Three point particles of masses $1.0\,kg, \,1.5\, kg$ and $2.5\,kg$ are placed at three corners of a right angle triangle of
sides $4.0\,cm, \,3.0\,cm$ and $5.0\,cm$ as shown in the figure. The centre of mass of the system is at a point:
a.
$0.6\,cm$ right and $2.0\,cm$ above $1\,kg$ mass.
b.
$1.5\,cm$ right and $1.2\,cm$ above $1\,kg$ mass.
c.
$2.0\,cm$ right and $0.9\,cm$ above $1\,kg$ mass.
d.
$0.9\,cm$ right and $2.0\,cm$ above $1\,kg$ mass.
Correct Answer
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Q2mcq
The time period of revolution of electron in its ground state orbit in a hydrogen atom is $1.6 \times 10^{-16}s.$ The
frequency of revolution of the electron in its first excited state (in $s^{-1}$ ) is:
a.
$1.6 \times 10^{14}$
b.
$7.8 \times 10^{14}$
Correct Answer
c.
$6.2 \times 10^{15}$
d.
$5.6 \times 10^{12}$
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Q3mcq
If we need a magnification of $375$ from a compound microscope of tube length $150mm$ and an objective of
focal length $5mm,$ the focal length of the eye-piece, should be close to:
a.
$22mm$
Correct Answer
b.
$2mm$
c.
$4mm$
d.
$33mm$
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Q4mcq
As shown in the figure, a bob of mass $m$ is tied to a massless string whose other end portion is wound on a fly
wheel (disc) of radius $r$ and mass $m.$ When released from rest the bob starts falling vertically. When it has
covered a distance of $h,$ the angular speed of the wheel will be:
A parallel plate capacitor has plates of area A separated by distance $d$ between them. It is filled with a
dielectric which has a dielectric constant that varies as $K(x) =K_{0}(1 +\alpha x)$ where $x$ is the distance measured
from one of the plates. If $(\alpha d) << 1,$ the total capacitance of the system is best given by the expression:
Two infinite planes each with uniform surface charge density $+\sigma$ are kept in such a way that the angle between
them is $30^{o}$ . The electric field in the region shown between them is given by:
If the magnetic field in a plane electromagnetic wave is given by
$\vec{B}= 3 \times 10^{-8} sin(1.6 \times 10^{3}x+ 48 \times 10^{10}t)ˆjT,$ then what will be expression for electric field ?
Consider a circular coil of wire carrying constant current $I,$ forming a magnetic dipole. The magnetic flux
through an infinite plane that contains the circular coil and excluding the circular coil area is given by $\phi _{i}$ The
magnetic flux through the area of the circular coil area is given by $\phi _{0}$ . Which of the following option is
correct?
a.
$\phi _{i} =\phi _{0}$
b.
$\phi _{i} >\phi _{0}$
c.
$\phi _{i} <\phi _{0}$
d.
$\phi _{i} = -\phi _{0}$
Correct Answer
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Q9mcq
Two moles of an ideal gas, with $\frac{C_{P}}{C_{V}} = \frac{5}{3}$ , are mixed with three moles of another ideal gas $\frac{C_{P}}{C_{V}} = \frac{4}{3}$ . The value
of $\frac{C_{P}}{C_{V}}$ for the mixture is
a.
$1.45$
b.
$1.50$
c.
$1.47$
d.
$1.42$
Correct Answer
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Q10mcq
A long solenoid of radius $R$ carries a time $(t)$ dependent current $I(t) =I_{0}t(1 - t)$ . A ring of radius $2R$ is
placed coaxially near its middle. During the time interval $0 \leq t\leq 1,$ the induced current $(I_{R})$ and the induced
$EMF(V_{R})$ in the ring change as:
a.
Direction of $I_{R}$ remains unchanged and $V_{R}$ is
maximum at $t= 0.5$
b.
At $t= 0.25$ direction of $I_{R}$ reverses and $V_{R}$ is
maximum
c.
Direction of $I_{R}$ remains unchanged and $V_{R}$ is
zero at $t= 0.25$
d.
At $t= 0.5$ direction of $I_{R}$ reverses and $V_{R}$ is
zero
Correct Answer
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Visible light of wavelength $6000 \times 10^{-8}cm$ falls normally on a single slit and produces a diffraction pattern. It
is found that the second diffraction minimum is at $60^{o}$ from the central maximum. If the first minimum is
produced at $\theta _{1},$ then $\theta _{1}$ is close to
a.
$20^{o}$
b.
$30^{o}$
c.
$25^{o}$
d.
$45^{o}$
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A satellite of mass $M$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After
it reaches height $R$ ( $R=$ radius of the earth), it ejects a rocket of mass $\frac{M}{10}$ so that subsequently the satellite
moves in a circular orbit. The kinetic energy of the rocket is ( $G$ is the gravitational constant; $M_{e}$ is the mass of
the earth):
A non-isotropic solid metal cube has coefficients of linear expansion as: $5 \times 10^{-5}/^{o}C$ along the x-axis and
$5 \times 10^{-6}/^{o}C$ along the y and the z-axis. If the coefficient of volume expansion of the solid is $C \times 10^{-6}/^{o}C$
then the value of $C$ is _____________
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A litre of dry air at STP expands adiabatically to a volume of $3$ litres. If $\gamma = 1.40,$ the work done by air is:
$(3^{1.4} = 4.6555)$ [Take air to be an ideal gas]
a.
$60.7J$
b.
$90.5J$
c.
$100.8J$
d.
$48J$
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A loop $ABCDEFA$ of straight edges has six corner points
$A(0, 0, 0),B(5, 0, 0),C(5, 5, 0),D(0, 5, 0),E(0, 5, 5)$ and $F(0, 0, 5)$ . The magnetic field in this region is
$\vec{B}=(3ˆi+ 4ˆk)T$ . The quantity of flux through the loop $ABCDEFA$ (in $Wb$ ) is _____________
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Speed of a transverse wave on a straight wire (mass $6.0\,g,$ length $60\,cm$ and area of cross-section $1.0\,mm^{2}$ is
$90\,m\,s^{-1}$ . If the Young's modulus of wire is $16 \times 10^{11}\,N\,m^{-2}$ , the extension of wire over its natural length is:
a.
$0.03\,mm$
b.
$0.02\,mm$
c.
$0.04\,mm$
d.
$0.01\,mm$
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A Carnot engine operates between two reservoirs of temperatures $900K$ and $300K$ . The engine performs
$1200J$ of work per cycle. The heat energy (in $J$ ) delivered by the engine to the low temperature reservoir, in a
cycle, is _____________
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A $60HP$ electric motor lifts an elevator having a maximum total load capacity of $2000\,kg.$ If the frictional force
on the elevator is $4000\,N,$ the speed of the elevator at full load is close to : $(1\,HP = 746\,W, \,g= 10\,m\,s^{-2})$
a.
$1.7\,m\,s^{-1}$
b.
$1.9\,m\,s^{-1}$
c.
$1.5\,m\,s^{-1}$
d.
$2.0\,m\,s^{-1}$
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A polarizer - analyser set is adjusted such that the intensity of light coming out of the analyser is just $36%$ of
the original intensity. Assuming that the polarizer - analyser set does not absorb any light, the angle by which
the analyser needs to be rotated further, to reduce the output intensity to zero, is $(sin^{-1}(\frac{3}{5} )= 37^{\circ})$
a.
$53^{o}$
b.
$37^{o}$
c.
$90^{o}$
d.
$45^{o}$
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A $LCR$ circuit behaves like a clamped harmonic oscillator. Comparing it with a physical spring-mass damped
oscillator having damping constant $'b',$ the correct equivalence would be:
A particle $(m = 1kg)$ slides down a frictionless track $(AOC)$ starting from rest at a point $A$ (height $2m$ ). After
reaching $C,$ the particle continues to move freely in air as a projectile. When it reaching its highest point $P$
(height $1m$ ), the kinetic energy of the particle (in $J$ ) is: (Figure drawn is schematic and not to scale; take
$g= 10ms^{-2}$ ) ______________.
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The radius of gyration of a uniform rod of length $l,$ about an axis passing through a point $\frac{l}{4}$ away from the
centre of the rod, and perpendicular to it, is:
a.
$\frac{1}{4}$
b.
$\frac{1}{8}$
c.
$\frac{l}{7}$
$\sqrt{\,} _{48} l$
d.
$\frac{l}{3}$
$\sqrt{\,} _{8} l$
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A beam of electromagnetic radiation of intensity $6.4 \times 10^{-5}W/cm^{2}$ is comprised of wavelength, $\lambda = 310nm$ .
It falls normally on a metal (work function $\phi = 2eV$ ) of surface area of $1cm^{2}$ . If one in $10^{3}$ photons ejects an
election, total number of electrons ejected in $1s$ is $10^{x}$ . $(hc= 1240eVnm, 1eV = 1.6 \times 10^{-19}J),$ then $x$ is
___________
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Consider the following reactions:
$-(a) (CH_{3})_{3}CCH(OH)CH_{3}^{conc.H_{2}SO}\rightarrow \rightarrow^{4}$
$-(b) (CH_{3})_{2}CHCH(Br)CH_{3}^{alc.KOH}$
$-(c) (CH_{3})_{2}CHCH(Br)CH_{3}^{(CH_{3})_{3}\vec{O}^{\ominus }K}\rightarrow \rightarrow^{\oplus }$
$(d)$
Which of the reaction(s) will not produce Saytzeff product?
a.
$(a), (c)$ and $(d)$
b.
$(d)$ only
c.
$(c)$ only
d.
$(b)$ and $(d)$
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Two solutions, $A$ and $B,$ each of $100L$ was made by dissolving $4g$ of $NaOH$ and $9.8g$ of $H_{2}SO_{4}$ in water,
respectively. The $pH$ of the resultant solution obtained from mixing $40L$ of solution $A$ and $10L$ of solution $B$ is
____________. $(log \,2 = 0. 3)$
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During the nuclear explosion, one of the products is $^{90}Sr$ with half life of $6.93$ years. If $1$ $\mu$ $g$ of $^{90}Sr$ was
absorbed in the bones of a newly born baby in place of $Ca,$ how much time, in years, is required to reduce it
by $90%$ if it is not lost metabolically ____________.
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At $35\,^{o}C,$ the vapour pressure of $CS_{2},$ is $512mm$ Hg and that of acetone is $144\,mmHg.$ A solution of $CS_{2}$ in
acetone has a total vapour pressure of $600\,mmHg$ . The false statement amongst the following is:
a.
Raoult's law is not obeyed by this system
b.
a mixture of $100mLCS_{2}$ and $100mL$ acetone has
a volume $< 200mL$
c.
$CS_{2}$ and acetone are less attracted to each other
than to themselves
d.
heat must be absorbed in order to produce the
solution at $35^{o}C$
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Chlorine reacts with hot and concentrated $NaOH$ and produces compounds $(X)$ and $(Y)$ . Compound $(X)$ gives
white precipitate with silver nitrate solution. The average bond order between $Cl$ and $O$ atoms in $(Y)$ is
_______________.
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Given that the standard potentials $(E^{o})$ of $Cu^{2+} / Cu$ and $Cu^{+} / Cu$ are $0.34V$ and $0.522V$ respectively, the $E^{o}$
of $Cu^{2+} / Cu^{+}$ is:
a.
$0.182V$
b.
$+0.158V$
c.
$-0.182V$
d.
$-0.158V$
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For the reaction;
$A(l) \rightarrow 2B(g)\rightarrow$
$\Delta U = 2.1kcal, \Delta S = 20calK^{-1}$ at $300K.$
Hence $\Delta G$ in $kcal$ is _____________.
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A solution of $m-$ chloroaniline, $m-$ chlorophenol and $m-$ chlorobenzoic acid in ethyl acetate was extracted
initially with a saturated solution of $NaHCO_{3}$ to give fraction $A$ . The left over organic phase was extracted
with dilute $NaOH$ solution to give fraction $B$ . The final organic layer was labelled as fraction $C$ . Fractions
$A, B$ and $C,$ contain respectively:
a.
$m-$ chlorobenzoic acid, $m-$ chloroaniline and
$m-$ chlorophenol
b.
$m-$ chlorobenzoic acid, $m-$ chlorophenol and
$m-$ chloroaniline
c.
$m-$ chlorophenol, $m-$ chlorobenzoic acid and
$m-$ chloroanilinc
d.
$m-$ chloroaniline, $m-$ chlorobenzoic acid and
$m-$ chlorophenol
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Let the function $,f : [-7, 0] \rightarrow \rightarrow R$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0).$ If $f(-7) = -3$ and
$f'(x) \leq 2$ for all $x\in (-7, 0),$ then for all such functions $f, \,f(-1) +f(0)$ lies in the interval
a.
$(-\infty , 20]$
b.
$[-3, 11]$
c.
$(-\infty , 11]$
d.
$[-6, 20]$
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The area of the region (in sq. units), enclosed by the circle $x^{2} +y^{2} = 2$ which is not common to the region
bounded by the parabola $y^{2} =x$ and the straight line $y=x$ , is
a.
$\frac{1}{6}(24\pi -1)$
b.
$\frac{1}{3}(6\pi -1)$
c.
$\frac{1}{3}(12\pi -1)$
d.
$\frac{1}{6}(12\pi -1)$
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An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive
heads are obtained for $k= 3, 4, 5,$ otherwise $X$ takes the value $-1.$ Then the expected value of $X,$ is
a.
$\frac{3}{16}$
b.
$\frac{1}{8}$
c.
$-\frac{3}{16}$
d.
$- \frac{1}{8}$
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If $y(\alpha ) =\sqrt{2}(\frac{tan\alpha +cot\alpha}{1+tan^{2}\alpha})+ \frac{1}{sin^{2}\alpha} , \,\alpha \in (\frac{3\pi}{4},\pi )$ , then $\frac{dy}{d\alpha}$ at $\alpha = \frac{5\pi}{6}$ is
a.
$4$
b.
$\frac{4}{3}$
c.
$-4$
d.
$- \frac{1}{4}$
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If $f(a+b+ 1 - x) =f(x),$ for all $x,$ where $a$ and $b$ are fixed positive real numbers, then
$b$
$\frac{1}{a+b} \int x(f(x) +f(x+ 1))dx$ is equal to
$a$
a.
$b-1$
$\int f(x+ 1)dx$
$a-1$
b.
$b-1$
$\int f(x)dx$
$a-1$
c.
$b+1$
$\int f(x)dx$
$a+1$
d.
$b+1$
$\int f(x+ 1)dx$
$a+1$
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A vector $\vec{a}=\alpha ˆi+ 2ˆj+\beta ˆk(\alpha ,\beta \in R)$ lies in the plane of the vectors, $\vec{b}= ˆi+ ˆj$ and $\vec{c}= ˆi-ˆj+ 4ˆk.$ If $\vec{a}$
bisects the angle between $\vec{b}$ and $\vec{c},$ then
a.
$\vec{a}\cdot ˆi+ 3 = 0$
b.
$\vec{a}\cdot ˆi+ 1 = 0$
c.
$\vec{a}\cdot ˆk+ 2 = 0$
d.
$\vec{a}\cdot ˆk+ 4 = 0$
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Let $A(1, 0),B(6, 2)$ and $C(\frac{3}{2}, 6)$ be the vertices of a triangle $ABC.$ If $P$ is a point inside the triangle $ABC$
such that the triangles $APC,APB$ and $BPC$ have equal areas, then the length of the line segment $PQ,$ where
$Q$ is the point $(-\frac{7}{6}, - \frac{1}{3} ),$ is
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If the system of linear equations
$2x+ 2ay+az= 0$
$2x+ 3by+bz= 0$
$2x+ 4cy+cz= 0,$
where $a,b,c\in R$ are non-zero and distinct; has a non-zero solution, then
a.
$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in $A.P.$
b.
$a,b,c$ are in $G.P.$
c.
$a+b+c= 0$
d.
$a,b,c$ are in $A.P.$
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$1 1 1$
$[ ]$
Let $\alpha$ be a root of the equation $x^{2} +x+ 1 = 0$ and the matrix $A= \frac{1}{\sqrt{3}} 1 \alpha \alpha ^{2} ,$ then the matrix $A^{31}$ is
$[_{2}_{4}]$
$1 \alpha \alpha$
equal to
a.
$A^{3}$
b.
$I_{3}$
c.
$A^{2}$
d.
$A$
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Let $P$ be a plane passing through the points $(2, 1, 0), (4, 1, 1)$ and $(5, 0, 1)$ and R be any point $(2, 1, 6)$ .Then
the image of $R$ in the plane $P$ is
a.
$(6, 5, 2)$
b.
$(6, 5, -2)$
c.
$(4, 3, 2)$
d.
$(3, 4, -2)$
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Five numbers are in $A. P. ,$ whose sum is $25$ and product is $2520.$ If one of these five numbers is $-\frac{1}{2} ,$ then the
greatest number amongst them is
a.
$27$
b.
$7$
c.
$\frac{21}{2}$
d.
$16$
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Let $\alpha$ and $\beta$ be two real roots of the equation $(k+ 1)tan^{2}x- \sqrt{2}\cdot \lambda tanx= (1 - k),$ where $k(\neq -1)$ and $\lambda$
are real numbers. If $tan^{2}(\alpha +\beta ) = 50,$ then a value of $\lambda$ is
a.
$10\sqrt{2}$
b.
$10$
c.
$5$
d.
$5\sqrt{2}$
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If the variance of the first $n$ natural numbers is $10$ and the variance of the first $m$ even natural numbers is $16,$
then the value of $m+n$ is equal to
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If the sum of the coefficients of all even powers of $x$ in the product
$(1 +x+x^{2} + … +x^{2n})(1 - x+x^{2} - x^{3} + … +x^{2n})$ is $61,$ then n is equal to
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