Two particles of the same mass $m$ are moving in circular orbits because of force, given by $F\,(r) = -\frac{16}{r} - r^{3}$ .
The first particle is at a distance $r= 1$ , and the second, at $r= 4$ . The best estimate for the ratio of kinetic
energies of the first and the second particle is closest to
a.
$3\, \times 10^{-3}$
b.
$6\, \times 10^{2}$
c.
$6\, \times 10^{-2}$
Correct Answer
d.
$10^{-2}$
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Q2mcq
A charge $q$ is spread uniformly over an insulated loop of radius $r$ . If it is rotated with an angular velocity $\omega$
with respect to normal axis then magnetic moment of the loop is:
a.
$\frac{3}{2} q\omega r^{2}$
b.
$\frac{1}{2} q\omega r^{2}$
Correct Answer
c.
$q\omega r^{2}$
d.
$\frac{4}{3} q\omega r^{2}$
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Q3mcq
An oscillator of mass $M$ is at rest in its equilibrium position in a potential, $V = \frac{1}{2}k(x\,-X)^{2}$ . A particle of
mass $m$ comes from the right with speed $u$ and collides completely inelastic with $M$ and sticks to it. This
process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after $13$
collisions is: $(M = 10, \,m= 5, \,u= 1, \,k= 1)$
a.
$\frac{2}{3}$
b.
$\frac{1}{\sqrt{3\,}}$
Correct Answer
c.
$\sqrt{\frac{3}{5}}$
d.
$\frac{1}{2}$
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Q4mcq
A heating element has a resistance of $100\,\Omega$ at room temperature. When it is connected to a supply of $220\,V$ , a
steady current of $2\,A$ passes in it and temperature is $500\,^{o}C$ more than the room temperature. The temperature
coefficient of resistance of the heating element is
a.
$5\, \times 10^{-4}^{o}C^{-1}$
b.
$2\, \times 10^{-4}^{o}C^{-1}$
Correct Answer
c.
$1\, \times 10^{-4}^{o}C^{-1}$
d.
$0.5\, \times 10^{-4}^{o}C^{-1}$
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Q5mcq
Unpolarized light of intensity $I$ is incident on a system of two polarizers, $A$ followed by $B$ . The intensity of
emergent light is $\frac{I}{2}$ . If a third the polarizer $C$ is placed between $A$ and $B$ the intensity of emergent light is
reduced to $\frac{I}{3}$ . The angle between the polarizers $A$ and $C$ is $\theta$ , then
A plane electromagnetic wave of wavelength $\lambda$ has an intensity $I$ . It is propagating along the positive $Y-$
direction. The allowed expressions for the electric and magnetic fields are given by:
A galvanometer with its coil resistance $25\,\Omega$ requires a current of $1\, mA$ for its full deflection. In order to
construct an ammeter to read up to a current of $2\,A$ the approximate value of the shunt resistance should be:
a.
$1.25\, \times 10^{-2\,}\Omega$
Correct Answer
b.
$2.5\, \times 10^{-3\,}\Omega$
c.
$2.5\, \times 10^{-2\,}\Omega$
d.
$1.25\, \times 10^{-3\,}\Omega$
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Q8mcq
The relative uncertainty in the period of a satellite orbiting around the earth is $10^{-2}$ . If the relative uncertainty
in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is:
a.
$2\, \times \,10^{-2}$
Correct Answer
b.
$6\, \times \,10^{-2}$
c.
$3\, \times \,10^{-2}$
d.
$10^{-2}$
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Q9mcq
The de-Broglie wavelength $(\lambda _{B})$ associated with the electron orbiting in the second excited state of hydrogen
atom is related to that in the ground state $(\lambda _{G})$ by:
a.
$\lambda _{B} = \,3\lambda _{G}$
Correct Answer
b.
$\lambda _{B} = 2\lambda _{G}$
c.
$\lambda _{B} =\lambda \frac{G}{3}$
d.
$\lambda _{B} =\lambda \frac{G}{2}$
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Q10mcq
In the following circuit the switch $S$ is closed at $t\, = 0$ . The charge on the capacitor $C_{1}$ as a function of time
will be given by $(C_{eq} = \, \frac{C_{1}C_{2}}{C_{1}+\,C_{2}})$
A particle executes simple harmonic motion and it is located at $x=a, \,b$ and $c$ at time $t_{0}, \,2t_{0}\,and\,3t_{0}$
respectively. The frequency of the oscillation is:
a.
$\frac{1}{2\pi t_{0}}cos^{-1}( \frac{a+c}{2b})$
b.
$\frac{1}{2\pi t_{0}} cos^{-1}(\frac{a+2b}{3c})$
c.
$\frac{1}{2\pi t_{0}}cos^{-1}( \frac{a+b}{2c})$
d.
$\frac{1}{2\pi t_{0}}cos^{-1}(\frac{2a+3c}{b})$
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At some instant, a radioactive sample $S_{1}$ having an activity $5\mu Ci$ has twice the number of nuclei as another
sample $S_{2}$ which has an activity of $10\mu Ci$ . The half lives of $S_{1}$ and $S_{2}$ are :
a.
$4 : 1$
b.
$1 : 4$
c.
$1 : 2$
d.
$2 : 1$
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The end correction of a resonance column is $1\, cm$ . If the shortest length resonating with the tuning fork is
$10\, cm$ , the next resonating length should be
a.
$32\, cm$
b.
$40\, cm$
c.
$28\, cm$
d.
$36\, cm$
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Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited
by emitting photons of wavelengths $\lambda _{N}, \,\lambda _{A}$ respectively. The ratio $\frac{\lambda _{N}}{\lambda _{A}}$ is closest to:
a.
$10^{-1}$
b.
$10^{-6}$
c.
$10$
d.
$10^{-10}$
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A body of mass $m$ starts moving from rest along $x-$ axis so that its velocity varies as $v=a\sqrt{s}$ where $a$ is a
constant and $s$ is the distance covered by the body. The total work done by all the forces acting on the body in
the first $t$ second after the start of the motion is
a.
$8\,ma^{4}t^{2}$
b.
$\frac{1}{4} ma^{4}t^{2}$
c.
$4\,ma^{4}t^{2}$
d.
$\frac{1}{8}ma^{4}t^{2}$
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In a circuit for finding the resistance of a galvanometer by half deflection method, a $6\,V$ battery and a high
resistance of $11\,k\Omega$ are used. The figure of merit of the galvanometer is $60\,$ $\mu$ $A\, division^{-1}$ . In the absence of
shunt resistance, the galvanometer produces a deflection of $\theta = 9$ divisions when current flows in the circuit.
The value of the shunt resistance that can cause the deflection of $\frac{\theta}{2}$ , is closest to:
a.
$550\,\Omega$
b.
$220\,\Omega$
c.
$55\,\Omega$
d.
$110\,\Omega$
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Two sitar strings, $A$ and $B$ playing the note ' $Dha$ ' are slightly out of tune and produce beats of frequency $5\, Hz$
. The tension of the string $B$ is slightly increased and the beat frequency is found to decrease by $3\, Hz$ . If the
frequency of $A$ is $425\, Hz$ . The original frequency of $B$ is
a.
$428\, Hz$
b.
$430\, Hz$
c.
$422\, Hz$
d.
$420\, Hz$
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A ray of light is incident at an angle of $60^{\circ }$ on one face of a prism of angle $30^{\circ }$ . The emergent ray of light
makes an angle of $30^{\circ }$ with incident ray. The angle made by the emergent ray with second face of prism will
be:
a.
$0^{\circ}$
b.
$90^{\circ}$
c.
$30^{\circ}$
d.
$45^{\circ}$
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The percentage errors in quantities $P$ , $Q$ , $R$ and $S$ are $0. 5%$ , $1%$ , $3%$ and $1. 5%$ respectively in the
measurement of a physical quantity $A= \frac{P^{3}Q^{2}}{\sqrt{R}S}$ . The maximum percentage error in the value of $A$ will be:
a.
$6.5\,%$
b.
$7.5\,%$
c.
$6.0\,%$
d.
$\,8.5\,%$
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A coil of cross-sectional area $A$ having $n$ turns is placed in a uniform magnetic field $B$ . When it is rotated with
an angular velocity $\omega$ $,$ the maximum e.m.f. induced in the coil will be:
a.
$\frac{3}{2}nBA\omega$
b.
$3nBA\omega$
c.
$nBA\omega$
d.
$\frac{1}{2}nBA\omega$
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One mole of an ideal monatomic gas is taken along the path $ABCA$ as shown in the $PV$ diagram. The
maximum temperature attained by the gas along the path $BC$ is given by:
a.
$\frac{25}{4} \frac{P_{0}V_{0}}{R}$
b.
$\frac{5}{8} \frac{P_{0}V_{0}}{R}$
c.
$\frac{25}{8} \frac{P_{0}V_{0}}{R}$
d.
$\frac{25}{16} \frac{P_{0}V_{0}}{R}$
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A power transmission line feeds input power at $\,2300\,V$ to a step-down transformer with its primary windings
having $4000\,$ turns giving the output power at $230\,V$ . If the current in the primary coil of the transformer is $5\,A$
and its efficiency is $90\,%$ , the output current would be:
a.
$45\,A$
b.
$50\,A$
c.
$20\,A$
d.
$25\,A$
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A carrier wave of peak voltage $14\,V$ is used for transmitting a message signal. The peak voltage of the
modulating signal given to achieve a modulation index of $80%$ will be:
a.
$22 .4 \,V$
b.
$7\,V$
c.
$11 .2 \,V$
d.
$28\,V$
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Let $\vec{A}= ˆi\, + \,ˆj \,and\,\vec{B}= 2ˆi-\,ˆj$ . The magnitude of a coplanar vector $\vec{C}$ such that
$( ) ( )$
$\vec{A}\,. \,\vec{C} \vec{B}. \,\vec{C} = \,\vec{A}\,. \,\vec{B}$ is given by:
a.
$\frac{=}{9}$
$\sqrt{\,}$
b.
$\sqrt{\frac{20}{9}}$
c.
$\frac{12}{5}$
$\sqrt{\,} _{9}$
d.
$\sqrt{\frac{10}{9}}$
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Two identical conducting spheres $A$ and $B$ carry an equal charges. They are separated by a distance much
larger than their diameters, and the force between them is $F$ . A third identical conducting sphere, $C$ , is
uncharged. Sphere $C$ is first touched to $A$ , then to $B$ , and then removed. As a result, the force between $A$ and
$B$ would be equal to:
a.
$\frac{3F}{4}$
b.
$\frac{F}{2}$
c.
$\frac{3F}{8}$
d.
$F$
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A small soap bubble of radius 4cm is trapped inside another bubble of radius 6cm without any contact. Let $P_{2}$
be the pressure inside the inner bubble and $P_{0}$ , the pressure outside the outer bubble. Radius of another bubble
with pressure difference $P_{2}- P_{0}$ between its inside and outside would be:
a.
$2 .4 \, cm$
b.
$12\, cm$
c.
$4 .8 \, cm$
d.
$6\, cm$
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When $XO_{2}$ is fused with an alkali metal hydroxide in presence of an oxidizing agent such as $KNO_{3}$ ; a dark
green product is formed which disproportionate in acidic solution to afford a dark purple solution. $X$ is:
a.
$Mn$
b.
$Cr$
c.
$V$
d.
$Ti$
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Assuming ideal gas behavior, the ratio of density of ammonia to that of hydrogen chloride at same temperature
and pressure is: (Atomic weight of Cl is 35.5 u)
a.
$1.46$
b.
$1.64$
c.
$0.46$
d.
$0.64$
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In a complexometric titration of metal ion with ligand $M\,(Metal\,ion) +L(Ligand) \rightarrow \rightarrow C\,(Complex)$ end
point is estimated spectrophotometrically (through light absorption). If 'M' and 'C' do not absorb light and only
'L' absorbs then the titration plot between absorbed light $(A)$ versus volume of ligand 'L' $(V)$ would look like:
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The mass of a non-volatile, non-electrolyte solute (molar mass $= \,50\,g\, mol^{-1}$ ) needed to be dissolved in 114 g
octane to reduce its vapour pressure by 75 %, is:
a.
$37.5\,g$
b.
$75\,g$
c.
$150\,g$
d.
$50\,g$
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At 320 K, a gas $A_{2}$ is 20 % dissociated to $A(g)$ . The standard Gibbs free energy change at
$320\,K\,and\,1\,atm\,in\,J mol^{-1}$ is approximately:
$(R\, = \,8 .314 \, JK^{-1} \, mol^{-1} \,; ln 2\, = \,0 .693 \,; ln 3\, = \,1 .098)$
a.
1844
b.
2068
c.
4281
d.
4763
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A group 13 element $'X'$ reacts with chlorine gas to produce a compound $XCl_{3}. \, XCl_{3}$ is electron deficient and
easily reacts with $NH_{3}\, to \, form \, Cl_{3}X ←NH_{3}$ adduct; however, $XCl_{3}$ does not dimerize. X is:
a.
B
b.
Al
c.
In
d.
Ga
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The gas phase reaction $2 NO_{2}(g)\, \,N_{2}O_{4}\,(g)$ is an exothermic reaction. The decomposition of $N_{2}O_{4}$ , in
equilibrium mixture of $NO_{2}\,(g)\, a\vec{n}d \,N_{2}O_{4}\,(g)$ can be increased by:
a.
Addition of an inert gas at constant pressure
b.
Lowering the temperature
c.
Increasing the pressure
d.
Addition of an inert gas at constant volume
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The correct match between items of List-I and List-II is:
List -I List - II
(A) Phenelzine (P) Pyrimidine
(B) Chloroxylenol (Q) Furan
(C) Uracil (R) Hydrazine
(D) Ranitidine (S) Phenol
a.
$(A) -(S), \,(B) -(R), \,(C) -(Q), \,(D) -(P)$
b.
$(A) -(R), \,(B) -(S), \,(C) -(P), \,(D) -(Q)$
c.
$(A) -(R), \,(B) -(S), \,(C) -(Q), \,(D) -(P)$
d.
$(A) -(S), \,(B) -(R), \,(C) -(P), \,(D) -(Q)$
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An unknown chlorohydrocarbon has 3.55 % of chlorine. If each molecule of the hydrocarbon has one chlorine
atom only; chlorine atoms present in 1 g of chlorohydrocarbon are :(Atomic wt. of $Cl = 35 .5 \,u;$ Avogadro
constant $= 6 .023 \times 10^{23}\, mol^{-1}$ )
a.
$6.023\, \times 10^{9}$
b.
$6.023\, \times 10^{23}$
c.
$6.023\, \times 10^{21}$
d.
$6.023\, \times 10^{20}$
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If the angle between the lines $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{5-x}{-2} = \frac{7y-14}{P} = \frac{z-3}{4}$ is $cos^{-1}(\frac{2}{3}),$ then $P$ is equal to
a.
$\frac{2}{7}$
b.
$\frac{7}{2}$
c.
$-\frac{4}{7}$
d.
$-\frac{7}{4}$
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The differential equation representing the family of ellipses having foci either on the $x$ -axis or on the $y$ -axis,
center at the origin and passing through the point $(0, 3)$ is
a.
$xyy' - y^{2} + 9 = 0$
b.
$xyy'' +x(y')^{2} - yy' = 0$
c.
$xyy' +y^{2} -9 = 0$
d.
$x+yy'' = 0$
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The mean and the standard deviation $(S. D. )$ of five observations are $9$ and $0$ , respectively. If one of the
observation is increased such that the mean of the new set of five observations becomes $10$ , then their $S. D.$ is
a.
$0$
b.
$2$
c.
$4$
d.
$1$
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Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as
$R_{1} = {(x,y) \in N \times N : 2x+y= 10}$ and $R_{2} = {(x,y) \in N \times N :x+ 2y= 10}$ . Then
a.
both $R_{1}$ and $R_{2}$ are transitive relations
b.
range of $R_{2}$ is ${1, 2, 3, 4}$
c.
range of $R_{1}$ is ${2, 4, 8}$
d.
both $R_{1}$ and $R_{2}$ are symmetric relations
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The locus of the point of intersection of the lines $\sqrt{2}x- y+ 4\sqrt{2}k= 0$ and $\sqrt{2}kx+ky-4 = 0\sqrt{2}$ ( $k$ is any
non-zero real parameter) is
a.
an ellipse whose eccentricity is $\frac{1}{\sqrt{3}}$
b.
a hyperbola whose eccentricity is $\sqrt{3}$
c.
a hyperbola with length of its transverse axis $8\sqrt{2}$
d.
an ellipse with length of its major axis $8\sqrt{2}$
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If a circle $C$ , whose radius is $3$ , touches externally the circle $x^{2} +y^{2} + 2x-4y-4 = 0$ at the point $(2, 2)$ ,
then the length of the intercept cut by this circle $C$ on the $x$ -axis is equal to
a.
$2\sqrt{3}$
b.
$\sqrt{5}$
c.
$3\sqrt{2}$
d.
$2\sqrt{5}$
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Let $\frac{1}{x_{1}}, \,\frac{1}{x_{2}}, … , \frac{1}{x_{n}}(x_{i} \neq 0$ for $i= 1, \,2, … . , \,n)$ be in A.P. such that $x_{1} = 4$ and $x_{21} = 20.$ If $n$ is the least
positive integer for which $x_{n} > 50$ , then $\sum ^{n}_{i=1}(\frac{1}{x_{i}})$ is equal to
a.
$3$
b.
$\frac{1}{8}$
c.
$\frac{13}{4}$
d.
$\frac{13}{8}$
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If the function $f$ defined as $f(x) = \frac{1}{x} - \frac{k-1}{e^{2}^{x}-1}, \,x\neq 0$ is continuous at $x= 0$ , then ordered pair $(k,f(0))$ is
equal to
a.
$(2, 1)$
b.
$(3, 1)$
c.
$(3, 2)$
d.
$(\frac{1}{3}, 2)$
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If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex
on the major axis is $\frac{3}{2}$ units, then its eccentricity is
a.
$\frac{2}{3}$
b.
$\frac{1}{2}$
c.
$\frac{1}{9}$
d.
$\frac{1}{3}$
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$|$
The sum of the intercepts on the coordinate axes of the plane passing through the point $(-2, -2, 2)$ and
containing the line joining the points $(1, -1, 2)$ and $(1, 1, 1)$ is
a.
$4$
b.
$12$
c.
$-8$
d.
$-4$
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If $\int \frac{tanx}{1+tanx+tan^{2}x}dx=x- \frac{K}{\sqrt{A}}tan^{-1}(\frac{Ktanx+1}{\sqrt{A}})+C$ , ( $C$ is a constant of integration), then the ordered pair
$(K,A)$ is equal to
a.
$(2, 1)$
b.
$(2, 3)$
c.
$(-2, 1)$
d.
$(-2, 3)$
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The number of numbers between $2, 000$ and $5, 000$ that can be formed with the digits $0, \,1, \,2, \,3, \,4$ (repetition
of digits is not allowed) and are multiple of $3$ is
a.
$36$
b.
$30$
c.
$24$
d.
$48$
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Let $P$ be a point on the parabola $x^{2} = 4y$ . If the distance of $P$ from the center of the circle
$x^{2} +y^{2} + 6x+ 8 = 0$ is minimum, then the equation of the tangent to the parabola at $P$ is
a.
$x+y+ 1 = 0$
b.
$x+ 4y-2 = 0$
c.
$x+ 2y= 0$
d.
$x- y+ 3 = 0$
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A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a
horizontal road. If it takes $18\, min$ for the angle of depression of the car to change from $30^{\circ}$ to $45^{\circ}$ , then the
time taken (in $min$ ) by the car to reach the foot of the tower is
a.
$\frac{9}{2}(\sqrt{3}-1)$
b.
$18(1 + \sqrt{3})$
c.
$18(\sqrt{3}-1)$
d.
$9(1 + \sqrt{3})$
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Two different families $A$ and $B$ are blessed with equal number of children. There are $3$ tickets to be distributed
amongst the children of these families so that no child gets more than one ticket. If the probability that all the
tickets go to the children of the family $B$ is $\frac{1}{12},$ then the number of children in each family is
a.
$6$
b.
$5$
c.
$3$
d.
$4$
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Let $\vec{a}= ˆi+ ˆj+ˆk, \,\vec{c}= ˆj- ˆk$ and a vector $\vec{b}$ be such that $\vec{a}\times \vec{b}= \vec{c}$ and $\vec{a}\cdot \vec{b}= 3$ . Then $\vec{b}$ equals
a.
$\frac{11}{3}$
b.
$\frac{11}{\sqrt{3}}$
c.
$\sqrt{\frac{11}{3}}$
d.
$\frac{\sqrt{11}}{3}$
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If the area of the region bounded by the curves, $y=x^{2}, \,y= \frac{1}{x}$ and the lines $y= 0$ and $x=t(t> 1)$ is
$1\, sq. \, unit$ , then $t$ is equal to
a.
$e\frac{2}{3}$
b.
$e\frac{3}{2}$
c.
$\frac{3}{2}$
d.
$\frac{4}{3}$
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Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum values of the function,
$f(x) = 2x^{3} -9x^{2} + 12x+ 5$ in the interval $[0, 3]$ . Then $M - m$ is equal to
a.
$9$
b.
$4$
c.
$1$
d.
$5$
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Let $p,q$ and $r$ be real numbers $(p\neq q,r\neq 0)$ , such that the roots of the equation $\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}$ are equal in
magnitude but opposite in sign, then the sum of squares of these roots is equal to
a.
$p^{2} +q^{2}$
b.
$\frac{p^{2}+q^{2}}{2}$
c.
$2(p^{2} +q^{2})$
d.
$p^{2} +q^{2} +r^{2}$
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$|$
$-$
Let $A, \,B$ and $C$ be three events, which are pair-wise independent and $E$ denotes the complement of an event
$-_{E. IfP(A\cap }_{B\cap }_{C) = 0andP(C) > 0,thenP[(A}_{\cap }_{B)}_{C]is equal to}$
$--$
a.
$P(A )- P(B)$
b.
$P(A)- P(B)$
$\bar{-}--$
c.
$P(A)+P(B)$
d.
$P(A) +P(B)$
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