183.A parabola with ais on the ais passes thr" G 3,6C and its &erte is at

oriin. 7etermine the lenth of its lat"s re't"m. Ans. 12 184.
81

18!.A paraboli' ar'h has width of 18 m a'ross the bottom. At a &erti'al distan'e of 3 m abo&e the bottom, the width a'ross the ar' is 12 m. #hat is the heiht of the ar' in m Ans. ).4 1!0.
1!3.
b"lbMs 'enter as fo'"s. Find the depth of the headliht, in 'm, if the b"lbMs 'enter is 2 'm from the &erte and the radi"s of the headliht is 10 'm. Ans. 12.) 1!4.
f"n'tion y E 2 2 = * E ). Ans. 4 = * E 0 202.Find the e$"ation of the lo'"s of the 'enter of the 'ir'le whi'h mo&es so that it is tanent to the y = ais and to the 'ir'le of radi"s one G1C with 'enter at G2,0C. Ans. y2 = 6 O 3 E 0 203.Find the e$"ation of the parabola with &erte at G4,3C and fo'"s at G4,1C. Ans. 2 = 8 O 16y = 32 E0 204.Find the area bo"nded by the '"r&es  2 O 8y O 16 E 0,  = 4 E 0, the  = ais, and the y = ais. Ans. 10.6* s$. "nits 20).Find the area in s$. "nits bo"nded by the parabolas  2 = 2y E 0 and  2 O 2y = 8 E 0. Ans. 10.* 206.#hat is the lenth of the lat"s re't"m of the parabola  2 E 20y. Ans. 20 20*.
208.Find the e$"ations of th dire'tri of the parabola 2 E 16 y. Ans. y E 4 20!.Find the e$"ation of the dire'tri of the parabola y2 E 16. Ans.  E 4 210.
Ans. 2 E !00y 214.-f an a"tomobile headliht reBe'tor is '"t by a plane thro"h its ais, the se'tion is a parabola ha&in the liht 'enter as a fo'"s. -f the liht is 18 mm from the &erte and the diameter of the liht is 2)0 mm, nd the depth of the headliht. Ans. 21* mm 21).An ar'h is 18 m hih has the form of a parabola with &erte ais.
21*.A D"onset h"t has a paraboli' 'ross = se'tion whose heiht is 30 m and whose base is 60 m wide. -f a 'eilin 40 m wide is to be pla'ed inside the h"t, how hih will it be abo&e the base Ans. 16.6* m 218.An ar'h in the form of a paraboli' '"r&e with &erti'al ais is 40 m a'ross the bottom. A Bat horiontal beam 2).2!8 m lon is pla'ed 12 m abo&e the base. 5omp"te the heiht of he ar'h at the 'enter. Ans. 20 m

MATHEMATICS SET 44

21!.Find the e$"ation of a parabola with aes &erti'al and passin thro"h G0,0C, G1,0C and G),20C. Ans. 2 =  O y E 0 220.Find the lo'ation of the fo'"s of the parabola y 2 O 4y = 4 = 8 E 0. Ans. G2,2C

221.#here is the &erte of the parabola  2 E 4 Gy = 2C. Ans. 0,2 222.#here is the fo'"s of the parabola  2 E 4 Gy = 2C Ans. G0,3C 223.Find the fo'"s of the parabola G O 2C 2 E 16 Gy = 3C. Ans. G2,1C 224.Find the lo'ation of the &erte of the parabola G O 2C2 E 16 Gy = 3C. Ans. G2,3C 22).Find the e$"ation of the parabola with &erte at G),2C and fo'"s at G), 4C. Ans. 2 = 10 O 8y O 41 E0 226.5omp"te the lenth of the lat"s re't"m of parabola y 2 O 8 = 6y O 2) E 0. Ans. 8 22*.A parabola has its ais parallel to the y = ais, one end of its lat"s re't"m is at G!,6C and the &erte is at G),4C. Find the lenth of the lat"s re't"m. Ans. 8

228.Find the dire'tri of the parabola  2 O 2 O 12y O 3* E 0. Ans. yE0 22!.Find the distan'e of the dire'tri of the parabola G O 2C2 E 16 Gy = 3C from the  = ais. Ans. * 230.5omp"te the distan'e of dire'tri from the lat"s re't"m of the parabola G = )C 2 E 8 Gy O 2C Ans. 4 231.Find the e$"ation of the dire'tri for a parabola ha&in its &erte at G6,0C and passin thro"h G2,1C whose ais is parallel to the y = ais. Ans. yO4E0 232.
alon y = ais and &erte at the oriin. Ans. 2 E 16y 23).Find the e$"ation of a parabola with &erte at G3,2C and the fo'"s at G0,2C. Ans. y2 = 4y = 12 = 32 E0 236.Find the e$"ation of a parabola ha&in its dire'tri of y E * and its &erte at G2,6C. Ans. 2  4 O 4y = 20 E 0 23*.Find the e$"ation of the ais of symmetry of  the f"n'tion y E 2 2 = * O ). Ans. 4 = * E 0 238.A parabola has an e$"ation of  2 E 9y O 5 whose &erte is at G0, 2C and passes thr" G8,6C. 7etermine the &al"e of 9 and 5 and the lat"s re't"m. Ans. 16, 32, 16 23!.A '"r&e has an e$"ation of  2 E 5y O d.
240.#hat is the heiht of the paraboli' ar'h, whi'h has a span of 48 ft and ha&in a heiht of 20 ft at a distan'e of 16 ft from the 'enter of the span Ans. 36 ft 241.Find the distan'e from the &erte of a parabola 2 E 4 Gy =2C to the line 3 O 4y O 2 E 0. Ans. 2 242.Find the shortest distan'e from the &erte of the parabola G = 4C 2 E 16y to the line 3 = 4y = 1* E 0. Ans. + 243.A point mo&es so that its distan'e from the point G2,1C is e$"al to its distan'e from the  = ais.
24).o'ate the 'enter of the '"r&e 2 O 16y 2 = 16 O !6y O 144 E 0. Ans. G8,3C 246.#hat is the lenth of the ma@or ais of the '"r&e 2 O 16y 2 = 16 O !6y O 144 E 0. Ans. 4 24*.o'ate the &erte, 'losest to the oriin, of the ellipse  2 O 16y = 16 O !6y O 144 E 0. Ans. G2,3C 248.Find the distan'e between the fo'i of the '"r&e 2)2 O !y2 = 300  = 144y O 12)1 E 0. Ans. 8 24!.#hat is the lenth of the line 'onne'tin the &erti'es of the '"r&e 2)2 O !y2 = 300 = 144y O 12)1 E 0 Ans. 10 2)0.ow far apart are the dire'tresses of the '"r&e 2)2 O !y2 = 300 = 144y O12)1 E 0 Ans. 12.) MATHEMATICS SET 45

ELLIPSE

2)1.7etermine the lenth of the lat"s re't"m of

the '"r&e 2) 2 O !y2 = 300 = 144y O 12)1 E 0. Ans. 3.6 2)2.Find the e''entri'ity of the '"r&e 2) 2 O !y2 = 300 = 144y O 12)1 E 0. Ans. 0.8 2)3.Find the lenth of the lat"s re't"m of the '"r&e !2 O 2)y 2 =22) E 0. Ans. 3.6 2)4.
Ans.

)

2)8.A point mo&es s"'h that the s"m of its distan'e from G4,0C and G4,0C is always 10. Find the lenth of the minor ais of the '"r&e form"la. Ans. 6 2)!.A 'losed '"r&e is formed by the lo'"s of a point whi'h mo&es s"'h that the s"m of its distan'es from G4,0C and G4,0C is always 10. Find the distan'e from one dire'tri to the 'enter of the '"r&e. Ans. 6.2) 260.A point mo&es s"'h that the s"m of its distan'es from G8,0C and G8,0C is always 20. Find the distan'e from one &erte to the 'enter of the '"r&e formed. Ans. 10 261.
Ans. 2)!0.6 m 262.An ellipti'al lot has a semi = ma@or ais of 8 m.  
whose diameters are 10 m and 8 m lon. Ans. 0.6 268.
2*1.
is the distan'e between the dire'tri'es Ans. 20 2*3.
85

2**.
282.
283.
Ans. 1.) 28!.
e$"ation of 4 2 O !y2 = 64 O )4y O 301 E 0. Ans. 8J3 2!).5omp"te the distan'e between the dire'tress of the '"r&e ! 2 O 2)y 2 = )4 = 2)0y O 481 E 0. Ans. 12.) 2!6.7etermine the distan'e between the fo'i of the '"r&e !2 O 18 O 2)y 2 = 100y E 116 Ans. 8 2!*.i&en the e$"ation of the ellipse ! 2 O 4y2 = 24y = *2 O 144 E 0. 5omp"te the lo'ation of the &erti'es. Ans. 4,6 and 4,0 2!8.
2!!.An ellipse has an e''entri'ity of 1J3. Find the distan'e between the two dire'tri if the distan'e between the fo'i is 4. Ans. 36

86

300.
MATHEMATICS SET 47

306.Find the area in the se'ond $"adrant bo"nded the '"r&e  2 J 4 O y2 J ! E 1 and the 'oordinate aes. Ans. 3J2π 30*.Find the area bo"nded by the ellipse  2 O 16y 2 = 16 O !6y O 144 E 0. Nse area E π ab. Ans. 16π 308.Find the area en'losed by the '"r&e ! 2 O 18 O 2)y2 = 100y E 116 Ans. 1)π s$. "nits 30!.A semi = ellipse and a parabola rest on the same 60 meters wide and 20 meters hih. Nsin the 'ommon base as  = ais, 'omp"te the di(eren'e of  ordinates at points 2) meters from the 'enter of  the base. Ans. 4.!) m 310.A satellite orbits the earth in an ellipti'al path of e''entri'ity 0.6 and semi = minor ais of lenth 12,000 miles. -f the 'enter of the earth is at one of the fo'i, nd the

maim"m altit"de of the satellite. Ans. 24,000 311.
31).Find the e$"ation of the "pward asymptote of the hyperbola whose e$"ation is G = 2C 2 J ! = Gy O 4C2 J16 Ans. 4 = 3y = 20 E0 316.
from the oriin. #hat is the e$"ation of the hyperbola Ans. y E 16 321.
32).#hat is the &erti'al asymptote of the '"r&e y E 3 J G3 = 32 O  = 3C Ans. =3E0 326.Find the lo'ation of the &erte of the hyperbola 16y2 = !2 O 36 O !6y = 36 E 0. Ans. G2,0C and G2, 6C 32*.Find the fo'"s of the hyperbola 16y 2 = !2 O 36 O !6y = 36 E 0. Ans. G2,2C and G2, 8C 328.From the i&en e$"ation of the hyperbola 16y 2 = !2 O 36 E 0. Find the area of the re'tanle on the aes of the hyperbola. Ans. 48 32!.Find the e$"ation of a hyperbola with a lat"s re't"m of 1 and slope of asymptotes of ± 1J2 . Ans. 2 = 4y2 E 4 330.Find the e''entri'ity of a hyperbola ha&in distan'e between fo'i e$"al to 18 and the distan'e between dire'tri'es e$"al to 2. Ans. 3 331.From the i&en e$"ation of the '"r&e 16 2 = !y2 =

128 = !0y = 113 E 0 lo'ate the 'enter of the '"r&e Gh,/C. Ans. G4,)C MATHEMATICS SET 4/

332.Find the e''entri'ity of the '"r&e 16 2 = !y2 = 128 = !0y = 113 E 0. Ans. 1.6* 333.-f the e''entri'ity of a hyperbola is e$"al to 2, 'omp"te the 'on@"ate. Ans. 1.1)) 334.Find the e$"ation of a hyperbola ha&in a lenth of lat"s re't"m of  18 and the distan'e between fo'i is 12. Ans. 32 = y2 E 2* 33).Find the e$"ation of a hyperbola with e''entri'ity 3 and lat"s re't"m of 8J3. Ans. *22 = !y2 E 2 336.Fow far from the  = ais is the fo'"s F of the hyperbola  2 = 2y2 O 4 O 4y O 4 E 0 Ans. 2.*3 33*.5omp"te the lenth of the lat"s re't"m of the

hyperbola ! 2 = 4y2 O !0 O 18! E 0. Ans. ! 338.Find the e$"ation of the asymptotes for a hyperbola Gy = )C 2 = G O )C2 E 36. Ans. y = ) E ± G O )C 33!.Find the e$"ation of the n"mbers of the family of hyperbola that passes thro"h G2,0C and whose fo'i are G4,0C and G4,0C. Ans. 32 = y2 E 12 340.Find the e$"ation of the path of a point that mo&es so that is distan'e from G4,0C is twi'e its distan'e from the line  E 1. Ans. 32 = y2 E 12 341.5omp"te the e''entri'ity of a hyperbola ha&in one &erte at G6,)C, 'on@"ate ais alon the  =a is and asymptotes

88

of ) = 6y = 30 E 0 and ) O 6y = 30 E 0. Ans. 1.)62 342.A point mo&es so that the di(eren'e between its distan'es from G0,)C and G0,)C is 8. Find the e$"ation of the lo'"s. Ans. !y = 16 2 E 144 343.Find the e$"ation of the hyperbola ha&in 'enter at oriin with trans&erse ais on the  = ais a E 4 and lat"s re't"m E 32. Ans. 42 = y2 E 64 344.Find the e''entri'ity of  a hyperbola whose trans&erse and 'on@"ate aes are e$"al in lenth. Ans. √ 2 34).For what &al"e of / will the line y E  O / be tanent to the hyperbola  2 = 4y2 E 48 Ans. ± 6 346.Find the e$"ation of the hyperbola whose asymptotes are y E ±

2 and whi'h passes thr" G)J2,3C. Ans. 42 = y2 E 16 34*.#hat is the e$"ation of the asymptote of the hyperbola  2 J y = y 2 J 4 E 1 Ans. 2 = 3y E 0 348. 5omp"te the a'"te anle formed by the asymptote of the hyperbola  2 = 3y2 = 8 = 14 E 0. Ans. 600 34!.5omp"te the lo'ation of the fo'i of an e$"ilateral hyperbola  2 = y2 E !. Ans. G3 √2,0C andG3√2,0C 3)0.5omp"te the lo'ation of the &erti'es of the e$"ilateral hyperbola y 2 = 2 E 16. Ans. G0,4C and G0, 4C 3)1.o'ate the &erti'es of the e$"ilateral hyperbola y E  18. Ans.  E 3 √2 ; y E 3√2 3)2.
passes thro"h points G2,2C. 5omp"te the e$"ation of the hyperbola. Ans. y E 4 3)3.
3)4.what is the e$"ation of the tanent to the '"r&e !2 O 2)y2 = 22) E 0 at G0,3C Ans. y=3E0 3)).Find the e$"ation of a line normal to the '"r&e 2 E 16y at G4,1C. Ans. 2 O y = ! E 0 3)6.#hat is the e$"ation of  the normal to the '"r&e 2 O y2 E 2) at G4,3C Ans. 3 = 4y E 0 3)*.Find the slope of the '"r&e 2 O y2 = 6 O 10y O ) at point G1,0C. Ans. 2J) 3)8.Find the e$"ation of the line normal to the tanent at G4,0C on the '"r&e 2 O 16y E 32 = 4 = y2

Ans. 4 = 3y E 16 3)!.Find the slope of the '"r&e 2 O y2 E 2) at G4,3C. Ans.  4J3 360.Find the slope of the '"r&e 2 E 6y O 10 at point G4,1C. Ans. 4J3 361.A '"r&e, in re'tan"lar 'oordinates is to ha&e slope e$"al to the ratio Jy at any of its point. -f this '"r&e m"st pass thro"h G1,0C, determine the e$"ation of the '"r&e. Ans. 2 = y2 E  362.7etermine the e$"ation of the line tanent to the raph y E 2 2 O 1 at the point G1,3C. Ans. y E 4 = 1 363.Find the e$"ation of the tanent to the parabola y2 E 8 at G2,4C. Ans. =yO2E0 MATHEMATICS SET 4

364.Find the e$"ation of the tanent to the parabola 2 E 6y O 10 thro"h point G*,)C.

89

Ans. and

10 = 3y E ))

4 = 3y E 13 36).Find the e$"ation of the tanents to the 'ir'le 2 O y2 E 2) passin thr" G2)J3,0C. Ans. 3 O 4y E 2) ; 3 = 4y E 2) 366.Find the e$"ation of the normal to the tanent of the parabola 2 E 16y at G8,4C. Ans.  O y E 12 36*.Find the slope of the '"r&e 2 E 6y O 10 at point G4,1C. Ans. 4J3 368.Find the e$"ation of the tanent to a 'ir'le 2 O y2 E 2) ha&in a slope of 1. Ans. y E  ± )√2 36!.Find the e$"ation of the tanents to the '"r&e !2 O 2)y 2 E 22) ha&in a slope of 2. Ans. y E 2 ± √10! 3*0.Find the lenth of the s"b = tanent and s"b = normal of the '"r&e  2 O y2 E 2) at point G 3,4C. Ans. G3,16J3C

3*1.Find the lenth of lenth of the s"b = tanent of the '"r&e y2 E 8 at G2,4C. Ans. 4 3*2.Find the lenth of the tanent from point G*,8C to the 'ir'le  2 O y2 = ! E 0. Ans. 2√26 3*3.Find the &al"e of / for whi'h the lenth of the tanent from point G),4C to the 'ir'le  2 O y2 O 2/y E 0 is one. Ans. ) 3*4.Find the lonest distan'e from G10,*C to the 'ir'le  2 O y2 = 4 = 2y = 20 E 0. Ans. 1) DIAMETER OF CONICS

3*).i&en the e$"ation of ellipse 64 2 O 2)y 2 E 1600 and the slope of the 'hord is 1J). Find the e$"ation of the diameter of ellipse. Ans. 64 O )y E 0 3*6.Find the e$"ation of the diameter of the ellipse 2 J 2) O y 2 J ! E 1 whi'h bise'ts all 'hords of slope m E 2. Ans. ! O )0y E 0. 3**.Find the e$"ation of the diameter of a parabola

y2 E 64 whi'h bise'ts 'hords parallel to the line 2 = 3y E 6. Ans. y = 48 E 0 3*8.Find the e$"ation of the diameter of the hyperbola y E ! whi'h bise't 'hords whose slope is =3. Ans. y E 3 3*!.Find the slope of the 'hords whi'h are bise'ted by the diameter of ellipse whose e$"ation is 2 = 3y E 0 if the ellipse has an e$"ation of 4 2 O !y2 E 36. Ans.  2J3 380.Find the slope of the 'hords whi'h are bise'ted by the diameter of hyperbola whose e$"ation is 3 = 2y E 0 if the e$"ation of the hyperbola is !y 2 = 42 E 36. Ans. 8J2* 381.Find the slope of the 'hords whi'h are bise'ted by the diameter of a parabola  E 3 if the e$"ation of the parabola is  2 E  6y. Ans. 1

382.Find the e$"ation of the diameter 'on@"ate to the diameter of hyperbola  = )y E 0 if the hyperbola has an e$"ation of 2 2 = )y2 E 10. Ans. 0 383.Find the e$"ation of the diameter 'on@"ate to the diameter 3 O y E 0 of the hyperbola y E 0. Ans. y E 3 384.Find the e$"ation of the polar of the point P G1,)C with respe't to the hyperbola 4 2 = !y2 E 36. Ans. 4 = 4)y = 36 E0 38).Find the e$"ation of the polar of the point A G ),)C with respe't to the ellipse 16 2 O 2)y 2 E 400. Ans. 16 = 2)y O 80 E0

POLAR COORDINATES

386.Nsin re'tan"lar 'oordinates, a point is lo'ated at G3,4C. Find its polar 'oordinates. Ans. ), )3.130 90

38*.-n re'tan"lar 'oordinates, the lo'ation of a point is G8,6C. Find its polar 'oordinates. Ans. 10, 36.!0 388.Nsin re'tan"lar 'oordinates, the lo'ation of a point is G),6C. Find its polar 'oordinates. Ans. *.81, )0.1! 0 38!.An point is at G),48 0C "sin polar 'oordinates. Find its re'tan"lar 'oordinates. Ans. 3.3),3.*2 3!0.Nsin polar 'oordinates, a point is at G*, 38 0C. Find its re'tan"lar 'oordinates. Ans. ).), 4.3 3!1.Find the 'enter and radi"s of the sphere  2 O y2 O2 = 4 = 6y = 2 = 11 E 0 Ans. G2,3,1C , ) 3!2.o'ate the 'enter of the sphere  2 O y2 O 2 O 8 = 2y O 1 E 0 Ans. 4 3!3.#here is the 'enter of the sphere  2 O y2 O 2 E 6y = 8 Ans. G0,3,4C

3!4.#hat is the radi"s of the sphere  2 O y2 O 2 E 6y = 8 Ans. ) MATHEMATICS SET 5

3!).
Ans. r G3 'os θ O 4 sin θC E 20 400.5hane the e$"ation G2 O y2C3 E 42 y2 to polar 'oordinates. Ans. r E sin 2 θ 401.Nsin polar 'oordinates, nd the polar e$"ation of the path of a point whi'h is e$"idistant from 2 points whose polar 'oordinates are G2a, 0C and Ga, πJ2C. Ans. 3a E 2r G2 'os θ sin θC

402.Find the area of the trianle whose &erti'es ha&e polar 'oordinates of G0,0 0C, G6,200C and G8, )00C. Ans. 12 403.#ha are the  and y 'oordinates of the fo'"s of the 'oni' se'tion des'ribe by the followin e$"ation r sin 2 θ E 'os θ GAnle θ 'orresponds to a riht trianle with ad@a'ent side  and

opposite side y and hypoten"se rC. Ans. G1J4, 0C 404.ransform r 2 sin 2 θ E 6 into re'tan"lar 'oordinates. Ans. y E 3 40).Find the area en'losed by the '"r&e r2 G4 sin 2 θ O ! 'os 2 θC E 36. Ans. 6π 406.>ne of the fo'"s of the '"r&e r2 G16 'os2 θ  ! sin2 θC E 144 Ans. G),0C 40*.Find the lenth of the lat"s re't"m of the '"r&e r 'os2 θ  4 'os θ E 16 sin θ. Ans. 16 408.
91

G,yC of a point in the '"r&e when θ E 300. Ans. G2.1*,1.2)C DIAMETER OF ELLIPSE

410.7etermine the e$"ation of the diameter of the ellipse !2 O 2)y 2 E 22) whi'h bise'ts all 'hords ha&in e$"al slopes of 3J). Ans. ! O 1)y E 0 411.
CENTROIDS OR INTERSECTION OF MEDIANS OF A TRIANGLE

414.
Ans.

2

41*.
Ans.

G2,1C the third &erte.

9!),8C and 5G10,1C. #hat is the abs'issa and ordinate of the 'entroid of the trianle Ans.

),4

420.
41).-n trianle A95, A is Ans.

)

G10,8C and G12,4C.
at G4,4C, 9G8,6C and its 'entroid is at G1,3C. 7etermine the 'oordinates of 5.

418.
Ans.

G1, 1 1J3C. Find the ordinate of 

416.
the third &erte.

trianle are at G2,6C, G3,

G12,4C and the semi = 'ir'le to"'hes the  = ais. Find the distan'e of  he 'entroid of the

8

2C and G4,2C. #hat is the

the trianle

'ir'le ends at G4,4C and

'omposite from the y = Ans.

ordinate of the 'entroid of 

diameter of a semi =

ais. 41!.
8.2*

trianle are at AG3,3C, 92