MATHEMATICS 1. These are are two lines lines that do not not intersect intersect and are are not parallel. parallel. skew lines 2. It is a plane plane curve traced traced by a point point on the perimeter perimeter of a circl circle e that is rolling around a xed circle circle of the same radius. cardioid 3. If f(x is continuo continuous us over a closed interval interval and di!eren di!erentiabl tiable e on the open interval (a"b" and if f(a#f(b#$" then there is at least one number c in (a"b such that f%(c#$. This statement is Rolle’s Theorem &. 'et the closed closed interval interval a" b) be the domain domain of function function f. f. The domain of f(x * 3 is given by the closed interval [a + 3 , + 3! +. The distance distance between between the points points ,- dened dened by ,(cos," ,(cos," *sin, *sin, and -(sin," cos, is eual to "#$"$ /. 0hat conic conic section section is described described by the the euation euation r # / (& 3cos4 3cos4 Elli%se 5. It is the derivati derivative ve with respect respect to to one of those those variables" variables" with the others held constant (as opposed to the total derivative" in which all variables are allowed to vary. &artial 'erivative 6. If f(x is continuo continuous us on the closed interval interval a"b) a"b) and di!erent di!erentiable iable on the open interval (a"b" then there is a number c in (a"b such that 7 f ( ( b )− f ( ( a ) = f ' ( c ) b− a
(aw o) Mean or Mean *ale Theorem 8. 9oint 9oint in the triangl triangle e in which thee thee :uler%s :uler%s line will will not pass pass incenter 1$. 1$. It is a typ type e of conc concen entr tric ic sphe spherres in a reg regul ular ar poly polyhe hedr dron on in whic which h
the sphere is tangent to all faces. Ins%here 11. It states that the surface are area of a sur surface of revolution generated by the revolution of a curve about an external axis is eual to the product of the arc length of the generating curve and the distance traveled by the curve;s geometric centroid.
irst &a%%s Theorem 12. It is one of the most basic curvilinear geometric shapes" the
surface formed by the points at a xed distance from a given line segment. The solid enclosed by this surface and by two planes perpendicular to the axis. C-linder 13. It is the solid generated by revolving a sector of a semicircle about its diameter as an axis. S%herical Sector 1&.
In suaring this matrix
" the output is an matrix itself. This
matrix is classied as Idem%otent matri/ 1/. The roots of the characteristic euation is called as ei0envales 15. In an x and y plane" =artesian coordinate plane is used to plot points. In plotting the real and imaginary axis of complex numbers" the complex plane used is called as Ar0and 'ia0ram 16. The >atrix Hermitian Matri/
is classied as
18.
√ 1 8
?ind the roots of ± 1, ± i,
2$.
± ( 1+i )
√ 2
?ind the principal root of
√ 3− 4 i 3
"#1245.#62.2
( ) +i 1− i
1
21.
0hat is the value of
2013
i 4 x iy ( + ) =−16 " then x must be If
22. ± √ 2
0hich of the following in not a real part of √ 1+ i 3
23.
√ 2cos 6
2&.
5 π 4
?ind the euation of the locus of the center of the circle which
moves so that it is tangent to the y*axis and to the circle of radius one (1 with center at (2"$ 2
y −6 x + 3= 0
2+.
?ind the length of the latus rectum of the following ellipse7 2
2
x + 9 y −300 x −144 y + 1251=0
3#1
2/.
2 2 The area of the ellipse 8 x + 25 y −36 x −189=0 is eual to7
" 5 π s7# nits
25.
The chords of the ellipse /&@2 A 2+y@2 # 1/$$ having eual
slopes of 1+ are bisected by its diameter. Betermine the euation of the diameter of the ellipse. 1$/ + 6- 8 . 26. , points move so that its distance from the point (2"*1 is eual to its distance from the x*axis. The euation of the locus is /2 9 $/ + 2- + 6 8 . 28. ?ind the locus of a point which moves in such a way that its distance from (3"$ is always eual to its distance from the origin. 2/ 9 3 8 . 3$. ?ind the circumcenter of the given triangle with coordinates of vertices as , (1"3" - (+"+" = (5"+. :1, ;2< 31. Betermine the coordinates of the point which is three*fths of the way from the point (2"*+ to the point (*3"+. :;","< 32. ?ind the area of one loop of the curve r2 # 6cos2C. $ 4 x
33.
:valuate
d cos u du dx x
∫ 2
$ cos $/ 9 2/ cos / 2 3 dx 2 3&. is 0 ( 1 − x )
∫
'iver0ent
3+.
Dolve for
54
1 2 2
( 9 + x )
arctan
dx x
x
1
3/.
∫
3 +
18 ( 9 + x
∫ (5 −4 xx− x ) / dx 2 3 2
2
) + c
is eual to
5 −2 x 9 √ 5 −4 x − x
35.
+ c
2
The orthogonal traEectories of the family of curves x2 A y2 # c2
are (ines 36. , tanF initially holds 1$$ gal of a brine solution containing 2$ Ib of salt. ,t t # $" fresh water is poured into the tanF at the rate of + galmin" while the well*stirred mixture leaves the tanF at the same rate. ?ind the amount of salt in the tanF at any time t. = 8 Ce;t>2. 38. Betermine the order and degree" respectively" of the di!erential 4
5
d b euation 5 ( d p 4 )
10
A
db ( ) dp
A
b
7
*
5
b
# p
)or, ?ve &$. Dolve (cos x cos y cot x dx sin x sin y dy # $ sin / cos - 8 ln :c sin /< &1. ?ind the general solution of y% # y sec x. - 8 C :sec / + tan /< &2. The population of a country doubles in +$ years.
years will it be ve times as much4 ,ssume that the rate of increase is proportional to the number of inhabitants. "2. -ears &3. 'ocate the centroid of the plane area bounded by y # x@2 and y # x. :.#6,.#$< &&. 'ocate the centroid of the plane area bounded by the euation y@2 # &x" x # 1 and the x*axis on the rst uadrant. :3>6, 3>$<
&+.
It is a solution not containing any arbitrary constant and is not
deducible from the general solution by giving particular values to the arbitrary constants in it. sin0lar soltion 2
&/.
The partial di!erential euation
2
∂ φ ∂ φ + 2 2 ∂ x ∂ y A
∂φ ∂φ + ∂ x ∂ y # $ has
de0ree " order 2 &5. , cylindrical tanF having a radius of 2m and a height of 6m is
lled with water at a depth of /m. =ompute the worF done in pumping all the liuid out of the top of the container. 3 14 23 @oles &6. , conical vessel 12m across the top and 1+m deep. If it contains water to a depth of 1$m nd the worF done in pumping the liuid to the top of the vessel. "2 32#6 kBm &8. , + lb. monFey is attached to a 2$ ft hanging rope that weighs $.3 lbft. the monFey climbs the rope up to the top.
2
from
( 0,0 ) to ( √ 6 , 6 ) about the y *
axis. 12%i>3 s7# nits +1.
?ind the area of the region bounded by the curves
¿
12 x 2 x + 4 " the
x*axis" x # 1" and x # &. 6 l n 4 s7are nits
+2. m$
The DI unit for moment of inertia is
E(ECTRBICS 1 0hat is the electric force between a strange and charm uarFs separated by 12 pico meters4 ;361 nC
2 , law stating that the current in a thermionic diode varies directly with the three*halves power of anode voltage and inversely with the suare of the distance between the electrodes" provided the operating conditions are such that the current is limited only by the space charge. Child’s (aw 3 ?ind the net charge enclosed in a cube 2m on an edge" parallel to the axes and centered at the origin" if the charge density is 2
ρ=50 x cos
( ) π 2
3
y ( μC / m )
.
$#4 C & It is an euation describing the magnetic eld generated by an electric
current. It relates the magnetic eld to the magnitude" direction" length" and proximity of the electric current Dio Savart (aw + The natural response of an RLC circuit is described by the di!erential dv ( 0 ) d v dv ( ) v v + + = = = 0 . The v (t is 2 0, 0 10, 2 dt dt d t 2
euation
( + t ) e−t V
10 1
/ G The current through or voltage across" an element in a linear bilateral networF is eual to the algebraic sum of the current or voltages produced independently in each source. H This is Fnown as S%er%osition Theorem 5 , 12 ? capacitor is allowed to discharge through its own leaFage resistance. Jsing an electronic voltmeter" it is observed that the voltage across the capacitor falls from 12$ K to 1$$ K in 3$$s. =alculate the leaFage resistance of the capacitor. "3 M 6 , capacitor of 1$ ? is connected to a dc supply through a resistance of 1.1 >Ω. =alculate the time taFen for the capacitor to reach 8$L of its nal charge. 26#33 s 8 0hen the transistor is saturated" K=: is approximately
.#2 * 1$ , transistor with M # 1$$ is connected as common base" was found to
have leaFage current I=-N # 1 ,. If the transistor is configured as common emitter" what is the approximate value of its I=:N4 ".. A 11 The maximum voltage that can be applied across the collector*emitter terminal for a given transistor is specied as *:DR
13 Dolve for the input impedance from the gure above. "#36 kF 1& =alculate the voltage gain from the gure above. 31#1 1+ The output voltage of the circuit shown below is
2*
1/ ,n op*amp has an open*loop gain of 5+"$$$ and a cuto! freuency of 1$$
D,
# 1.+P=0.
# $.+P =0" and the
Q=
# $./ P=0. 0hat is the maximum power
=D
that can be dissipated if the ambient temperature is +$P= and T Qmax # 2$$ P=4 6# G 16 , single*phase half*wave controlled converter is fed from a sinusoidal source. If the average output voltage is 2+L of the maximum possible average output voltage for a purely resistive load" then ring angle is %i>3 18 , thyristor has a maximum allowable Eunction temperature of 12$R= and the ambient temperature is &$R=. If thermal resistance is 1./R =0" the maximum allowable internal power dissipation is 6. G
EAS 1. , blocF having a mass of 2+$ Fg is placed on top of an inclined plane having a slope of 3 vertical to & horiOontal. If the coeScient of friction between the blocF and the inclined plane is $.1+" determine the force 9 that may be applied parallel to the inclined plane to Feep blocF from sliding down the plane. ""#2 B 2. ,n airtight closed box of weight 9 is suspended from a spring balance. , bird of weight 0 is place on the oor of the bow" and the balance reads 0 A 9. If the bird ies without accelerating what is the balance reading4 &+G 3. ,ny wave in which the vibration direction is perpendicular to the direction of propagation. transverse wave &. In a diverging lens" parallel rays will appear to originate from the virtal )ocs
+. ,n elevator weighing 2$$$ lb attains an upward velocity of 1/ fps in & sec with uniform acceleration. 0hat is the tension in supporting cables4 226. l /. The worF done by the net force on a particle euals the change in the particle%s Finetic energy.H This statement is Fnown as UUUUUUUUU. Gork;ener0- theorem 5. If the bodies sticF together after collision" the coeScient of restitution is eual to . 6. 'umen is a unit of measure for lminos / 8. , cable is to be strung from two towers at the same elevation and 6$$ ft apart. The weight of the cable is + lbft and the maximum sag can be 25$ ft. 0hat is the length of the cable4 ".". )t 1$. , +$$*g blocF is shot up the incline (shown below with an initial speed of 2$$ cms.
.#316 m 11. , car has wheels of radius 3$ cm. It starts from rest and
accelerates uniformly to a speed of 1+ ms in a time of 6 s. ?ind the angular acceleration of its wheels. 1#2 rad>s 12. , ball at the end of a 16$*cm long string swings as a pendulum as shown below. The ball%s speed is &$$ cms as it passes through its lowest point. To what height h above this position will it rise before stopping4
.#"1 m 13. Two baseballs are red into a pile of hay. If one has twice the
speed of the other" how much farther does the faster baseball penetrate4 (,ssume that the force of the haystacF on the baseballs is constant. )or times as mch 1&. 0hen a mass m is hung on a spring" the spring stretches /.$ cm. Betermine its period of vibration if it is then pulled down a little and released. .#$4 s 1+. , converging lens with +$ cm focal length forms a real image that is 2.+ times larger than the obEect. sec
EST 1. It is the process of converting a freuency or band of freuencies to another location in the total freuency spectrum.
re7enc- translation 2. It is the process of extracting a phase*coherent reference carrier from a
receiver signal. Carrier recover3. This type of coding compares the 9,> signal to a ramp waveform while a binary counter is being advanced at a uniform rate. (evel at a time &. It is a broad general term applied to any form of wireless communication from one point to another by converting them into electric and magnetic elds that propagate readily over long distances. Radio +. There are UUUUUUU primary unshielded twisted pair cables classied by the :I,TI, +/6 standard. Seven /. ?or a single channel 9=> system with a sample rate f s#/$$$ samples per second and a seven*bit compressed 9=> code" determine the line speed. $2 ... %s 5. , varactor has a maximum capacitance of 6$ p? and is used in tuned circuit with a 1$$ V< inductor. ?ind the resonant freuency with no tuning voltage applied. "# MHJ 6. UUUUUUUUUU are those with some form of conductor that provides a conduit in which electromagnetic signals are contained. ided Transmission Media 8. The band of light freuencies that is too high to be seen by the human eye with wavelengths ranging between 55$ nm and 1$/ nm In)rared 1$. In UUUUUUUU" the serial binary data is mixed with a higher freuency pseudorandom binary code at a faster rate and the result is used to phase modulate a carrier. 'irect se7ence 11. ?or a carrier freuency of / W
1&. , signal would have a bandwidth of 2$$ F
the mounting insulation has C =D # $./ o=0. 0hat is the maximum power dissipated if the ambient temperature is &$ o= and T Qmax # 2$$ o =4 1"#6 G
It is a geometry figure formed by two half-times emanating from a common point.
,ngle _______ are angles whose terminal sides coincide when the angles are in standard position. co-terminal angles The word calculus is derived from the latin word for?
Stone
c. Pebble
Who is the ather of !nalytic "eometry?
#ene $escartes
! _____ is the set of prints that are e%uidistant from a fi&ed point and a fi&ed line. Parabola
! rench mathematician who discovered matrices in the year '()*.
,rthur =ayley
ind the directed distance from line+&,y-)* to the point/-+0'1.
*11/
dx 2 = 3 ind the complete solution of ln & dy xy x
2
x
2
1
+ =c lnx 2 4 y 2 2
If there e&ist a functions
c 1 y 1 + c 2 y 2+ … + c n y n=0
y 1 ,
y 2 ,…, y n∧ c1 , c 2 , … c n
are constants not all 3ero such that
. Identically in some interval a4&4b0 then the functions y'
0y50..0yn are said to be0 . 6inearly dependent
! television antenna is on the roof of a building. rom a point on the ground +) ft from the building the angles of elevation of the top and bottom of the antenna are 7'o *8 and 5o *8 respectively. 9ow tall is the antenna? . '5.' ft ind dy if y /+& 5 , '15:+. 4 xdx
√ 3 x 2+ 1 3
ind the area of the triangle whose vertices are /-50-10 /'0)1 and /7051.
5) ;&press /sec & 2 tan &15 in terms of sin &. 1 −sinx 1+ sinx
! triangular trough is '* ft long0 + ft wide at the top an d 5 ft deep. If water flows in at the rate of ( cubic ft per minute0 how fast the surface is rising when the water in the trough is ' ft deep?
(:'7 ft:min
When a shot-putter e&erts a net force of '* < on a shot0 the shot has an acceleration of '= m:s5. What is the mass of the shot?
>. g
! 5* < stone rests on a table. What is the force the table e& erts on the store? In what direction? 5* <0 upward
It is defined as a definite amount of matter the parts of which are fi&ed in position relative to each other.
#igid @ody
___________ merely tells us how much flu& is leaving a small volume on a per-unit basisA no basis is associated with it.
$ivergence The __________ of a particle during some time interval is defined as the ra tio of change in its velocity0 and the time interval. average acceleration
. the line integral of 9 about any closed path is e&actly e%ual to the direct current enclosed by that path !mpere8s circuital law
It states that0 BThe speed of light in vacuu m is the same in all inertial frames of reference and is dependent of the motion of the source.C
;instein8s Second Postulate
This law states that the pressure of a fi&ed amount o f gas confined to a fi&ed volume is directly proportional to the Delvin temperature. . !monton8s 6aw
Eonvert /*.)(>71'* to binary *.'*''5
The _____ process was developed specifically to handle e&cess activated sludge from aerobic treatment plants without primary clarifiers. !naerobic digestion
Two o&ygen molecules in air are moving toward a head-on collision against each other. @efore collision0 molecule ! is moving to the right with a velocity of 5** m:s and @ is moving to the left with a velocity of '05** m:s5. ind the velocity of @ after collision if ! bounces bac with velocity of '05** m:s. 5** m:s !n adiabatic e&pansion of air through a no33le from (5( Pa and >' oE to '+( Pa. The initial inetic energy is negligible. or an isotropic e&pansion0 co mpute the specific volume at the e&it section. F. *.5= m+:g
! rubber ball weighs = <. What is the acceleration of the ball if an upward force )= < is applied. m:s5
Simplify the &y , &83 , y3 to a minimum number of literals. &y , &83 Gsing '*8s complement0 subtract >57+5 2 +57*. )=5(5
