A heavy box is to be dragged along a rough horizontal floor. To do so, the person $A$ pushes it at an angle $30^{\circ}$
from the horizontal and requires a minimum force $F_{A}$ , while the person $B$ pulls the box at an angle $60^{\circ}$ from
the horizontal and needs minimum force $F_{B}$ . If the coefficient of friction between the box and the floor is $\frac{\sqrt{3}}{5}$
, the ratio $\frac{F_{A}}{F_{B}}$ is
a.
$\sqrt{\frac{3}{2}}$
b.
$\frac{2}{\sqrt{3}}$
Correct Answer
c.
$\sqrt{3}$
d.
$\frac{5}{\sqrt{3}}$
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Q2mcq
The diameter of the objective lens of the microscope makes an angle $\beta$ at the focus of the microscope. Further,
the medium between the object and the lens is the oil of the refractive index $n$ . Then the resolving power of the
microscope.
a.
Increases with decreasing value of $\beta$
b.
Increases with increasing value of $nsin 2\beta$
Correct Answer
c.
Increases with increasing value of $\frac{1}{nsin2\beta}$
d.
Increases with decreasing value of $n$
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Q3mcq
A black coloured solid sphere of radius $R$ and mass $M$ is inside a cavity with a vacuum inside. The walls of
the cavity are maintained at temperature $T_{0}$ . The initial temperature of the sphere is $3T_{0}$ . If the specific heat of
the material of the sphere varies as $\alpha T^{3}$ per unit mass with the temperature $T$ of the sphere, where $\alpha$ is a
constant, then the time taken for the sphere to cool down to temperature $2T_{0}$ will be
( $\sigma$ is Stefan Boltzmann constant)
Given, $A$ and $B$ are input terminals
Logic $1$ is $> 5\,V$
Logic $0$ is $< 1\,V$
Which logic gate operation, the following circuit does?
Note: This question was awarded a bonus. $C$ option changed.
a.
OR gate.
b.
NOR gate.
c.
Output will always be one.
Correct Answer
d.
XOR gate.
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Q5mcq
In the diagram shown, the difference in the two tubes of the manometer is $5\, cm$ , the cross-section of the tube
at $A$ and $B$ is $6\, mm^{2}$ and $10\, mm^{2}$ respectively. The rate at which water flows through the tube is
$(g= 10\,m\,s^{-2})$
a.
$7. 5\, cc \,s^{-1}$
Correct Answer
b.
$12. 5\, cc \,s^{-1}$
c.
$8. 0\, cc \,s^{-1}$
d.
$10. 0\, cc \,s^{-1}$
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Q6mcq
The gap between the plates of a parallel plate capacitor of area $A$ and distance between plates $d$ , is filled with a
dielectric whose relative permittivity varies linearly from $\epsilon _{1}$ at one plate to $\epsilon _{2}$ at the other. The capacitance of
the capacitor is
Long-range radio transmission is possible when the radio waves are reflected from the ionosphere. For this to
happen the frequency of the radio waves must be in the range:
a.
$150 -500\, kHz$
b.
$80 -150\, MHz$
c.
$1 -3\, MHz$
d.
$8 -25\, MHz$
Correct Answer
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Q8mcq
A particle is released on a vertical smooth semicircular track from point $X$ so that, $OX$ makes angle from the$\theta$
vertical (see figure). The normal reaction of the track on the particle vanishes at the point $Y$ where $OY$
makes an angle with the horizontal. Then$\phi$
The electric field in a region of space is given by, $\vec{E} =E_{0}ˆi + 2E_{0}ˆj$ where $E_{0} = 100\,N\,C^{-1}$ . The flux of this
field through a circular surface of radius $0. 02\,m$ parallel to the $Y$ ‐ $Z$ plane is nearly
a.
$0. 02\,N\,m^{2}\,C^{-1}$
b.
$0. 005\,N\,m^{2}\,C^{-1}$
c.
$0. 125\,N\,m^{2}\,C^{-1}$
Correct Answer
d.
$3. 14\,N\,m^{2}\,C^{-1}$
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Q10mcq
Consider a cylinder of mass M resting on a rough horizontal rug that is pulled out from under it with
acceleration 'a' perpendicular to the axis of the cylinder. What is F at point P ? It is assumed that the
friction
cylinder does not slip.
a.
$Ma$
b.
$\frac{Ma}{2}$
c.
$\frac{Ma}{3}$
Correct Answer
d.
$Mg$
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A ray of light is incident from a denser to a rarer medium. The critical angle for total internal reflection is $\theta$ $_{iC}$
and Brewster's angle of incidence is $\theta$ $_{iB}$ , such that $\frac{sin$\theta$_{iC}}{sin$\theta$_{iB}} =$ $\eta$ $= 1.28$ . The relative refractive index of the two
media is
a.
$0.4$
b.
$0.2$
c.
$0.9$
d.
$0.8$
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A bullet loses $(_{n} )$ of its velocity passing through one plank. Considering uniform retardation, the number of
such planks that are required to stop the bullet can be:
a.
Infinite
b.
$n$
c.
$\frac{n^{2}}{(2n-1)}$
d.
$\frac{2n^{2}}{(n-1)}$
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The figure shows a circular area of tthe radius $R$ where a uniform magnetic field $\vec{B}$ is going into the plane of
the paper and increasing in magnitude at a constant rate. In that case, which of the following graphs, drawn
schematically, correctly shows the variation of the induced electric field $E(r)$ ?
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The total length of a sonometer wire fixed between two bridges is $110\, cm$ . Now, two more bridges are placed
to divide the length of the wire in the ratio $6 : 3 : 2$ . If the tension in the wire is $400\,N$ and the mass per unit
length of the wire is $0. 01\, kg \,m^{-1}$ , then the minimum common frequency with which all the three parts can
vibrate, is
a.
$1000 Hz$
b.
$1100 Hz$
c.
$100 Hz$
d.
$110 Hz$
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A large number of liquid drops each of radius $r$ coalesce to form a single drop of the radius $R$ . The energy
released in the process is converted into kinetic energy of the big drop so formed. The speed of the big drop is
(given surface tension of the liquid $T$ , density $\rho$ )
A ball of mass $160\,g$ is thrown up at an angle of $60^{\circ}$ to the horizontal at a speed of $10\,m\,s^{-1}$ . The angular
momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is
thrown is nearly $(g = 10\,m\,s^{-2})$
a.
$1. 73\, kg \,m^{2}\,s^{-1}$
b.
$3. 46\, kg \,m^{2}\,s^{-1}$
c.
$3. 0\, kg \,m^{2}\,s^{-1}$
d.
$6. 0\, kg \,m^{2}\,s^{-1}$
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An example of a perfect diamagnet is a superconductor. This implies that when a superconductor is put in a
magnetic field of intensity $B$ , the magnetic field $B_{s}$ inside the superconductor will be such that
a.
$B_{s} =B$ .
b.
$B_{s} = 0$ .
c.
$B_{s} <B$ but $B_{s} \neq 0$ .
d.
$B_{s} = -B$
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A square frame of side $10\, cm$ and a long straight wire carrying current $1\,A$ are in the plane of the paper.
Starting from close to the wire, the frame moves towards the right with a constant speed of $10\,m\,s^{-1}$ (see
figure). The e.m.f induced at the time the left arm of the frame is at $x= 10\, cm$ from the wire is
a.
$0.75\,$ $\mu$ $V$
b.
$1\,$ $\mu$ $V$
c.
$2\,$ $\mu$ $V$
d.
$0.5\,$ $\mu$ $V$
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In Young’s double-slit experiment, the distance between the two identical slits is $\,6. 1$ times larger than the slit
width. Then the number of intensity maxima observed within the central maximum of the single-slit diffraction
pattern is :
a.
$3$
b.
$6$
c.
$24$
d.
$12$
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Match List‐ $I$ (Event) with List‐ $II$ (Order of the time interval for the happening of the event) and select the
correct option from the options given below the lists.
List-I List-II
$(a)$ The rotation period of earth $(i) 10^{5}\,s$
$(b)$ Revolution period of earth $(ii) 10^{7}\,s$
$(c)$ Period of a light wave $(iii) 10^{-15}\,s$
$(d)$ Period of a sound wave $(iv) 10^{-3}\,s$
A monoatomic gas is compressed from a volume of $2\,m^{3}$ to a volume of $1\,m^{3}$ at a constant pressure of
$100\,N\,m^{2}$ . Then it is heated at constant volume by supplying $150\,J$ of energy. As a result, the internal energy
of the gas
a.
Decreases by $50\,J$
b.
Increases by $250\,J$
c.
Decreases by $250\,J$
d.
$0\,J$
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In an experiment to determine the gravitational acceleration $g$ of a place with the help of a simple pendulum,
the measured time period squared is plotted against the string length of the pendulum in the figure. What is the
value of $g$ at the place?
a.
$10. 0\,m\,s^{-2}$
b.
$9. 87\,m\,s^{-2}$
c.
$9. 91\,m\,s^{-2}$
d.
$9. 81\,m\,s^{-2}$
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A piece of wood from a recently cut tree shows $20$ decays per minute. A wooden piece of the same size placed
in a museum (obtained from a tree cut many years back) shows $2$ decays per minute. If the half-life of $C^{14}$ is
$5730\, years$ , then the age of the wooden piece placed in the museum is approximately
[This question was awarded a bonus and proper correction was made to avoid that]
a.
$10439\, years$
b.
$39049\, years$
c.
$19042\, years$
d.
$13094\, years$
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A gas molecule of mass $M$ at the surface of the earth has kinetic energy equivalent to $0\,^{\circ}C$ . If it were to go up
straight without colliding with any other molecules, how high it would rise? Assume that the height attained is
much less than the radius of the earth. ( $k_{B}$ is Boltzmann constant)
a.
$\frac{273k}{2Mg}^{B}$
b.
$\frac{819k}{2Mg}^{B}$
c.
$0$
d.
$\frac{546k_{B}}{3Mg}$
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Four bulbs $B_{1}$ , $B_{2}$ , $B_{3}$ and $B_{4}$ of $100\,W$ each are connected to $220\,V$ main as shown in the figure. The
reading in an ideal ammeter will be
a.
$0. 90\,A$
b.
$1. 35\,A$
c.
$0. 45\,A$
d.
$1. 80\,A$
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If microwaves, $X$ ‐rays, infrared, gamma rays, ultraviolet, radio waves and visible parts of the electromagnetic
spectrum are denoted respectively by $M$ , $X$ , $I$ , $G$ , $U$ , $R$ and $V$ the following is the arrangement in ascending
order of the wavelength
a.
$I$ , $M$ , $R$ , $U$ , $V$ , $X$ and $G$
b.
$R$ , $M$ , $I$ , $V$ , $U$ , $X$ and $G$
c.
$M$ , $R$ , $V$ , $X$ , $U$ , $G$ and $I$
d.
$G$ , $X$ , $U$ , $V$ , $I$ , $M$ and $R$
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A body is in simple harmonic motion with time period $T = 0.5 s$ and amplitude $A= 1 cm$ . Find the average
velocity in the interval in which it moves from equilibrium position to half of its amplitude.
a.
$16 cm/s$
b.
$6 cm/s$
c.
$4 cm/s$
d.
$12 cm/s$
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Match List‐ $I$ (Experiment performed) with List‐ $II$ (Phenomena discovered/associated)and select the correct
option from the options given below the lists
List-I List-II
$(a)$ Davisson and Germer experiment $(i)$ Wave nature of electrons
$(b)$ Millikan's oil drop experiment $(ii)$ Charge of an electron
$(c)$ Rutherford experiment $(iii)$ Quantisation of energy levels
$(d)$ Franck-Hertz experiment $(iv)$ Existence of the nucleus
The gravitational field in a region is given by $\vec{g}=(5ˆi + 12ˆj)\,N\, kg^{-1}$ . The change in the gravitational potential
energy of a particle of mass $2\, kg$ when it is taken from the origin to a point $(7\,m, -3\,m)$ is
a.
$71 J$
b.
$13\sqrt{58}\,J$
c.
$2 J$
d.
$1 J$
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The velocity of water in a river is $18\, km \,h^{-1}$ near the surface. If the river is $5\,m$ deep, find the shearing stress
between the horizontal layers of water. The coefficient of viscosity of water $= 10^{-2}\, poise$ .
a.
$10^{-4}\,N\,m^{-2}$
b.
$10^{-3}\,N\,m^{-2}$
c.
$10^{-2}\,N\,m^{-2}$
d.
$10^{-1}\,N\,m^{-2}$
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An octahedral complex with molecular composition M.5NH .Cl.SO has two isomers, A and B. The solution
3 4
of A gives a white precipitate with AgNO solution and the solution of B gives white precipitate with BaCl
3 2
solution .The type of isomerism exhibited by the complex is :
a.
Coordinate isomerism
b.
Geometrical isomerism
c.
Ionisation isomerism
d.
Linkage isomerism
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The observed osmotic pressure for a $0. 10\,M$ solution of $Fe(NH_{4})_{2}(SO_{4})_{2}$ at $25^{\circ}C$ is $10. 8\, atm$ . The expected
and experimental (observed) values of Van't Hoff factor ' $i$ ' will be respectively;
$(R\, = \,0. 082\,L\, atm \,k^{-}\, mol^{-1})$
a.
$4$ and $4. 00$
b.
$3$ and $5. 42$
c.
$5$ and $4. 42$
d.
$5$ and $3. 42$
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The ionization energy of gaseous $Na$ atoms is $495. 5\, kJ \, mol^{-1}$ . The lowest possible frequency of light that
ionizes a sodium atom is
$(h = 6.626 \times 10^{-34}\, Js, \, N_{A}= 6.022 \times 10^{23}\, mol^{-1})$
a.
$1.24 \times 10^{15}\, s^{-1}$
b.
$7.50 \times 10^{4}\, s^{-1}$
c.
$4.76 \times 10^{14}\, s^{-1}$
d.
$3.15 \times 10^{15}\, s^{-1}$
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How many electrons are involved in the following redox reaction ?
$Cr_{2}O^{2-}_{7} + Fe^{2+} + C_{2}O^{2-}_{4} \rightarrow Cr\rightarrow^{3 +} + Fe^{3+} + CO_{2}$ (Unbalanced)
a.
6
b.
5
c.
3
d.
4
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Sulphur dioxide and oxygen were allowed to diffuse through a porous partition. $20\, dm^{3}$ of $SO_{2}$ diffuses
through the porous partition in $60$ seconds. The volume of $O_{2}$ in $dm^{3}$ which diffuses under the similar
condition in $30$ seconds will be (atomic mass of sulphur $= \,32\,u$ );
a.
$28. 2$
b.
$10. 0$
c.
$14. 1$
d.
$7. 09$
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Consider the reaction :
$H_{2}SO_{3(aq)}+ Sn^{4+}_{(aq)} + H_{2}O_{(l)} \rightarrow Sn\rightarrow^{2+}_{(aq)} + HSO^{-}_{4(aq)} + 3H^{+}_{(aq)}$
Which of the following statements is correct ?
a.
Sn4+
is the oxidizing agent because it undergoes
oxidation
b.
Sn4+
is the reducing agent because it undergoes
oxidation
c.
H SO is the reducing agent because it
2 3
undergoes reduction
d.
H SO is the reducing agent because it
2 3
undergoes oxidation
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Nickel $(Z = 28)$ combines with a uninegative monodentate ligand to form a diamagnetic complex $[NiL_{4}]^{2-}.$
The hybridisation involved and the number of unpaired electrons present in the complex are respectively:
a.
$dsp^{2},$ zero
b.
$sp^{3},$ zero
c.
$dsp^{2},$ , one
d.
$sp^{3},$ two
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For the decomposition of the compound, represented as
$NH_{2}COONH_{4}(s) \rightleftharpoons 2NH_{3}(g) + CO_{2}(g)$
the $K_{p} = 2 .9 \times 10^{-5} atm^{3}$ . If the reaction is started with 1 mole of the compound, the total pressure at
equilibrium would be :
a.
$7 .66 \times 10^{-2} atm$
b.
$38 .8 \times 10^{-2} atm$
c.
$5 .82 \times 10^{-2} atm$
d.
$1 .94 \times 10^{-2}\, atm$
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Zirconium phosphate $[Zr_{3}(PO_{4})_{4}]$ dissociates into three zirconium cations of charge $+4$ and four phosphate
anions of charge $-3$ . If molar solubility of zirconium phosphate is denoted by $s$ and its solubility product by
$K_{sp}$ then which of the following relationship between $s$ and $K_{sp}$ is correct ?
a.
$S = {K_{sp}/6912}^{7}$
b.
$S = {K_{sp}/144}^{1/7}$
c.
$S ={K_{sp}/(6912)^{1/7}}$
d.
$S = (K_{sp}/6912)^{1/7}$
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Let $x$ , $M$ and $\sigma ^{2}$ be respectively the mean, mode and variance of $n$ observations $x_{1}, \,x_{2}, . ..., \,x_{n}$ and
$d_{i} \bar{= -}x_{i}- a, \,i= 1, \,2, . ..., \,n,$ where $a$ is any number.
Statement I:Variance of $d_{1}, \,d_{2}, . . . , \,d_{n}$ is $\sigma ^{2}$ .
$-$ Statement II:Mean and mode of $d_{1}, \,d_{2}, . . . . , \,d_{n}$ are $-x- a$ and $-M - a$ , respectively.
a.
Statement and Statement are both true
b.
Statement
c.
Statement
d.
Statement
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$r 2r-1 3r-2$
If $\Delta _{r} = \frac{n}{2} n-1 a$ , then the value of $\sum ^{n}_{r=1}^{-1} \Delta _{r}$
$\frac{1}{2}n(n-1) \,(n-1)^{2} \frac{1}{2}(n-1)(3n+ 4)$
a.
Is independent of both $a$ and $n$
b.
Depends only on $a$
c.
Depends only on $n$
d.
Depends both on $a$ and $n$
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Two women and some men participated in a chess tournament in which every participant played two games
with each of the other participants. If the number of games that the men played between them-selves exceeds
the number of games that the men played with the women by $\,66$ , then the number of men who participated in
the tournament lies in the interval
a.
$(11, 13]$
b.
$(14, 17)$
c.
$[10, 12)$
d.
$[8, 9]$
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Let $f(n) =[\frac{1}{3} + \frac{3n}{100} ]n$ , where $[n]$ denotes the greatest integer less than or equal to $n$ . Then $\sum ^{56}_{n=1} f(n)$ is
equal to
a.
$56$
b.
$1287$
c.
$1399$
d.
$689$
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A chord is drawn through the focus of the parabola $y^{2} = 6x$ such that its distance from the vertex of this
parabola is $\frac{\sqrt{5}}{2}$ , then its slope can be
a.
$\frac{\sqrt{5}}{2}$
b.
$\frac{2}{\sqrt{3}}$
c.
$\frac{\sqrt{3}}{2}$
d.
$\frac{2}{\sqrt{5}}$
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If a line $L$ is perpendicular to the line $5x- y= 1,$ and the area of the triangle formed by the line $L$ and the
coordinate axes is $5$ sq units, then the distance of the line $L$ from the line $x+ 5y= 0$ is
a.
$\frac{7}{\sqrt{13}}$ units
b.
$\frac{7}{\sqrt{5}}$ units
c.
$\frac{5}{\sqrt{13}}$ units
d.
$\frac{5}{\sqrt{7}}$ units
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The area of the region $($ in square units $)$ above the $x$ -axis bounded by the curve $y= tanx, \,0 \leq x\leq \frac{\pi}{2}$ and the
tangent to the curve at $x= \frac{\pi}{4}$ is
a.
$\frac{1}{2}(log 2 - \frac{1}{2})$
b.
$\frac{1}{2}(1 + log 2)$
c.
$\frac{1}{2}(1 -log 2)$
d.
$\frac{1}{2}(log 2 + \frac{1}{2})$
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If the volume of a spherical ball is increasing at the rate of $4\pi \, cc / sec$ then the rate of increase of its radius
$(in \, cm / sec),$ when the volume is $288\pi \, cc$ is
a.
$\frac{1}{9}$
b.
$\frac{1}{6}$
c.
$\frac{1}{24}$
d.
$\frac{1}{36}$
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The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the
points $(a^{2} + 1, \,a^{2} + 1)$ and $(2a,\,-\,2a), \,a\neq 0$ . Then for any $a,$ the orthocentre of this triangle lies on the line
a.
$y- (a^{2} + 1)x= 0$
b.
$y-2ax= 0$
c.
$y+x= 0$
d.
$(a-1)^{2}x-(a+ 1)^{2}y= 0$
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If non-zero real numbers $b$ and $c$ are such that $min f(x) >max g(x)$ , where $f(x) =x^{2} + 2bx+ 2c^{2}$ and
$g(x) = -x^{2} -2cx+b^{2}, \,(x\in R)$ ; then $\frac{c}{b}$ lies in the interval
a.
$(\sqrt{2},\,\infty )$
b.
$[\frac{1}{2},\, \frac{1}{\sqrt{2}})$
c.
$(0,\,\frac{1}{2})$
d.
$[\frac{1}{\sqrt{2}},\,\sqrt{2}]$
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If $\vec{x}= 3ˆi-6ˆj- ˆk,\, \vec{y}= ˆi+ 4ˆj-3ˆk$ and $\vec{z}= 3ˆi-4ˆj-12ˆk$ , then the magnitude of the projection of $\vec{x}\times \vec{y}$ on
$\vec{z}$ is
a.
$14$
b.
$12$
c.
$15$
d.
$10$
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The number of terms in an $A.P.$ is even, the sum of the odd terms in it is $24$ and that the even terms is $30.$ If
the last term exceeds the first term by $10 \frac{1}{2},$ then the number of terms in the $A.P.$ is
a.
$4$
b.
$8$
c.
$16$
d.
$12$
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$|$
Equation of the line of the shortest distance between the lines $\frac{x}{1} = \frac{y}{-1} = \frac{z}{1}$ and $\frac{x-1}{0} = \frac{y+1}{-2} = \frac{z}{1}$ is
a.
$\frac{x}{-2} = \frac{y}{1} = \frac{z}{2}$
b.
$\frac{x}{1} = \frac{y}{-1} = \frac{z}{-2}$
c.
$\frac{x-1}{1} = \frac{y+1}{-1} = \frac{z}{-2}$
d.
$\frac{x-1}{1} = \frac{y+1}{-1} = \frac{z}{1}$
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The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{5} = 1$ , meets the
$x$ -axis and $y$ -axis at $A$ and $B$ , respectively. Then $OA^{2} - OB^{2}$ , where $O$ is the origin, equals
a.
$-\frac{20}{9}$
b.
$\frac{16}{9}$
c.
4
d.
$- \frac{4}{3}$
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Let, the function $F$ be defined as $F(x) =\int _{1}^{x} \frac{e^{t}}{t}dt, \,x> 0,$ then the value of the integral $\int _{1}^{x} \frac{e^{t}}{t+a}dt,$ where
$a> 0,$ is
a.
$e^{a}[F(x) - F(1 +a)]$
b.
$e^{-a}[F(x+a) - F(a)]$
c.
$e^{a}[F(x+a) - F(1 +a)]$
d.
$e^{-a}[F(x+a) - F(1 +a)]$
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