A disc rotates about its axis of symmetry in a hoizontal plane at a steady rate of $3.5$ revolutions per second. A
coin placed at a distance of $1.25\,cm$ from the axis of rotation remains at rest on the disc. The coefficient of
friction between the coin and the disc is $(g = 10\,m/s^{2})$
a.
$0.5$
b.
$0.7$
c.
$0.3$
d.
$0.6$
Correct Answer
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Q2mcq
A constant voltage is applied between two ends of a metallic wire. If the length is halved and the radius of the
wire is doubled, the rate of heat developed in the wire will be:
a.
Increased 8 times
Correct Answer
b.
Doubled
c.
Halved
d.
Unchanged
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Q3mcq
The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be
determined from a suitable combination of the fundamental physical constants $G, h$ and $c$ . Which of the
following correctly gives the Planck length?
a.
$G^{2}hc$
b.
$( \frac{Gh}{c^{3}})\frac{1}{2}$
Correct Answer
c.
$G\frac{1}{2}\,h^{2}c$
d.
$G(h^{2}c^{3}$
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Q4mcq
A body takes 10 minutes to cool from $60^{\circ }C$ to $50^{\circ }C$ . The temperature of surroundings is constant at $25^{\circ }C$ .
Then, the temperature of the body after next 10 minutes will be approximately
a.
$43^{\circ }C$
Correct Answer
b.
$47^{\circ }C$
c.
$41^{\circ }C$
d.
$45^{\circ }C$
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Q5mcq
5 beats/ second are heard when a turning fork is sounded with a sonometer wire under tension, when the length
of the sonometer wire is either $0.95\,m$ or $1\,m$ . The frequency of the fork will be:
a.
$195\,Hz$
Correct Answer
b.
$251\,Hz$
c.
$150\,Hz$
d.
$300\,Hz$
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Q6mcq
A man in a car at location Q on a straight highway is moving with speed $v$ . He decides to reach a point $P$ in a
field at a distance $d$ from highway (point $M$ ) as shown in the figure.Speed of the car in the field is half to that
on the highway. What should be the distance RM, so that the time taken to reach $P$ is minimum?
a.
$\frac{d}{\sqrt{3}}$
Correct Answer
b.
$\frac{d}{2}$
c.
$\frac{d}{\sqrt{2}}$
d.
$d$
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Q7mcq
The value closest to the thermal velocity of a Helium atom at room temperature $(300\,K)$ in $ms^{-1}$ is:
$[k_{B} = 1.4 \times 10^{-23}\,J/K; m_{He} = 7 \times 10^{-27}\,kg]$
a.
$1.3 \times 10^{4}$
b.
$1.3 \times 10^{5}$
c.
$1.3 \times 10^{2}$
d.
$1.3 \times 10^{3}$
Correct Answer
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Q8mcq
A proton of mass $m$ collides elastically with a particle of unknown mass at rest. After the collision, the proton
and the unknown particle are seen moving at an angle of $90^{\circ }$ with respect to each other. The mass of unknown
particle is:
a.
$\frac{m}{\sqrt{3}}$
b.
$\frac{m}{2}$
c.
$2\,m$
d.
$m$
Correct Answer
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Q9mcq
A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of
$10\,cm$ . The separation between the two lenses is $2\,cm$ . The focal lengths of the component lenses
Muon $(\mu ^{-1})$ is negatively charged $(|q| = |e|)$ with a mass $m_{\mu } = 200\,m_{e}$ , where $m_{e}$ is the mass of the electron
and e is the electronic charge. If $\mu ^{-1}$ is bound to a proton to form a hydrogen like atom, identify the correct
statements (A) Radius of the muonic orbit is 200 times smaller than that of the electron (B) the speed of the
$\mu ^{-1}$ in the $n$ th orbit is $\frac{1}{200}$ times that of the election in the nth orbit (C) The lonization energy of muonic atom
is 200 times more than that of an hydrogen atom (D) The momentum of the muon in the nth orbit is 200 times
more than that of the electron
a.
(A), (B), (D)
b.
(B), (D)
c.
(C),(D)
d.
(A), (C), (D)
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Two simple harmonic motions, as shown, are at right angles. They are combined to form Lissajous figures.
$x(t) =Asin(at+\delta )y(t) =Bsin(bt)$ Identify the correct match below
(1) Parameters: $A = B, a = 2\,b;\delta = \frac{\pi}{2}$ ; Curve: (2) Parameters: $A = B, a = b;\delta = \frac{\pi}{2};$ Curve: Line
Circle
(3) Parameters: $A \neq B, a = b;\delta = \frac{\pi}{2} ;$ Curve: Ellipse(4) Parameters: $A \neq B, a = b;\delta = 0$ ; Curve:
Parabola
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As shown in the figure, forces of $10^{5}\,N$ each are applied in opposite directions, on the upper and lower faces of
a cube of side $10\,cm$ , shifting the upper face parallel to itself by $0.5\,cm$ . If the side of another cube of the same
material is, $20\,cm$ then under similar conditions as above, the displacement will be:
a.
$1.00\,cm$
b.
$0.25\,cm$
c.
$0.37\,cm$
d.
$0.75\,cm$
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A body of mass $2\,kg$ slides down with an acceleration of $3\,m/s^{2}$ on a rough inclined plane having a slope of $30^{\circ }$
. The external force required to take the same body up the plane with the same acceleration will be:
$(g = 10\,m/s^{2})$
a.
$4\,N$
b.
$14\,N$
c.
$6\,N$
d.
$20\,N$
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A thin rod $MN$ , free to rotate in the vertical plane about the fixed end $N$ , is held horizontal. When the end $M$ is
released the speed of this end, when the rod makes an angle $\alpha$ with the horizontal, will be proportional to: (see
figure)
a.
$\sqrt{cos\alpha}$
b.
$cos\alpha$
c.
$sin\alpha$
d.
$\sqrt{sin\alpha}$
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A capacitor $C_{1}$ is charged up to a voltage $V = 60\,V$ by connecting it to battery $B$ through switch (1), Now $C_{1}$
is disconnected from battery and connected to a circuit consisting of two uncharged capacitors $C_{2} = 3.0\mu F$
and $C_{3} = 6.0\mu F$ through a switch $(2)$ as shown in the figure. The sum of final charges on $C_{2}$ and $C_{3}$ is:
a.
$36\mu C$
b.
$20\mu C$
c.
$54\mu C$
d.
$40\mu C$
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If the de Broglie wavelengths associated with a proton and an $\alpha$ -particle are equal, then the ratio of velocities
of the proton and the $\alpha$ -particle will be:
a.
$1 : 4$
b.
$1 : 2$
c.
$4 : 1$
d.
$2 : 1$
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A plane polarized light is incident on a polariser with its pass axis making angle $\theta$ with $x$ -axis, as shown in the
figure. At four different values of $\theta ,\theta = 8^{\circ }, 38^{\circ }, 188^{\circ }$ and $218^{\circ }$ , the observed intensities are same. What is the
angle between the direction of polarization and $x$ -axis
a.
$203^{\circ }$
b.
$45^{\circ }$
c.
$98^{\circ }$
d.
$128^{\circ }$
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$||$
An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of $8 : 27$
. The ratio of the radii of the nuclei (assumed to be spherical ) is:
a.
$8 : 27$
b.
$2 : 3$
c.
$3 : 2$
d.
$4 : 9$
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[12] A solid ball of radius $R$ has a charge density $\rho$ given by $\rho =\rho _{0}(1 - \frac{r}{R})$ for $0 \leq r\leq R$ . The electric field
outside the ball is:
a.
$\frac{\rho _{0}R^{3}}{\epsilon _{0}r^{2}}$
b.
$\frac{4\rho _{0}R^{3}}{3\epsilon _{0}r^{2}}$
c.
$\frac{3\rho _{0}R^{3}}{4\epsilon _{0}r^{2}}$
d.
$\frac{\rho _{0}R^{3}}{12\epsilon _{0}r^{2}}$
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Two Carnot engines $A$ and $B$ are operated in series. Engine A receives heat from a reservoir at $600\,K$ and
rejects heat to a reservoir at temperature T. Engine B receives heat rejected by engine A and in turn rejects it to
a reservoir at $100\,K$ . If the efficiencies of the two engines $A$ and $B$ are represented by $\eta _{A}$ and $\eta _{B}$ respectively,
then what is the value of $\frac{\eta _{A}}{\eta _{B}}$
a.
$\frac{12}{7}$
b.
$\frac{12}{5}$
c.
$\frac{5}{12}$
d.
$\frac{7}{12}$
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At the centre of a fixed large circular coil of radius $R$ , a much smaller circular coil of radius $r$ is placed. The
two coils are concentric and are in the same plane. The larger coil carries a current I. The smaller coil is set to
rotate with a constant angular velocity $\omega$ about an axis along their common diameter. Calculate the emf
induced in the smaller coil after a time $t$ of its start of rotation.
When an air bubble of radius $r$ rises from the bottom to the surface of a lake, its radius becomes $\frac{5r}{4}$ . Taking the
atmospheric pressure to be equal to $10\,m$ height of water column, the depth of the lake would approximately be
(ignore the surface tension and the effect of temperature):
a.
$10.5\,m$
b.
$8.7\,m$
c.
$11.2\,m$
d.
$9.5\,m$
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A copper rod of mass $m$ slides under gravity on two smooth parallel rails, with separation $l$ and set at an angle
of $\theta$ with the horizontal. At the bottom, rails are joined by a resistance R. There is a uniform magnetic field $B$
normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is:
a.
$\frac{mgR cos\theta}{B^{2}l^{2}}$
b.
$\frac{mgR sin\theta}{B^{2}l^{2}}$
c.
$\frac{mgR tan\theta}{B^{2}l^{2}}$
d.
$\frac{mgR cot\theta}{B^{2}l^{2}}$
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A copper rod of cross-sectional area A carries a uniform current I through it. At temperature T, if the volume
charge density of the rod is $\rho$ , how long will the charges take to travel a distance $d$ ?
a.
$\frac{2\rho dA}{IT}$
b.
$\frac{2\rho dA}{I}$
c.
$\frac{\rho dA}{I}$
d.
$\frac{\rho dA}{IT}$
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A parallel plate capacitor with area $200\,cm^{2}$ and separation between the plates $1.5\,cm$ , is connected across a
battery of emf V. If the force of attraction between the plates is $25 \times 10^{-6}\,N$ , the value of $V$ is approximately:
$2 2$
$(\epsilon _{0} = 8.85 \times 10^{-12} \frac{C}{\,N.m\,}) )$
a.
$150\,V$
b.
$100\,V$
c.
$250\,V$
d.
$300\,V$
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A current of $1\,A$ is flowing on the sides of an equilateral triangle of side $4.5 \times 10^{-2}\,m$ . The magnetic field at
the centre of the triangle will be:
a.
$4 \times 10^{-5}\,Wb/m^{2}$
b.
Zero
c.
$2 \times 10^{-5}\,Wb/m^{2}$
d.
$8 \times 10^{-5}\,Wb/m^{2}$
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A thin uniform bar of length $L$ and mass $8\,m$ lies on a smooth horizontal table. Two point masses $m$ and $2\,m$
moving in the same horizontal plane from opposite sides of the bar with speeds $2v$ and $v$ respectively. The
masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the centre of the bar. If the bar
starts rotating about its center of mass as a result of collision, the angular speed of the bar will be:
a.
$\frac{v}{6\,L}$
b.
$\frac{6v}{5\,L}$
c.
$\frac{3v}{5\,L}$
d.
$\frac{v}{5\,L}$
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The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance $49\mu H$ and a
capactiance of $2.5nF$ . It is modulated by an audio signal of $12kHz$ . The frequency range occupied by the side
bands is:
a.
$18kHz -30kHz$
b.
$63kHz -75kHz$
c.
$442kHz -466kHz$
d.
$13482kHz -13494kHz$
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A plane polarized monochromatic EM wave is travelling a vacuum along $z$ direction such that at $t = t_{1}$ it is
found that the electric field is zero at a spatial point $z_{1}$ . The next zero that occurs in its neighbourhood is at $z_{2}$ .
The frequency of the electromagnetic wave is:
a.
$\frac{3\times 10^{8}}{|z_{2}-z_{1}|}$
b.
$\frac{6\times 10^{8}}{|z_{2}-z_{1}|}$
c.
$\frac{1.5\times 10^{8}}{|z_{2}-z_{1}|}$
d.
$\frac{1}{t+^{z}^{2-}^{z}^{1}}$
$1 3\times 108$
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If $x$ gram of gas is adsorbed by $m$ gram of adsorbent at pressure $P$ the plot of $log \frac{x}{\,m}$ versus $log P$ is linear.
The slope of the plot is: ( $m$ and $k$ are constants and $n> 1$ )
a.
$log k$
b.
$\frac{1}{n}$
c.
$2k$
d.
$n$
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Two 5 molal solutions are prepared by dissolving a non-electrolyte, non-volatile solute separately in the
solvents $X$ and $Y$ . The molecular weights of the solvents are $M_{X}$ and $M_{Y}$ , respectively where $M_{X} = \frac{3}{4}M_{Y}$ .
The relative lowering of vapour pressure of the solution in $X$ is " $m$ " times that of the solution in Y. Given that
the number of moles of solute is very small in comparison to that of solvent, the value of "m" is:
a.
$\frac{3}{4}$
b.
$\frac{1}{2}$
c.
$\frac{1}{4}$
d.
$\frac{4}{3}$
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At a certain temperature in a 5L vessel, 2 moles of carbon monoxide and 3 moles of chlorine were allowed to
reach equilibrium according to the reaction, $CO + Cl_{2} \rightleftharpoons COCl_{2}$ . At equilibrium, if one mole of $CO$ is
present then equilibrium constant $(K_{c})$ for the reaction is:
a.
$2.5$
b.
4
c.
2
d.
3
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When 2-butyne is treated with $H_{2}$ /Lindlar's catalyst, compound $X$ is produced as the major product and when
treated with $Na$ /liq. $NH_{3}$ it produces $Y$ as the major product. Which of the following statements is correct?
a.
Y will have higher dipole moment and higher
boiling point than $X$ Y
b.
$Y$ will have higher dipole moment and lower
boiling point than $X$
c.
X will have lower dipole moment and lower
boiling point than $Y$
d.
$X$ will have higher dipole moment and higher
boiling point than Y
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In the leaching method, bauxite ore is digested with a concentrated solution of $NaOH$ that produces ' $X$ '.
When $CO_{2}$ gas is passed through the aqueous solution of ' $X'$ , a hydrated compound ' $Y'$ ' is precipitated. ' $X$ '
and ' $Y$ ' respectively are.
a.
$Na [Al(OH)_{4}]$ and $Al_{2}(CO_{3})_{3}\cdot xH_{2}O$
b.
$Al(OH)_{3}$ and $Al_{2}O_{3}\cdot xH_{2}O$
c.
$NaAlO_{2}$ and $Al_{2}(CO_{3})_{3}\cdot xH_{2}O$
d.
$Na [Al(OH)_{4}]$ and $Al_{2}O_{3}\cdot xH_{2}O$
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$\Delta _{f}G^{\circ }$ at $500\,K$ for substance ' $S$ ' in liquid state and gaseous state are $+100.7kcalmol^{-1}$ and $+103 kcalmol^{-1}$ ,
respectively. Vapour pressure of liquid ' $S$ ' at $500\,K$ is approximately equal to:
$(R = 2calK^{-1}\,mol^{-1}).\,$
a.
$100\,atm$
b.
$1\,atm$
c.
$10\,atm$
d.
$0.1\,atm$
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For per gram of reactant, the maximum quantity of $N_{2}$ gas is produced in which of the following thermal
decomposition reactions? (Given: Atomic wt. : $Cr = 52u, Ba = 137u$ ).
a.
$Ba(N_{3})_{2}$ (s) $\longrightarrow Ba(C) + 3\,N_{2}(\,g)$
Following four solutions are prepared by mixing different volumes of $NaOH$ and $HCl$ of different
concentrations, $pH$ of which one of them will be equal to 1 ?
For a first order reaction, $A \rightarrow P, t\rightarrow_{1/2}$ (half-life) is 10 days. The time required for $\frac{1}{4}^{th\,}$ conversion of $A$ (in
days) is: $(ln 2 = 0.693, ln 3 = 1.1)$ .
a.
$3.2$
b.
$2.5$
c.
$4.1$
d.
5
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If the position vectors of the vertices $A,B$ and $C$ of a $\triangle ABC$ are respectively $4\hat{i}+ 7\hat{j}+ 8\hat{k}, 2\hat{i}+ 3\hat{j}+ 4\hat{k}$ and
$2\hat{i}+ 5\hat{j}+ 7\hat{k}$ , then the position vector of the point, where the bisector of $\angle A$ meets $BC$ is
a.
$\frac{1}{2}(4\hat{i}+ 8\hat{j}+ 11\hat{k})$
b.
$\frac{1}{3}(6\hat{i}+ 13\hat{j}+ 18\hat{k})$
c.
$\frac{1}{4}(8\hat{i}+ 14\hat{j}+ 9\hat{k})$
d.
$\frac{1}{3}(6\hat{i}+ 11\hat{j}+ 15\hat{k})$
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The foot of the perpendicular drawn from the origin, on the line, $3x+y=\lambda (\lambda \neq 0)$ is $P$ . If the line meets $x$ -
axis at $A$ and $y$ -axis at $B$ , then the ratio $BP :PA$ is
a.
$9 : 1$
b.
$1 : 3$
c.
$1 : 9$
d.
$3 : 1$
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Tangents drawn from the point $(-8, 0)$ to the parabola $y^{2} = 8x$ touch the parabola at $P$ and $Q$ . If $F$ is the
focus of the parabola, then the area of the triangle $PFQ$ (in sq. units) is equal to
a.
48
b.
32
c.
24
d.
64
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Consider the following two statements. Statement $p$ : The value of $sin 120^{\circ }$ can be divided by taking $\theta = 240^{\circ }$
in the equation
$2 sin \frac{\theta}{2} =\sqrt{1 + sin\theta}- \sqrt{1 -sin\theta}.$
Statement $q$ : The angles $A,B,C$ and $D$ of any quadrilateral $ABCD$ satisfy the equation
$cos(\frac{1}{2}(A+C))+ cos( \frac{1}{2}(B+D))= 0$
Then the truth values of $p$ and $q$ are respectively.
a.
$F, T$
b.
$T, T$
c.
F, F
d.
T, F
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A tower $T_{1}$ of height $60\,m$ is located exactly opposite to a tower $T_{2}$ of height $80\,m$ on a straight road. From the
top of $T_{1}$ , if the angle of depression of the foot of $T_{2}$ is twice the angle of elevation of the top of $T_{2}$ , then the
width (in $m$ ) of the road between the feet of the towers $T_{1}$ and $T_{2}$ is
a.
$20\sqrt{2}$
b.
$10\sqrt{2}$
c.
$10\sqrt{3}$
d.
$20\sqrt{3}$
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A player $X$ has a biased coin whose probability of showing heads is $p$ and a player $Y$ has a fair coin. They start
playing a game with their own coins and play alternately. The player who throws a head first is a winner. If $X$
starts the game, and the probability of winning the game by both the players is equal, then the value of ' $p$ ' is
a.
$\frac{1}{3}$
b.
$\frac{1}{5}$
c.
$\frac{1}{4}$
d.
$\frac{2}{5}$
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Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3I)(A-5I) =O$ , where $I =I_{3}$ and $O=O_{3}$ . If $\alpha A+$
$\beta A^{-1} = 4I$ , then $\alpha +\beta$ is equal to
a.
8
b.
12
c.
13
d.
7
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Let $f(x)$ be a polynomial of degree 4 having extreme values at $x= 1$ and $x= 2$ . If $lim_{x\rightarrow 0\rightarrow } (\frac{f(x)}{x^{2}} + 1)= 3$
then $f(-1)$ is equal to
a.
$\frac{1}{2}$
b.
$\frac{3}{2}$
c.
$\frac{5}{2}$
d.
$\frac{9}{2}$
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$2x+ 5_{-1}$
$\int =A\sqrt{2}7 -6x- x +Bsin ( \frac{x+ 3}{4})+C$
$dx$
$\sqrt{7 -6x-x^{2}}$
(where $C$ is a constant of integration), then the ordered pair $(A,B)$ is equal to
a.
$(-2, -1)$
b.
$(2, -1)$
c.
$(-2, 1)$
d.
$(2, 1)$
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Let $A = -^{2}+^{3}-… + (-1)n^{-1}^{n}$ and $B = 1 - A$ . Then, the least odd natural number $p$
$_{n} (\frac{3}{4}) (\frac{3}{4}) (\frac{3}{4}) (\frac{3}{4})_{n}_{n}$
, so that $B_{n} >A_{n}$ , for all $n\geq p$ is
a.
5
b.
7
c.
11
d.
9
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The sides of a rhombus $ABCD$ are parallel to the lines, $x- y+ 2 = 0$ and $7x- y+ 3 = 0$ . If the diagonals of
the rhombus intersect at $P(1, 2)$ and the vertex $A$ (different from the origin) is on the $y$ axis, then the ordinate
of $A$ is
a.
2
b.
$\frac{7}{4}$
c.
$\frac{7}{2}$
d.
$\frac{5}{2}$
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The tangent to the circle $C_{1} :x^{2} +y^{2} -2x-1 = 0$ at the point $(2, 1)$ cuts off a chord of length 4 from a
circle $C_{2}$ whose centre is $(3, -2)$ . The radius of $C_{2}$ is
a.
$\sqrt{6}$
b.
2
c.
$\sqrt{2}$
d.
3
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A normal to the hyperbola, $4x^{2} -9y^{2} = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$ , respectively. If the
parallelogram $OABP(O$ being the origin) is formed, then the locus of $P$ is
a.
$4x^{2} -9y^{2} = 121$
b.
$4x^{2} + 9y^{2} = 121$
c.
$9x^{2} -4y^{2} = 169$
d.
$9x^{2} + 4y^{2} = 169$
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