A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $x$ ‐ $y$ plane centred at origin. A point
charge $q$ is moving towards the ring along the $z-$ axis and has speed $v$ at $z= 4a$ . The minimum value of $v$
such that it crosses the origin is:
A current of $5\,A$ passes through a copper conductor (resistivity $= 1.7 \times 10^{-8}\,\Omega \,m$ ) of radius of cross-section
$\,5\,mm$ . Find the mobility of the charges if their drift velocity is $\,1.1 \times 10^{-3}\, ms^{-1}$ .
a.
$1.5\,m^{2}\,V^{-1}\,s^{-1}$
b.
$1.8\,m^{2}\,V^{-1}\,s^{-1}$
c.
$1.0\,m^{2}\,V^{-1}\,s^{-1}$
Correct Answer
d.
$1.3\,m^{2}\,V^{-1}\,s^{-1}$
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Q3mcq
A ball is thrown upward with an initial velocity $V_{0}$ from the surface of the earth. The motion of the ball is
affected by a drag force equal to $\,m$ $\gamma$ $v^{2}$ (where $m$ is mass of the ball, $v$ is its instantaneous velocity and is a$\gamma$
constant). Time taken by the ball to rise to its zenith is:
Two particles of masses $M$ and $2M are\,$ moving with speeds of $10\,m\,s^{-1}$ and $5\,m\,s^{-1}$ , as shown in the figure.
They collide at the origin and after that they move along the indicated directions with speeds $v_{1}$ and $v_{2}$ ,
respectively. The values of $v_{1}$ and $v_{2}$ are, nearly
a.
$6.5\,m\,s^{-1}$ and $3.2\,m\,s^{-1}$
b.
$3.2\,m\,s^{-1}$ and $12.6\,m\,s^{-1}$
c.
$13.02\,m\,s^{-1}$ and $19. 7\,m\,s^{-1}$
Correct Answer
d.
$3.2\,m\,s^{-1}$ and $6.3\,m\,s^{-1}$
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Q5mcq
An $NPN$ transistor operates as a common emitter amplifier, with a power gain of $\,60\,dB$ . The input circuit
resistance is $\,100\,\Omega$ and the output load resistance is $\,10\,k\Omega$ . The common emitter current gain is:$\beta$
a.
$6 \times 10^{2}$
b.
$10^{2}$
Correct Answer
c.
$10^{4}$
d.
$60$
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Q6mcq
The displacement of a damped harmonic oscillator is given by $x(t) = e^{-0.1t} cos(10$ $\pi$ $t +$ $\phi$ $).$ Here $t$ is in
seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to:
a.
$27\,s$
b.
$4\,s$
c.
$13\,s$
d.
$7\,s$
Correct Answer
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Q7mcq
The electric field of a plane electromagnetic wave is given by
$\vec{E}= E_{0}ˆicos(kz)cos($ $\omega$ $t)$
The corresponding magnetic field $\vec{B}$ is then given by:
In the given circuit, an ideal voltmeter connected across the $10\,\Omega$ resistance reads $\,2\,V$ . The internal resistance
$\,r$ , of each cell is:
a.
$0\,\Omega$
b.
$1.5\,\Omega$
c.
$0.5\,\Omega$
Correct Answer
d.
$1\,\Omega$
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Q9mcq
A message signal of frequency $100\,MHz$ and peak voltage $100\,V$ is used execute amplitude modulation on a
carrier wave of frequency $300\,GHz$ and peak voltage $\,400\,V$ . The modulation index and difference between
the two side band frequencies are:
a.
$4; 2 \times 10^{8}\,Hz$
b.
$0.25; 2 \times 10^{8}\,Hz$
Correct Answer
c.
$\,4; 1 \times 10^{8}\,Hz$
d.
$\,0.25; 1 \times 10^{8}\,Hz$
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Q10mcq
A transformer consisting of $300$ turns in the primary and $150$ turns in the secondary gives output power of
$\,2.2\,kW$ . If the current in the secondary coil is $\,10\,A$ , then the input voltage and current in the primary coil
are:
a.
$440\,V$ and $20\,A$
b.
$220\,V$ and $20\,A$
c.
$440\,V$ and $5\,A$
Correct Answer
d.
$220\,V$ and $10\,A$
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A proton, an electron, and a Helium nucleus, have the same energy. They are in circular orbits in a plane due to
magnetic field perpendicular to the plane. Let $r_{p}, r_{e}$ and $r_{He}$ be their respective radii, then,
a.
$r_{e} > r_{p} = r_{He}$
b.
$r_{e} < r_{p} = r_{He}$
c.
$r_{e} < r_{p} < r_{He}$
d.
$r_{e} > r_{p} > r_{He}$
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A moving coil galvanometer allows a full scale current of $10^{-4}\,A$ . A series resistance of $2\, \times 10^{4}\,\Omega$ is required
to convert the galvanometer into a voltmeter of range $0 -5\,V$ . Therefore, the value of shunt resistance
required to convert the above galvanometer into an ammeter of range $0 -10\,mA$ is:
a.
$100\,\Omega$
b.
$200\,\Omega$
c.
$300\,\Omega$
d.
$10\,\Omega$
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In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown
in the figure, it is a straight line.
One may conclude that
a.
$R(T) = R_{0}e^{T^{2}/T^{2}}0$
b.
$R(T) = \frac{R_{0}}{T^{2}}$
c.
$R(T) = R_{0}e^{-T^{2}/T^{2}}0$
d.
$R(T) = R_{0}e^{-T^{2}}0^{/T^{2}}$
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A stationary source emits sound waves of frequency $\,500\,Hz$ . Two observers moving along a line passing
through the source detect sound to be of frequencies $480\,Hz$ and $\,530\,Hz$ . Their respective speeds are, in
$\,m\,s^{-1}$ ,
(Given speed of sound $\, = 300\,m/s$ )
a.
$16, \,14$
b.
$12, \,16$
c.
$12, \,18$
d.
$8, \,18$
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One plano-convex and one plano-concave lens of the same radius of curvature $R$ but of different materials are
joined side by side as shown in the figure. If the refractive index of the material of $1$ is $\mu$ $_{1}$ and that of $2$ is $\mu$ $_{2}$ ,
then the focal length of the combination is:
a.
$\frac{R}{2-($\mu$_{1}-$\mu$_{2})}$
b.
$\frac{R}{2($\mu$_{1}-$\mu$_{2})}$
c.
$\frac{2R}{$\mu$_{1}-$\mu$_{2}}$
d.
$\frac{R}{$\mu$_{1}-$\mu$_{2}}$
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Given below in the left column are different modes of communication using the kinds of waves given in the
right column.
(1) Optical Fibre Communication (P) Ultrasound
(2) Radar (Q) Infrared Light
(3) Sonar (R) Microwaves
(4) Mobile Phones (S) Radio Waves
From the options given below, find the most appropriate match between entries in the left and the right
column.
a.
$1 -Q, \,2 -S, \,3 -P, \,\,4 -R$
b.
$1 -\,S, \,\,2 -Q, \,\,3 -R, \,\,4 -P$
c.
$1 -Q, \,\,2 -S, \,\,3 -R, \,\,4 -P$
d.
$1 -R, \,\,2 -P, \,\,3 -S, \,\,4 -Q$
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$n$ moles of an ideal gas with constant volume heat capacity $C_{v}$ undergo an isobaric expansion by certain
volume. The ratio of the work done in the process, to the heat supplied is:
a.
$\frac{4nR}{C_{v}+nR}$
b.
$\frac{4nR}{C_{v}-nR}$
c.
$\frac{nR}{C_{v}+nR}$
d.
$\frac{nR}{C_{v}-nR}$
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In a photoelectric effect experiment, the threshold wavelength of light is $\,380\,nm$ . If the wavelength of
incident light is $\,260\,nm$ , the maximum kinetic energy of emitted electrons will be
Given $E$ (in $\,eV$ ) $= \frac{1237}{$\lambda$(in\,nm)}$
a.
$4.5\,eV$
b.
$3.0\,eV$
c.
$1.5\,eV$
d.
$15.1\,eV$
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In a meter bridge experiment, the circuit diagram and the corresponding observation table are shown in figure.
S. No. $R(\Omega ) l(cm)$
1. $1000 60$
2. $100 13$
3. $10 1.5$
$1 1.0$
4.
Which of the reading is inconsistent?
a.
$1$
b.
$3$
c.
$2$
d.
$4$
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The ratio of surface tensions of mercury and water is given to be $7.5$ , while the ratio of their densities is $13.6$
. Their contact angles, with glass, are close to $135^{\circ}$ and $0^{\circ}$ , respectively. If it is observed that mercury gets
depressed by an amount $h$ in a capillary tube of radius $r_{1}$ , while water rises by the same amount $h$ in a capillary
tube of radius $r_{2}$ , then the ratio $\frac{r_{1}}{r_{2}}$ is close to
a.
$\frac{3}{5}$
b.
$\frac{2}{3}$
c.
$\frac{4}{5}$
d.
$\frac{2}{5}$
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A particle of mass $m$ is moving along a trajectory given by
$x=x_{0}+ a\,cos$ $\omega$ $_{1}t$
$y=y_{0}+ b\,sin$ $\omega$ $_{2}t$
The torque, acting on the particle about the origin, at $t = 0$ is:
The value of acceleration due to gravity at Earth's surface is $\,9.8\,m\,s^{-2}$ . The altitude above its surface at which
the acceleration due to gravity decreases to $\,4.9\,m\,s^{-2}$ , is close to: (Radius of earth $\, = 6.4 \times 10^{6}\,m$ )
a.
$1.6 \times 10^{6}\,m$
b.
$2.6 \times 10^{6}\,m$
c.
$6.4 \times 10^{6}\,m$
d.
$9.0 \times 10^{6}\,m$
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A thin disc of mass $M$ and radius $R$ has mass per unit area $\sigma$ $(r) = kr^{2}$ where $r$ is the distance from its centre. Its
moment inertia about an axis going through its centre of mass and perpendicular to its plane is:
a.
$\frac{MR^{2}}{3}$
b.
$\frac{MR^{2}}{2}$
c.
$\frac{MR^{2}}{6}$
d.
$\frac{2MR^{2}}{3}$
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Two radioactive materials $A$ and $B$ have decay constants $10$ $\lambda$and , respectively. If initially they have the$\lambda$
same number of nuclei, then the ratio of the number of nuclei of $A$ to that of $B$ will be $1/e$ after a time:
a.
$\frac{1}{10$\lambda$}$
b.
$\frac{1}{9$\lambda$}$
c.
$\frac{1}{11$\lambda$}$
d.
$\frac{11}{10$\lambda$}$
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A ray of light $AO$ in vacuum is incident on a glass slab at angle $60^{\circ}$ and refracted at angle $30^{\circ}$ along $OB$ as
shown in the figure. The optical path length of light ray from $A$ to $B$ is:
a.
$\frac{2\sqrt{3}}{a} + 2b$
b.
$2a + 2b$
c.
$2a + \frac{2b}{\sqrt{3}}$
d.
$2a + \frac{2b}{3}$
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Two wires $A$ & $\,B$ are carrying currents $I_{1}$ and $I_{2}$ as shown in the figure. The separation between them is $\,d$ . A
third wire $C$ carrying a current $I$ is to be kept parallel to them at a distance $x$ from $A$ such that the net force
acting on it is zero. The possible values of $x$ are:
a.
$x= \pm \frac{Id}{(I_{1}-I_{2})}^{1}$
b.
$x=(\frac{I}{I_{1}+I_{2}}^{1} )d$ and $x= \frac{I}{(I_{1}-I_{2})}^{2} d$
c.
$x=(\frac{I_{2}}{I_{1}+I_{2}})d$ and $x=(\frac{I_{2}}{I_{1}-I_{2}})d$
d.
$x=(\frac{I_{1}}{I_{1}-I_{2}} )d$ and $x= \frac{I_{2}}{(I_{1}+I_{2})} d$
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A $25 \times 10^{-3}\,m^{3}$ volume cylinder is filled with $1$ mol of $O_{2}$ gas at room temperature $(300\,K)$ . The molecular
diameter of $O_{2}$ , and its root mean square speed, are found to be $0.3\,nm$ and $200\,m/s$ , respectively. What is
the average collision rate (per second) for an $O_{2}$ molecule?
a.
$~10^{11}$
b.
$~10^{12}$
c.
$~10^{10}$
d.
$~10^{13}$
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A cylinder with fixed capacity of $67.2$ litre contains helium gas at STP. The amount of heat needed to raise the
temperature of the gas by $20^{\circ}C$ is:
[Given that $R = 8.31\,J\,mol^{-1}\,K^{-1}]$
a.
$748\,J$
b.
$700\,J$
c.
$350\,J$
d.
$374\,J$
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Two coaxial discs, having moments of inertia $I_{1}$ and $\frac{I_{1}}{2}$ , are rotating with respective angular velocities $\omega$ $_{1}$ and
$\frac{$\omega$_{1}}{2}$ , about their common axis. They are brought in contact with each other and thereafter they rotate with a
common angular velocity. If $E_{f}$ and $E_{i}$ are the final and initial total energies, then $(E_{f} - E_{i})$ is:
a.
$\frac{I_{1}$\omega$^{2}_{1}}{6}$
b.
$\frac{3}{8} I_{1}$ $\omega$ $^{2}_{1}$
c.
$- \frac{I_{1}$\omega$^{2}_{1}}{12}$
d.
$- \frac{I_{1}$\omega$^{2}_{1}}{24}$
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Match the refining methods (Column I) with metals (Column II).
Column I Column II
(Refining methods) (Metals)
(I) Liquation (a) $Zr$
(II) Zone Refining (b) $Ni$
(III) Mond Process (c) $Sn$
(IV) Van Arkel Method (d) $Ga$
Consider the following table:
Gas $a/(k\,Pa\,dm^{6}mol^{-1}) b/(dm^{3}mol^{-1})$
A $642.32 0.05196$
B $155.21 0.04136$
Gas $a/(k\,Pa\,dm^{6}mol^{-1}) b/(dm^{3}mol^{-1})$
C $431.91 0.05196$
D $155.21 0.4382$
$a\,$ and $b$ are vander Waals constants. The correct statement about the gases is:
a.
Gas $C$ will occupy more volume than gas $A$ ; gas
$B$ will be lesser compressible than gas $D$
b.
Gas $C$ will occupy lesser volume than gas $A$ ; gas
$B$ will be lesser compressible than gas $D$
c.
Gas $C$ will occupy more volume than gas $A$ ; gas
$B$ will be more compressible than gas $D$
d.
Gas $C$ will occupy lesser volume than gas $A$ ; gas
$B$ will be more compressible than gas $D$
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Consider the following statements
$(a)$ The pH of a mixture containing $400\,mL$ of $0.1\,M\,H_{2}SO_{4}$ and $400\,mL$ of $0.1\,M\,NaOH$ will be
approximately $1.3.$
$(b)$ Ionic product of water is temperature dependent.
$(c)$ A monobasic acid with $K_{a} = 10^{-5}$ has a $pH = 5$ . The degree of dissociation of this acid is $50%$ .
$(d)$ The Le Chatelier's principle is not applicable to common-ion effect.
The correct statements are:
a.
$(b)\,$ and $\,(c)$
b.
$(a), \,(b)\,$ and $\,(d)$
c.
$(a)\,$ and $\,(b)$
d.
$(a), \,(b)\,$ and $\,(c)$
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Three complexes,
$[CoCl(NH_{3})_{5}]^{2+}(I), \,\,[Co(NH_{3})_{5}H_{2}O]^{3+}(II)$ and $[Co(NH_{3})_{6}]^{3+}(III)$
absorb light in the visible region. The correct order of the wavelength of light absorbed by them is:
a.
$(II) > (I) > (III)$
b.
$(III) > (II) > (I)$
c.
$(I) > (II) > (III)$
d.
$(III) > (I) > (II)$
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At $300\,K$ and $1$ atmospheric pressure, $10\,mL$ of a hydrocarbon required $55\,mL\,$ of $O_{2}$ for complete
combustion, and $40\,mL$ of $CO_{2}$ is formed. The formula of the hydrocarbon is:
a.
$C_{4}H_{8}$
b.
$C_{4}H_{10}$
c.
$C_{4}H_{6}$
d.
$C_{4}H_{7}Cl$
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A bacterial infection in an internal wound grows as $N'(t) = N_{0}exp(t)$ , where the time $t$ is in hours. A dose of
antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes
down as $\frac{dN}{dt} = -5N^{2}$ . What will be the plot of $\frac{N_{0}}{N}$ vs $t$ after $1$ hour?
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Consider the statements $S1$ and $S2$ :
$S1$ : Conductivity always increases with decreases in the concentration of electrolyte.
$S2$ : Molar conductivity always increases with decreases in the concentration of electrolyte.
The correct option among the following
a.
$S1$ is wrong and $S2$ is correct
b.
$S1$ is correct and $S2$ is wrong
c.
Both $S1$ and $S2$ are correct
d.
Both $S1$ and $S2$ are wrong
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At room temperature, a dilute solution of urea is prepared by dissolving $0.60\,g$ of urea in $360\,g$ of water. If the
vapour pressure of pure water at this temperature is $35\,mm\,Hg$ , lowering of vapour pressure will be:
(molar mass of urea $= 60\,g\,mol^{-1}$ )
a.
$0.028\,mm\,Hg$
b.
$0.027\,mm\,Hg$
c.
$0.031\,mm\,Hg$
d.
$0.017\,mm\,Hg$
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A gas undergoes physical adsorption on a surface and follows the given Freundlich adsorption isotherm
equation
$\frac{x}{m} = kp^{0.5}$
Adsorption of the gas increases with:
a.
Decrease in $p$ and increase in $T$
b.
Increase in $p$ and decrease in $T$
c.
Increase in $p$ and increase in $T$
d.
Decrease in $p$ and decrease in $T$
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The sum $\frac{3\times 1^{3}}{1^{2}} + \frac{5\times (1^{3}+2^{3})}{1^{2}+2^{2}} + \frac{7\times (1^{3}+2^{3}+3^{3})}{1^{2}+2^{2}+3^{2}}+. . . . .$ upto $10^{th}$ term is
a.
$660$
b.
$600$
c.
$620$
d.
$680$
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If $Q(0, -1, -3)$ is the image of the point P in the plane $3x- y+ 4z= 2$ and $R$ is the point $(3, -1, -2)$ , then
the area (in sq. units) of $\Delta PQR$ is
a.
$\frac{\sqrt{91}}{4}$
b.
$\frac{\sqrt{91}}{2}$
c.
$2\sqrt{13}$
d.
$\frac{\sqrt{65}}{2}$
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Let $f(x) = e^{x} -x$ and $g(x) = x^{2} -x, \,\forall \,x\,$ $\epsilon$ $\,R$ . Then the set of all $x\,$ $\epsilon$ $\,R$ , where the function
$h(x) = (fog)(x)$ is increasing, is:
a.
$[-1, - \frac{1}{2}] ⋃[\frac{1}{2}, \infty )$
b.
$[0, \infty )$
c.
$[0, \frac{1}{2}]\cup [1, \infty )$
d.
$[- \frac{1}{2}, \,0]\cup [1, \infty )$
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If the length of the perpendicular from the point $(\beta , 0,\beta ), (\beta \neq 0)$ to the line, $\frac{x}{1} = \frac{y-1}{0} = \frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$ , then $\beta$
is equal to
a.
$2$
b.
$-1$
c.
$-2$
d.
$1$
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$\{\frac{sin(p+1)x+sinx}{x} , x< 0$
If $f(x) = q , x= 0$ is continuous at $x= 0$ , then the ordered pair $(p, \,q)$ is equal to:
$\{$
$\frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3/2}} , x> 0$
a.
$(- \frac{3}{2}, - \frac{1}{2})$
b.
$(-\frac{1}{2} , \frac{3}{2})$
c.
$(\frac{5}{2} , \frac{1}{2})$
d.
$(-\frac{3}{2} , \frac{1}{2})$
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The number of $\,6$ digit number that can be formed using the digits $0, \,1, \,2, \,5, \,7$ and $9$ which are divisible by $11$
and no digit is repeated is:
a.
$36$
b.
$60$
c.
$72$
d.
$48$
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Assume that each born child is equally likely to be a boy or girl. If two families have two children each, then
the conditional probability that all children are girls given that at least two are girls is:
a.
$\frac{1}{12}$
b.
$\frac{1}{10}$
c.
$\frac{1}{11}$
d.
$\frac{1}{17}$
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If the circles $x^{2} +y^{2} + 5Kx+ 2y+K = 0$ and $2(x^{2} +y^{2})+ 2Kx+ 3y-1 = 0, (K \in R)$ , intersect at the
points P and Q, then the line $4x+ 5y- K = 0$ , passes through $P$ and $Q,$ for:
a.
exactly two values of $K$
b.
no value of $K$
c.
exactly one value of $K$
d.
infinitely many values of $K$
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If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2} +xsin\theta -2sin\theta = 0, \,\theta \in (0, \frac{$\pi$}{2})$ , then
$\frac{\alpha ^{12}+\beta ^{12}}{(\alpha ^{-12}+\beta ^{-12}).(\alpha -\beta )^{24}}$ is equal to :
a.
$\frac{2^{6}}{(sin\theta +8)^{12}}$
b.
$\frac{2^{12}}{(sin\theta -4)^{12}}$
c.
$\frac{2^{12}}{(sin\theta +8)^{12}}$
d.
$\frac{2^{12}}{(sin\theta -8)^{6}}$
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If the coefficients of $x^{2}$ and $x^{3}$ , are both zero, in the expansion of the expression $(1 +ax+bx^{2})(1 -3x)^{15}$ , in
powers of $x$ , then the ordered pair $(a,b)$ is equal to
a.
$(28, 315)$
b.
$(-21, 714)$
c.
$(28, 861)$
d.
$(-54, 315)$
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If a directrix of a hyperbola centered at the origin and passing through the point $(4, -2\sqrt{3})$ is $5x= 4\sqrt{5}$
and its eccentricity is $e$ , then:
a.
$4e^{4} + 8e^{2} -35 = 0$
b.
$4e^{4} -24e^{2} + 35 = 0$
c.
$4e^{4} -24e^{2} + 27 = 0$
d.
$4e^{4} -12e^{2} -27 = 0$
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If the line $x-2y= 12$ is a tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at the point $(3, -\frac{9}{2} )$ , then the length of the
latus rectum of the ellipse is
a.
$5$ units
b.
$12\sqrt{2}$ units
c.
$9$ units
d.
$8\sqrt{3}$ units
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If $\int \frac{dx}{(x^{2}-2x+10)^{2}} =A(tan^{-1}(\frac{x-1}{3} )+ \frac{f(x)}{x^{2}-2x+10} )+C$ , then (where $C$ is a constant of integration)
a.
$A= \frac{1}{27}$ and $f(x) = 9(x-1)$
b.
$A= \frac{1}{81}$ and $f(x) = 3(x-1)$
c.
$A= \frac{1}{54}$ and $f(x) = 9(x-1)^{2}$
d.
$A= \frac{1}{54}$ and $f(x) = 3(x-1)$
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If the system of linear equations $x+y+z= 5$ , $x+ 2y+ 2z= 6$ , $x+ 3y+\lambda z=$ µ $, \,(\lambda , \,$ µ $\, \in R)$ , has
infinitely many solutions, then the value of $\lambda +$ µis:
a.
$7$
b.
$10$
c.
$12$
d.
$9$
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Let $f(x) =x^{2}, \,x\in R$ . For any $A⊆ R,$ define $g(A) = {x\in R\, :f(x) \in A}$ . If $S = [0, \,4]$ , then which one
of the following statements is not true?
a.
$g(f(S)) \neq S$
b.
$f(g(S)) \neq f(S)$
c.
$f(g(S)) =S$
d.
$g(f(S)) =g(S)$
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$ABC$ is a triangular park with $AB=AC = 100$ metres. A vertical tower is situated at the mid-point of BC. If
the angles of elevation of the top of the tower at, $A$ and $B$ are $cot^{-1}(3\sqrt{2})$ and $cosec^{-1}(2\sqrt{2})$ respectively,
then the height of the tower (in metres) is
a.
$\frac{100}{3\sqrt{3}}$
b.
$20$
c.
$25$
d.
$10\sqrt{5}$
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If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx} =(tanx- y)sec^{2}x$ , $x\in (- \frac{\pi}{2}, \, \frac{\pi}{2} )$ , such that
$y(0) = 0$ , then $y(- \frac{\pi}{4})$ is equal to:
a.
$\frac{1}{e} -2$
b.
$2 + \frac{1}{e}$
c.
$e-2$
d.
$\frac{1}{2} -e$
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Let $A(3, 0, -1), \,B(2, 10, 6)$ and $C(1, 2, 1)$ be the vertices of a triangle and $M$ be the mid-point of $AC$ . If $G$
divides $BM$ in the ratio, $2 : 1$ , then $cos(\angle GOA)$ ( $O$ being the origin) is equal to
a.
$\frac{1}{\sqrt{30}}$
b.
$\frac{1}{6\sqrt{10}}$
c.
$\frac{1}{\sqrt{15}}$
d.
$\frac{1}{2\sqrt{15}}$
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If for some $x\in R$ , the frequency distribution of the marks obtained by $20$ students in a test is:
Marks 2 3 5 7
Frequency distribution $(x+ 1)^{2} (2x-5) x^{2} -3x x$
Then the mean of the marks is :
a.
$3.0$
b.
$2.5$
c.
$3.2$
d.
$2.8$
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