Water flows into a large tank with flat bottom at the rate of $10^{-4} m^{3}s^{-1}.$ Water is also leaking out of a hole of
area $1\,cm^{2}$ at its bottom. If the height of the water in the tank remains steady then this height is:
a.
$5.1\,cm$
Correct Answer
b.
$1.7\,cm$
c.
$2.9\,cm$
d.
$4\,cm$
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Q2mcq
A potentiometer wire $AB$ having length $L$ and resistance $12r$ is joined to a cell $D$ of emf $\epsilon$ and internal
resistance $r$ . A cell $C$ having EMF $\epsilon /2$ and internal resistance $3r$ is connected. The length $AJ$ , at which the
galvanometer, as shown in the figure, shows no deflection is
a.
$\frac{5}{12}L$
b.
$\frac{11}{24} L$
c.
$\frac{11}{12}L$
d.
$\frac{13}{24} L$
Correct Answer
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Q3mcq
A piece of wood of mass $0.03\,kg$ is dropped from the top of a $100\,m$ height building. At the same time, a bullet
of mass $0.02\,kg$ is fired vertically upward, with a velocity $100\,ms^{-1},$ from the ground. The bullet gets
embedded in the wood. Then the maximum height to which the combined system reaches above the top of the
building before falling below is: $(g= 10\,ms^{-2})$
a.
$40\,m$
Correct Answer
b.
$20\,m$
c.
$10\,m$
d.
$30\,m$
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Q4mcq
In the cube of side 'a' shown in the figure, the vector from the central point of the face $ABOD$ to the central
point of the face $BEFO$ will be:
a.
$\frac{1}{2} a (ˆk-ˆi)$
b.
$\frac{1}{2} a(ˆi- ˆk)$
c.
$\frac{1}{2} a(ˆj-ˆi)$
Correct Answer
d.
$\frac{1}{2} a(ˆj- ˆk)$
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Q5mcq
In a Young's double slit experiment slit separation $0.1\,mm,$ one observes a bright fringe at angle $\frac{1}{40} rad$ by
using light of wavelength $\lambda _{1}.$ When the light of wavelength $\lambda _{2}$ is used a bright fringe is seen at the same angle
in the same set up. Given that $\lambda _{1}$ and $\lambda _{2}$ are in visible range $(380\,nm\,to\,740\,nm),$ their values are:
a.
$400\,nm, \,500\,nm$
b.
$380\,nm, \,525\,nm$
c.
$625\,nm, \,500\,nm$
Correct Answer
d.
$380\,nm, \,500\,nm$
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Q6mcq
Three Carnot engines operate in series between a heat source at a temperature $T_{1}$ and a heat sink at
temperature $T_{4}$ (see figure). There are two other reservoirs at temperature $T_{2}$ and $T_{3},$ as shown, with
$T_{1} >T_{2} >T_{3} >T_{4}.$ The three engines are equally efficient if:
Using a nuclear counter the count rate of emitted particles from a radioactive source is measured. At $t= 0$ it
was $1600$ counts per second and $t= 8$ seconds it was $100$ counts per second. The count rate observed, as
counts per second, at $t= 6$ seconds is close to:
a.
$400$
b.
$200$
Correct Answer
c.
$150$
d.
$360$
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Q8mcq
A $2\,W$ carbon resistor is color coded with green, black, red and silver respectively. The maximum current
which can be passed through this resistor is:
a.
$100\,mA$
b.
$0.4\,mA$
c.
$20\,mA$
Correct Answer
d.
$63\,mA$
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Q9mcq
To mop-clean a floor, a cleaning machine presses a circular mop of radius $R$ vertically down with a total force
$F$ and rotates it with a constant angular speed about its axis. If the force $F$ is distributed uniformly over the mop
and if coefficient of friction between the mop and the floor is $\mu ,$ the torque, applied by the machine on the mop
is:
a.
$2\,\mu FR/3$
Correct Answer
b.
$\mu FR/3$
c.
$\mu FR/6$
d.
$\mu FR/2$
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Q10mcq
An insulating thin rod of length $l$ has a linear charge density $\rho (x) =\rho _{0} \frac{x}{l}$ on it. The rod is rotated about an axis
passing through the origin $(x= 0)$ and perpendicular to the rod. If the rod makes $n$ rotations per second, then
the time averaged magnetic moment of the rod is:
a.
$\frac{\pi}{4} n \rho _{0}l^{3}$
Correct Answer
b.
$n \rho _{0}l^{3}$
c.
$\pi n \rho _{0} l^{3}$
d.
$\frac{\pi}{3}n \rho _{0}l^{3}$
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In an electron microscope, the resolution that can be achieved is of the order of the wavelength of electrons
used. To resolve a width of $7.5 \times 10^{-12} m,$ the minimum electron energy required is close to:
a.
$100\,keV$
b.
$25\,keV$
c.
$1\,keV$
d.
$500\,keV$
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A TV transmission tower has a height of $140\,m$ and the height of the receiving antenna is $40\,m.$ What is the
maximum distance upto which signals can be broadcasted from this tower in LOS (Line of Sight) mode?
(Given: radius of earth $= 6.4 \times 10^{6} m).$
a.
$48\,km$
b.
$40\,km$
c.
$80\,km$
d.
$65\,km$
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A satellite is moving with a constant speed $v$ in circular orbit around the earth. An object of mass 'm' is ejected
from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the
kinetic energy of the object is:
a.
$\frac{3}{2} m v^{2}$
b.
$m v^{2}$
c.
$\frac{1}{2} m v^{2}$
d.
$\,2\,m v^{2}$
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A plano-convex lens of refractive index $\mu _{1}$ and focal length $f_{1}$ is kept in contact with another plano-concave
lens of refractive index $\mu _{2}$ and focal length $f_{2}.$ If the radius of curvature of their spherical faces is $R$ each and
$f_{1} = 2f_{2},$ the $\mu _{1}$ and $\mu _{2}$ are related as:
a.
$2\mu _{2}- \mu _{1} = 1$
b.
$3\mu _{2}-2\mu _{1} = 1$
c.
$\mu _{1}+\mu _{2} = 3$
d.
$2\mu _{1}- \mu _{2} = 1$
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A parallel plate capacitor is of area $6\,cm^{2}$ and a separation $3\,mm.$ The gap is filled with three dielectric
materials of equal thickness (see figure) with dielectric constant $K_{1} = 10, \,K_{2} = 12$ and $K_{3} = 14.$ The
dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would
be:
a.
$4$
b.
$36$
c.
$14$
d.
$12$
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A solid metal cube of edge length $2\, cm$ is moving in the positive $y$ -direction, at a constant speed of $6\,m\,s^{-1}$ .
There is a uniform magnetic field of $0.1\,T$ in the positive $z$ -direction. The potential difference between the two
faces of the cube, perpendicular to the $x$ -axis, is
a.
$12\, mV$
b.
$1\, mV$
c.
$2\, mV$
d.
$6\, mV$
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Two guns $A$ and $B$ can fire bullets at speeds $1\,km/s$ and $2\,km/s$ respectively. From a point on a horizontal
ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the
two guns, on the ground is:
a.
$1 : 16$
b.
$1 : 2$
c.
$1 : 4$
d.
$1 : 8$
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A heat source at $T = 10^{3}\,K$ is connected to another heat reservoir at $T = 10^{2}\,K$ by a copper slab which is $1\,m$
thick. Given that the thermal conductivity of copper is $0 .1 \,W\,K^{-1}\,m^{-1}$ , the energy flux through it in the
steady-state is:
a.
$65\,W\,m^{-2}$
b.
$120\,W\,m^{-2}$
c.
$90\,W\,m^{-2}$
d.
$200\,W\,m^{-2}$
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$--$
Two electric dipoles, $A,$ with respective dipole moments $B d_{\vec{A}} = -4qaˆi$ and $d_{\vec{B}} = -2qaˆi$ are placed on the $x$ -
axis with a separation $R,$ as shown in the figure
The distance from $A$ at which both of them produce the same potential is:
a.
$\frac{R}{\sqrt{2\,}-1}$
b.
$\frac{\sqrt{2}R}{\sqrt{2}-1}$
c.
$\frac{\sqrt{2}\,R}{\sqrt{2}+1}$
d.
$\frac{R}{\sqrt{2}+1}$
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A train moves towards a stationary observer with speed $34\,m/s.$ The train sounds a whistle and its frequency
registered by the observer is $f_{1}.$ If the speed of the train is reduced to $17\,m/s,$ the frequency registered is $f_{2}.$ If
speed of sound is $340\,m/s,$ then the ratio $f_{1}/f_{2}$ is:
a.
$21/20$
b.
$20/19$
c.
$19/18$
d.
$18/17$
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A charge $Q$ is distributed over three concentric spherical shells of radii $a, \,b, \,c\,(a<b<c)$ such that their
surface charge densities are equal to one another.
The total potential at a point at distance $r$ from their common centre, where $r<a,$ would be:
A homogeneous solid cylindrical roller of radius $R$ and mass $M$ is pulled on a cricket pitch by a horizontal
force. Assuming rolling without slipping, angular acceleration of the cylinder is:
a.
$\frac{F}{3\,mR}$
b.
$\frac{3F}{2\,mR}$
c.
$\frac{2F}{3\,mR}$
d.
$\frac{F}{2\,mR}$
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A block of mass $m$ is kept on a platform which starts from rest with a constant acceleration $g/2$ upwards, as
shown in the figure. Work done by normal reaction on block in time $t$ is
a.
$- \frac{mg^{2}t^{2}}{8}$
b.
$\frac{mg^{2}t^{2}}{8}$
c.
$0$
d.
$\frac{3\,mg^{2}t^{2}}{8}$
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A magnet of total magnetic moment $10^{-2} ˆi\,A\,m^{2}$ is placed in a time varying magnetic field, $Bˆi(cos\omega t)$ where
$B= 1$ Tesla and $\omega = 0.125\, rad \,s^{-1}$ . The work done for reversing the direction of the magnetic moment at
$t= 1$ second, is:
a.
$0.007\,J$
b.
$0.02\,J$
c.
$0.014\,J$
d.
$0.01\,J$
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A uniform metallic wire has a resistance of $18\,\Omega$ and is bent into an equilateral triangle. Then, the resistance
between any two vertices of the triangle is:
a.
$2\,\Omega$
b.
$12\,\Omega$
c.
$8\,\Omega$
d.
$4\,\Omega$
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In the given circuit the cells have zero internal resistance. The currents (in amperes) passing through resistance
$R_{1}$ and $R_{2}$ respectively, are:
a.
$0, \,1$
b.
$2, \,2$
c.
$0.5\,, \,0$
d.
$1, \,2$
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A string of length $1\,m$ and mass $5\,g$ is fixed at both ends. The tension in the string is $8 .0 \,N$ . The string is set
into vibration using an external vibrator of frequency $100\, Hz$ . The separation between successive nodes on the
string is close to
a.
$20. 0\, cm$
b.
$10. 0\, cm$
c.
$16 .6 \, cm$
d.
$33. 3\, cm$
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The density of a material in $SI$ units is $128\,kg m^{-3}.$ In certain units in which the unit of length is $25\,cm$ and the
unit of mass is $50g,$ the numerical value of density of the material is:
a.
$410$
b.
$16$
c.
$40$
d.
$640$
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If the magnetic field of a plane electromagnetic wave is given by (The speed of light $= 3 \times 10^{8} m/s$ )
$B= 100 \times 10^{-6} sin[2\pi \times 2 \times 10^{15}(t- \frac{x}{c})]$ then the maximum electric field associated with it is:
a.
$3 \times 10^{4}\,N\,C^{-1}$
b.
$4 \times 10^{4}\,N\,C^{-1}$
c.
$4.5 \times 10^{4}\,N\,C^{-1}$
d.
$6 \times 10^{4}\,N\,C^{-1}$
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A process has $\Delta H = 200J mol^{-1}$ and $\Delta S = 40JK^{-1}mol^{-1}.$ Out of the values given below choose the
minimum temperature above which the process will be spontaneous:
a.
$20\,K$
b.
$5K$
c.
$12K$
d.
$4K$
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Liquids $A$ and $B$ form an ideal solution in the entire composition range. At $350K,$ the vapour pressure of pure
A and pure B are $7 \times 10^{3} Pa$ and $12 \times 10^{3} Pa$ , respectively. The composition of the vapour in equilibrium
with a solution containing 40 mole percent of A at this temperature is:
a.
$x_{A} = 0.4;x_{B} = 0.6$
b.
$x_{A} = 0.76;x_{B} = 0.24$
c.
$x_{A} = 0.28;x_{B} = 0.72$
d.
$x_{A} = 0.37;x_{B} = 0.63$
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What are the values of $\frac{K_{p}}{K_{c}}$ for the following reactions at $300 K$ respectively?
(At $300 K, RT = 24 .62 dm^{2} atm mol^{-1}$ )
$N_{2}\,(g) + O_{2}\,(g) \rightleftharpoons 2 NO \,(g)$
$N_{2}O_{4}\,(g) \rightleftharpoons 2 NO \,(g)$
$N_{2}\,(g) + 3H_{2}\,(g) \rightleftharpoons 2 NH_{3}\,(g)$
A mixture of $100\,m\, mol$ of $Ca (OH)_{2}$ and $2\,g$ of sodium sulphate was dissolved in water and the volume was
made up to $100\, mL.$ What is the mass of calcium sulphate formed and the concentration of $OH^{-}$ in resulting
solution, respectively? (Molar mass of $Ca (OH)_{2}, \, Na_{2}SO_{4}$ and $CaSO_{4}$ are $74, \,143\,$ and $136\,g mol^{-1},$
respectively; $K_{sp}$ of $Ca (OH)_{2}\, is 5 .5 \times 10^{-6}$ )
a.
$1 .9 g, 0 .14 mol L^{-1}$
b.
$13 .6 g, 0 .28 mol L^{-1}$
c.
$1 .9 g, 0 .28 mol L^{-1}$
d.
$13 .6 g, 0 .14 mol L^{-1}$
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Consider the given plots for a reaction obeying Arrhenius equation $(0^{\circ}C <T < 300^{\circ}C) :$ ( $K$ and $E_{a}$ are rate
constant and activetion energy, respectively )
(I)
(II)
a.
I is wrong but II is right
b.
Both I and II are wrong
c.
I is right but II is wrong
d.
Both I and II are correct
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In a class of $140$ students numbered $1$ to $140$ , all even numbered students opted Mathematics course, those
whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted
Chemistry course. Then the number of students who did not opt for any of the three courses is:
a.
$42$
b.
$1$
c.
$38$
d.
$102$
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If a circle $C$ passing through the point $(4, \,0)$ touches the circle $x^{2} +y^{2} + 4x-6y= 12$ externally at the point
$(1, \, -1),$ then the radius of $C$ is:
a.
$4$ units
b.
$5$ units
c.
$2\sqrt{5}$ units
d.
$\sqrt{57}$ units
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Consider the quadratic equation $(c-5)x^{2} -2cx+ (c-4) = 0, \,c\neq 5.$ Let $S$ be the set of all integral values
of $c$ for which one root of the equation lies in the interval $(0, \,2)$ and its other root lies in the interval $(2, \,3).$
Then the number of elements in $S$ is
a.
$11$
b.
$12$
c.
$18$
d.
$10$
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Let $\vec{a}= 2ˆi+\lambda _{1}ˆj+ 3ˆk, \,\vec{b}= 4ˆi+ (3 - \lambda _{2})ˆj+ 6ˆk$ and $\vec{c}= 3ˆi+ 6ˆj+ (\lambda _{3}-1)ˆk$ be three vectors such that
$\vec{b}= 2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}.$ Then a possible value of $(\lambda _{1}, \,\lambda _{2}, \,\lambda _{3})$ is
a.
$(- \frac{1}{2}, \,4, \,0)$
b.
$(1, 5, 1)$
c.
$(\frac{1}{2} , \,4, -2)$
d.
$(1, 3, 1)$
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Let, $n\geq 2$ be a natural number and $0 <\theta < \frac{\pi}{2}.$ Then $\int \frac{(sin^{n}\theta -sin\theta )cos\theta}{sin^{n}^{+1}\theta}\frac{1}{n} d\theta ,$ is equal to
An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the
numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack
of nine cards numbered $1,\,2, \,3, … , \,9$ is randomly picked and the number on the card is noted. The probability
that the noted number is either $7$ or $8$ is
a.
$\frac{13}{36}$
b.
$\frac{19}{72}$
c.
$\frac{15}{72}$
d.
$\frac{19}{36}$
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Let $A$ be a point on the line $\vec{r}= (1 -3\mu )ˆi+ (\mu -1)ˆj+ (2 + 5\mu )ˆk$ and $B(3, \,2, \,6)$ be a point in the space.
$-$
Then the value of $\mu$ for which the vector $A\vec{B}$ is parallel to the plane $x-4y+ 3z= 1$ is
a.
$\frac{1}{2}$
b.
$\frac{1}{4}$
c.
$-\frac{1}{4}$
d.
$\frac{1}{8}$
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If the parabolas $y^{2} = 4b(x- c)$ and $y^{2} = 8ax$ have a common normal, then which one of the following is a
valid choice for the ordered triad $(a,b,c)$
a.
$(1, 1, 3)$
b.
$( \frac{1}{2}, 2, 0)$
c.
$(\frac{1}{2} , 2, 3)$
d.
All of above
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If the line $3x+ 4y-24 = 0$ intersects the $x$ -axis is at the point $A$ and the $y$ -axis at the point $B,$ then the
incentre of the triangle $OAB,$ where $O$ is the origin, is:
a.
$(4, \,4)\,$
b.
$(3, \,4)$
c.
$(4, \,3)$
d.
$(2, \,2)$
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The mean of five observations is $5$ and their variance is $9.20.$ If three of the given five observations are $1,\,3$
and $8,$ then a ratio of other two observations is
a.
$10 : 3$
b.
$4 : 9$
c.
$6 : 7$
d.
$5 : 8$
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A point $P$ moves on the line $2x-3y+ 4 = 0.$ If $Q(1, \,4)$ and $R(3, \, -2)$ are fixed points, then the locus of the
centroid of $\Delta PQR$ is a line:
a.
with slope $\frac{2}{3}$
b.
with slope $\frac{3}{2}$
c.
parallel to $y$ -axis
d.
parallel to $x$ -axis
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For each $t\in R,$ let $[t]$ be the greatest integer less than or equal to $t$ . Then, $_{x}lim_{\rightarrow 1\rightarrow }_{+} \frac{(1-|x|+sin|1-x|)sin([1-x])}{|1-x|[1-x]}\frac{\pi}{2}$
a.
equals $0$
b.
equals $-1$
c.
does not exist
d.
equal $1$
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$-2 4 +d (sin\theta ) -2$
$[ ]$
Let $d\in R,$ and $A= 1 (sin\theta ) + 2 d , \theta \in [0, \,2\pi ].$ If the minimum value of $det(A)$ is
$[ ]$
$5 (2 sin\theta ) - d (-sin\theta ) + 2 + 2d$
$8,$ then a value of $d$ is:
a.
$2(\sqrt{2}+ 2)$
b.
$2(\sqrt{2}+ 1)$
c.
$-5$
d.
$-7$
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$max(|x|, \,x^{2}), |x| \leq 2$
Let $f(x) ={ .$ Let $S$ be the set of points in the interval $(-4, \,4)$ at which $f$ is not
$8 -2|x|, 2 < |x| \leq 4$
differentiable. Then $S$
a.
equals ${-2, \, -1, \,0, \,1, \,2}$
b.
equals ${-2, \,2}$
c.
is an empty set
d.
equal ${-2, \, -1, \,1, \,2}$
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Consider a triangular plot $ABC$ with sides $AB= 7\,m,BC = 5\,m$ and $CA= 6\,m.$ A vertical lamp-post at the
mid-point $D$ of $AC$ subtends an angle $30^{\circ}$ at $B.$ The height (in $m$ ) of the lamp-post is:
a.
$2\sqrt{21}$
b.
$\frac{2}{3}\sqrt{21}$
c.
$\frac{3}{2}\sqrt{21}$
d.
$7\sqrt{3}$
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The plane passing through the point $(4, \, -1, \,2)$ and parallel to the lines $\frac{x+2}{3} = \frac{y-2}{-1} = \frac{z+1}{2}$ and
$\frac{x-2}{1} = \frac{y-3}{2} = \frac{z-4}{3}$ also passes through the point
a.
$(1, \,1, \, -1)$
b.
$(-1, \, -\,1, \, -1)$
c.
$(-1, \, -1,\,1)$
d.
$(1, \,1, \,1)$
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Let $z_{1}$ and $z_{2}$ be any two non-zero complex numbers such that $3|z_{1}| = 4|z_{2}|.$ If $z= \frac{3z_{1}}{2z_{2}} + \frac{2z_{2}}{3z_{1}}$ then maximum
value of $|z|$ is
Note: In actual paper value of $|z|$ was asked. Hence, none of the options given were correct. So we have
modified the question as well as options.
a.
$\frac{7}{2}$
b.
$\frac{9}{2}$
c.
$\frac{5}{2}$
d.
$\frac{1}{2}\sqrt{\frac{17}{2}}$
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