There is a small source of light at some depth below the surface of water (refractive index $= \frac{4}{3}$ ) in a tank of
large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water,
percentage of light that emerges out of surface is (nearly):
[Use the fact that surface area of a spherical cap of height $h$ and radius of curvature $r$ is $2\pi rh$ ]
a.
$21%$
b.
$34%$
c.
$17%$
Correct Answer
d.
$50%$
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Q2mcq
A plane electromagnetic wave is propagating along the direction $\frac{ˆi+ˆj}{\sqrt{2}} ,$ with its polarization along the direction
$kˆ.$ The correct form of the magnetic field of the wave would be (here $B_{0}$ is an appropriate constant):
Two gases - argon (atomic radius $0.07nm,$ atomic weight $40$ ) and xenon (atomic radius $0.1nm,$ atomic weight
$140$ ) have the same number density and are at the same temperature. The ratio of their respective mean free
times is closest to:
a.
$3.67$
b.
$1.83$
Correct Answer
c.
$2.3$
d.
$4.67$
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Q4mcq
Two identical capacitors $A$ and $B,$ charged to the same potential $5V$ are connected in two different circuits as
shown below at time $t= 0.$ If the charge on capacitors $A$ and $B$ at time $t=CR$ is $Q_{A}$ and $Q_{B}$ respectively,
then (Here $e$ is the base of natural logarithm)
a.
$Q_{A} = \frac{V C}{e},Q_{B} = \frac{CV}{2}$
b.
$Q_{A} =V C,Q_{B} =CV$
c.
$Q_{A} =V C,Q_{B} = \frac{V C}{e}$
Correct Answer
d.
$Q_{A} = \frac{CV}{2},Q_{B} = \frac{V C}{e}$
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Q5mcq
For the four sets of three measured physical quantities as given below. Which of the following options is
correct?
$(i)A_{1} = 24.36,B_{1} = 0.0724,C_{1} = 256.2$
$(ii)A_{2} = 24.44,B_{2} = 16.082,C_{2} = 240.2$
$(iii)A_{3} = 25.2,B_{3} = 19.2812,C_{3} = 236.183$
$(iv)A_{4} = 25,B_{4} = 236.191,C_{4} = 19.5$
An electron gun is placed inside a long solenoid of radius $R$ on its axis. The solenoid has $n$ turns/length and
carries a current $I.$ The electron gun shoots an electron along the radius of the solenoid with speed $v.$ If the
electron does not hit the surface of the solenoid, maximum possible value of $v$ is (all symbols have their
standard meaning):
a.
$\frac{e\mu _{0}nIR}{m}$
Correct Answer
b.
$\frac{e\mu _{0}nIR}{2m}$
c.
$\frac{e\mu _{0}nIR}{4m}$
d.
$\frac{2e\mu _{0}nIR}{m}$
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Q7nat
In the circuit shown below, is working as a $8\,V\,dc$ regulated voltage source. When $12\,V$ is used as an input,
the power dissipated (in $mW$ ) in each diode is (Considering both zener diodes are identical)
✓ Accepted Answer:40
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Q8mcq
Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their
energy stored per unit volume is $1 : 4,$ the ratio of their diameters is:
a.
$\sqrt{2}: 1$
Correct Answer
b.
$1 : 2$
c.
$2 : 1$
d.
$1 : \sqrt{2}$
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Q9mcq
An electron of mass $m$ and magnitude of charge $e$ at rest, gets accelerated by a constant electric field $E$ . The
rate of change of de-Broglie wavelength of this electron at a time $t$ is (ignore relativistic effects)
A particle of mass m is projected with a speed u from the ground at an angle $\theta = \frac{\pi}{3}$ w.r.t. horizontal (x-axis).
When it has reached its maximum height, it collides completely inelastically with another particle of the same
mass and velocity $uˆi.$ The horizontal distance covered by the combined mass before reaching the ground is:
An electric field $\vec{E} = 4xˆi- (y^{2} + 1)ˆjN/C$ passes through the box shown in figure. The flux of the electric
field through surfaces $ABCD$ and $BCGF$ are marked as $\phi _{I}$ and $\phi _{II}$ respectively. The difference between
$(\phi _{I} - \phi _{II})$ is (in $Nm^{2}/C$ ) ____________.
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Planet $A$ has mass $M$ and radius $R.$ Planet $B$ has half the mass and half the radius of Planet $A.$ If the escape
velocities from the Planets $A$ and $B$ are $v_{A}$ and $v_{B},$ respectively, then $\frac{v}{v_{B}}^{A} = \frac{n}{4}.$ The value of $n$ is:
a.
$4$
b.
$1$
c.
$2$
d.
$3$
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A uniformly thick wheel with moment of inertia $I$ and radius $R$ is free to rotate about its centre of mass (see
fig). A massless string is wrapped over its rim and two blocks of masses $m_{1}$ and $m_{2}(m_{1} >m_{2})$ are attached to
the ends of the string. The system Is released from rest. The angular speed of the wheel when $m_{1}$ descends by a
distance $h$ is:
In a Young's double slit experiment $15$ fringes are observed on a small portion of the screen when light of
wavelength $500\,nm$ is used. Ten fringes are observed on the same section of the screen when another light
source of wavelength $\lambda$ is used. Then the value of $\lambda$ is (in $nm$ ) __________.
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In $LC$ circuit the inductance $L= 40\, mH$ and capacitance $C = 100\,$ $\mu$ $F.$ If a voltage $V(t) = 10sin(314t)$ is
applied to the circuit, the current in the circuit is given as:
a.
$0.52 cos(314t)$
b.
$10 cos(314t)$
c.
$5.2 cos(314t)$
d.
$0.52 sin(314t)$
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A small circular loop of conducting wire has radius a and carries current $I.$ It is placed in a uniform magnetic
field $B$ perpendicular to its plane such that when rotated slightly about its diameter and released, it starts
performing simple harmonic motion of time period $T.$ The mass of the loop is $m$ then:
a.
$T = \sqrt{\frac{2m}{iB}}$
b.
$T =\sqrt{\frac{\pi m}{2iB}}$
c.
$T =\sqrt{\frac{2\pi m}{iB}}$
d.
$T =\sqrt{\frac{\pi m}{iB}}$
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In a meter bridge experiment $S$ is a standard resistance. $R$ is a resistance wire. It is found that balancing length
is $l= 25cm.$ If $R$ is replaced by a wire of hall length and half diameter that of $R$ of same material, then the
balancing distance $l'$ (in $cm$ ) will now be ______________.
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Starting at temperature $300K,$ one mole of an ideal diatomic gas $(\gamma = 1.4)$ is first compressed adiabatically
from volume $V_{1}$ to $V_{2} = \frac{V_{1}}{16}.$ It is then allowed to expand isobarically to volume $2V_{2}$ . If all the processes are
the quasi-static then the final temperature of the gas (in $^{o}K$ ) is (to the nearest integer) ___________.
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A small spherical droplet of density $d$ is floating exactly half immersed in a liquid of density $\rho$ and surface
tension $T.$ The radius of the droplet is (take note that the surface tension applies an upward force on the
droplet):
a.
$r=\sqrt{\frac{2T}{3(d+\rho )g}}$
b.
$r=\sqrt{\frac{T}{(d-\rho )g}}$
c.
$r=\sqrt{\frac{T}{(d+\rho )g}}$
d.
$r=\sqrt{\frac{3T}{(2d-\rho )g}}$
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A rod of length $l$ has non-uniform linear mass density given by $\rho (x) =a+b( \frac{x}{l})^{2},$ where $a$ and $b$ are constants
and $0 \leq x\leq l$ The value of $x$ for the centre of mass of the rod is at:
a.
$\frac{3}{2}(\frac{a+b}{2a+b})L$
b.
$\frac{3}{4}(\frac{2a+b}{3a+b})L$
c.
$\frac{4}{3}(\frac{a+b}{2a+3b})L$
d.
$\frac{3}{2}(\frac{2a+b}{3a+b})L$
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A spring mass system (mass $m,$ spring constant $k$ and natural length $l$ ) rests in equilibrium on a horizontal disc.
The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system rotates
about it's axis with an angular velocity $\omega ,(k>>m\omega ^{2})$ the relative change in the length of the spring is best
given by the option:
a.
$\sqrt{\frac{2}{3}}(\frac{m\omega ^{2}}{k})$
b.
$\frac{2m\omega ^{2}}{k}$
c.
$\frac{m\omega ^{2}}{k}$
d.
$\frac{m\omega ^{2}}{3k}$
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A particle starts from the origin at $t= 0$ with an initial velocity of $3.0ˆim/s$ and moves in the $x- y$ plane with a
constant acceleration $(6.0ˆi+ 4.0ˆj)m/s^{2}.$ The $x-$ coordinate of the particle at the instant when its $y-$
coordinate is $32m$ is $D$ meters. The value of $D$ is:
a.
$32$
b.
$50$
c.
$60$
d.
$40$
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A wire of length $L$ and mass per unit length $6.0 \times 10^{-3}\,kg\,m^{-1}$ is put under tension of $540\,N.$ Two
consecutive frequencies that it resonates at are: $420\,Hz$ and $490\,Hz.$ Then $L$ in meters is :
a.
$2.1m$
b.
$1.1m$
c.
$8.1m$
d.
$5.1m$
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The energy required to ionise a hydrogen like ion in its ground state is $9$ Rydbergs. What is the wavelength of
the radiation emitted when the electron in this ion jumps from the second excited stale to the ground state?
a.
$24. 2nm$
b.
$11.4nm$
c.
$35.8nm$
d.
$8.6nm$
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The first and second ionisation enthalpies of a metal are $496$ and $4560\,kJ\,mol^{-1}$ , respectively. How many
moles of $HCl$ and $H_{2}SO_{4},$ respectively, will be needed to react completely with $1$ mole of the metal
hydroxide?
a.
$1$ and $1$
b.
$2$ and $0.5$
c.
$1$ and $2$
d.
$1$ and $0.5$
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The reaction of $H_{3}N_{3}B_{3}Cl_{3}(A)$ with $LiBH_{4}$ in tetrahydrofuran gives inorganic benzene $(B)$ . Further, the
reaction of $(A)$ with $(C)$ leads to $H_{3}N_{3}B_{3}(Me)_{3}$ . Compounds $(B)$ and $(C)$ respectively, are:
a.
Bogging god $MeBr$
b.
Diborane and $MeMgBr$
c.
Boron nitride and $MeBr$
d.
Borazine and $MeMgBr$
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The correct order of the spin-only magnetic moments of the following complexes is:
$(I) [Cr(H_{2}O)_{6}]Br_{2}$
$(II) Na_{4}[Fe(CN)_{6}]$
$(III) Na_{3}[Fe(C_{2}O_{4})_{3}](\Delta _{0} > P)$
$(IV) (Et_{4}N)_{2}[CoCl_{4}]$
a.
$(III) > (I) > (IV) > (II)$
b.
$(III) > (I) > (II) > (IV)$
c.
$(I) > (IV) > (III) > (II)$
d.
$(II) ≈(I) > (IV) > (III)$
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$A, B$ and $C$ are three biomolecules. The results of the tests performed on them are given below:
Molisch's test Barfoed Test Biuret Test
$A$ Positive Negative Negative
$B$ Positive Positive Negative
$C$ Negative Negative Positive
$A, B$ and $C$ are respectively:
a.
$A =$ Glucose, $B =$ Fructose $C =$ Albumin
b.
$A =$ Lactose, $B =$ Glucose, $C =$ Albumin
c.
$A =$ Lactose, $B =$ Glucose, $C =$ Alanine
d.
$A =$ Lactose, $B =$ Fructose, $C =$ Alanine
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A cylinder containing an ideal gas ( $0.1mol$ of $1.0dm^{3}$ ) is in thermal equilibrium with a large volume of $0.5$
molal aqueous solution of ethylene glycol at its freezing point. If the stoppers $S_{1}$ and $S_{2}$ (as shown in the
figure) are suddenly withdrawn, the volume of the gas in litres after equilibrium is achieved will be
_____________.
(Given, $K_{f}(water) = 2.0Kkgmol^{-1}, R = 0.08dm^{3}atmK^{-1}mol^{-1})$
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The solubility product of $Cr(OH)_{3}$ at $298K$ is $6.0 \times 10^{-31}.$ The concentration of hydroxide ions in a saturated
solution of $Cr(OH)$ will be
a.
$(2.22 \times 10^{-31}) \frac{_{1}3}{4}$
b.
$(18 \times 10^{-31})\frac{1}{4}$
c.
$(18 \times 10^{-31})\frac{1}{2}$
d.
$(4.86 \times 10^{-29})\frac{1}{4}$
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Among the statements $(a) -(d),$ the correct ones are:
$(a)$ Lithium has the highest hydration enthalpy among the alkali metals.
$(b)$ Lithium chloride is insoluble in pyridine.
$(c)$ Lithium cannot form ethynide upon its reaction with ethyne.
$(d)$ Both lithium and magnesium react slowly with $H_{2}O$ .
a.
$(a), (b)$ and $(d)$ only
b.
$(a), (c)$ and $(d)$ only
c.
$(b)$ and $(c)$ only
d.
$(a)$ and $(d)$ only
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A mixture of gases $O_{2}, H_{2}$ and $CO$ are taken in a closed vessel containing charcoal The graph that represents
the correct behaviour of pressure with time is:
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In the figure shown below reactant $A$ (represented by square) is in equilibrium with product $B$ (represented by
circle). The equilibrium constant is (approx):
a.
$4$
b.
$8$
c.
$1$
d.
$2$
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A sample of milk splits after $60min$ . at $300K$ and after $40min$ . at $400K$ when the population of Iactobacillus
acidophilus in it doubles. The activation energy (in $kJ / mol$ ) for this process is closest to ______________.
(Given, $R = 8.3Jmol^{-1}K^{-1}, ln(\frac{2}{3} )= 0.4, e^{-3} = 4.0$ )
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Consider the following reactions $A\frac{(i)CH_{3}MBr}{(ii)H_{3}O^{+}}B\frac{Cu}{573K}2 -methyl -2 -butene$ . The mass percentage of
carbon in $A$ is _____________.
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$5g$ of zinc is treated separately with an excess of
$(a)$ dilute hydrochloric acid and
$(b)$ aqueous sodium hydroxide.
The ratio of the volumes of $H_{2}$ evolved in these two reactions is:
a.
$1 : 2$
b.
$1 : 1$
c.
$1 : 4$
d.
$2 : 1$
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Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a} = ,\sqrt{3} \vec{b} = 5, \vec{b}\cdot \vec{c}= 10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}.$ If $\vec{a}$
is perpendicular to the vector $\vec{b}\times \vec{c},$ then $\vec{a}\times (\vec{b}\times \vec{c})$ is equal to ____________.
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Let $a,b\in R,a\neq 0$ be such that the equation, $ax^{2} -2bx+ 5 = 0$ has a repeated root $\alpha ,$ which is also a root of
the equation, $x^{2} -2bx-10 = 0.$ If $\beta$ is the other root of this equation, then $\alpha ^{2} +\beta ^{2}$ is equal to:
a.
$25$
b.
$26$
c.
$28$
d.
$24$
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$||$
$||$
If the distance between the plane, $23x-10y-2z+ 48 = 0$ and the plane containing the lines
$\frac{x+1}{2} = \frac{y-3}{4} = \frac{z+1}{3}$ and $\frac{x+3}{2} = \frac{y+2}{6} = \frac{z-1}{\lambda}(\lambda \in R)$ is equal to $\frac{k}{\sqrt{633}},$ then $k$ is equal to ____________.
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The length of the minor axis (along $y$ -axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$ . If this ellipse touches the
line $x = 8$ then its eccentricity is:
a.
$\frac{+ 6y}{11}$
$\frac{1}{2}\sqrt{\,}$
b.
$\sqrt{\frac{5}{6}}$
c.
$\frac{3}{5}$
$\frac{1}{2}\sqrt{\,} _{3}$
d.
$\frac{1}{3}\sqrt{\frac{11}{3}}$
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Let $f$ and $g$ be differentiable functions on $R$ such that $fog$ is the identity function. If for some
$a,b\in R,g'(a) = 5$ and $g(a) =b,$ then $f'(b)$ is equal to:
a.
$\frac{1}{5}$
b.
$1$
c.
$5$
d.
$\frac{2}{5}$
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Let $[t]$ denote the greatest integer $\leq t$ and $_{x}lim_{\rightarrow 0\rightarrow } x[ \frac{4}{x}]=A.$ Then the function, $f(x) =[x^{2}]sin(\pi x)$ is
discontinuous, when $x$ is equal to:
a.
$\sqrt{A+ 1}$
b.
$\sqrt{A+ 5}$
c.
$\sqrt{A+ 21}$
d.
$\sqrt{A}$
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$\{ x, 0 \leq x< \frac{1}{2}$
Given: $f(x) = \frac{1}{2},x= \frac{1}{2}$
$\{$
$1 - x, \frac{1}{2} <x\leq 1$
and $g(x) =(x- \frac{1}{2} )^{2},x\in R.$ Then, the area (in sq. units) of the region bounded by the curves, $y=f(x)$ and
$y=g(x)$ between the lines $2x= 1$ and $2x= \sqrt{3},$ is:
a.
$\frac{1}{3} + \frac{\sqrt{3}}{4}$
b.
$\frac{\sqrt{3}}{4} - \frac{1}{3}$
c.
$\frac{1}{2} - \frac{\sqrt{3}}{4}$
d.
$\frac{1}{2} + \frac{\sqrt{3}}{4}$
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If the curves, $x^{2} -6x+y^{2} + 8 = 0$ and $x^{2} -8y+y^{2} + 16 - k= 0, (k> 0)$ touch each other at a point, then
the largest value of $k$ is ____________.
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In the expansion of $( \frac{x}{cos\theta} + \frac{1}{xsin\theta} )^{16},$ if $l_{1}$ is the least value of the term independent of $x$ when $\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}$
and $l_{2}$ is the least value of the term independent of $x$ when $\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8},$ then the ratio $l_{2} :l_{1}$ is equal to:
a.
$1 : 8$
b.
$16 : 1$
c.
$8 : 1$
d.
$1 : 16$
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$| |$
If $\int \frac{d\theta}{cos^{2}\theta (tan 2\theta +sec 2\theta )} =\lambda tan\theta + 2 log_{e}|f(\theta )| +C$ where $C$ is a constant of integration, then the ordered pair
$(\lambda ,f(\theta ))$ is equal to:
a.
$(1, 1 -tan\theta )$
b.
$(-1, 1 -tan\theta )$
c.
$(-1, 1 + tan\theta )$
d.
$(1, 1 + tan\theta )$
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Let a function $f : [0, 5] R$ be continuous, $f(1) = 3$ and $F$ be defined as:
$F(x) =\int _{1}^{x} t^{2}g(t)dt,$ where $g(t) =\int _{1}^{t} f(u)du.$
Then for the function $F(\vec{x}),$ the point $x= 1$ is:
a.
a point of local minima
b.
not a critical point
c.
a point of local maxima
d.
a point of inflection
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Let $a_{n}$ be the $n^{th}$ term of a G.P. of positive terms. If $\sum ^{100}_{n=1} a_{2n+1} = 200$ and $\sum ^{100}_{n=1} a_{2n} = 100,$ then $\sum ^{200}_{n=1} a_{n}$
is equal to:
a.
$300$
b.
$225$
c.
$175$
d.
$150$
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If $10$ different balls are to be placed in $4$ distinct boxes at random, then the probability that two of these boxes
contain exactly $2$ and $3$ balls is:
a.
$\frac{965}{2^{11}}$
b.
$\frac{965}{2^{10}}$
c.
$\frac{945}{2^{10}}$
d.
$\frac{945}{2^{11}}$
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