A string $2.0\,m$ long and fixed at its ends is driven by a $240\,Hz$ vibrator. The string vibrates in its third
harmonic mode. The speed of the wave and its fundamental frequency is
a.
$320 \,m s^{-1}, 120\, Hz$
b.
$320 \,m s^{-1}, 80\, Hz$
Correct Answer
c.
$180 \,m s^{-1}, 80\, Hz$
d.
$180 \,m s^{-1}, 120\, Hz$
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Q2mcq
A test particle is moving in a circular orbit in the gravitational field produced by a mass density $\rho (r) = \frac{K}{r^{2}}.$
Identify the current relation between the radius $R$ of the particle’s orbit and its period $T :$
a.
$TR$ is a constant
b.
$T^{2}/_{R^{3}}$ is a constant
c.
$^{T}/_{R}$ is a constant
Correct Answer
d.
$^{T}/_{R^{2}}$ is a constant
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Q3mcq
A wedge of mass $M = 4m$ lies on a frictionless plane. A particle of mass $m$ approaches the wedge with speed
$v.$ There is no friction between the particle and the plane or between the particle and the wedge. The maximum
height climbed by the particle on the wedge is given by:
a.
$\frac{v^{2}}{g}$
b.
$\frac{v^{2}}{2g}$
c.
$\frac{2v^{2}}{5g}$
Correct Answer
d.
$\frac{2v^{2}}{7g}$
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Q4mcq
A metal wire of resistance $3 \Omega$ is elongated to make a uniform wire of double its previous length. This new
wire is now bent and the ends joined to make a circle. If two points on this circle make an angle $60^{o}$ at the
center, the equivalent resistance between these two points will be:
a.
$\frac{5}{3} \Omega$
Correct Answer
b.
$\frac{12}{5} \Omega$
c.
$\frac{7}{2} \Omega$
d.
$\frac{5}{2} \Omega$
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Q5mcq
The resistance of a galvanometer is $50\,ohm$ and the maximum current which can be passed through it is
$0.002\,A.$ What resistance must be connected to it in order to convert it into an ammeter of range $0 -0.5\,A?$
a.
$0.2\,ohm$
Correct Answer
b.
$0.5\,ohm$
c.
$0.002\,ohm$
d.
$0.02\,ohm$
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Q6mcq
A thin smooth rod of length $L$ and mass $M$ is rotating freely with angular speed $\omega$ $_{0}$ about an axis perpendicular
to the rod and passing through center. Two beads of mass $m$ and negligible size are at the center of the rod
initially. The beads of mass $m$ and negligible size are at the center of the rod initially. The beads are free to slide
along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be:
a.
$\frac{M$\omega$_{0}}{M+3m}$
b.
$\frac{M$\omega$_{0}}{M+2m}$
c.
$\frac{M$\omega$_{0}}{M+6m}$
Correct Answer
d.
$\frac{M$\omega$_{0}}{M+m}$
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Q7mcq
The parallel combination of two air filled parallel plate capacitors of capacitance $C$ and $nC$ is connected to a
battery of voltage, $V.$ When the capacitors are fully charged, the battery is removed and after that a dielectric
material of dielectric constant $K$ is placed the two plates of the first capacitor. The new potential difference of
the combined system is:
a.
$\frac{(n+1)V}{(K+n)}$
Correct Answer
b.
$\frac{nV}{K+n}$
c.
$\frac{V}{K+n}$
d.
$V$
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Q8mcq
The specific heats, $C_{p}$ and $C_{v}$ of a gas of diatomic molecules, $A,$ are given (in units of $J\,mol^{-1}\,K^{-1}$ ) by $29$
and $22,$ respectively. Another gas of diatomic molecules, $B,$ has the corresponding values $30$ and $21.$ If they are
treated as ideal gases, then:
a.
$A$ has one vibrational mode and $B$ has two
b.
$A$ has a vibrational mode but $B$ has none.
Correct Answer
c.
Both $A$ and $B$ have a vibrational mode each.
d.
$A$ is rigid but $B$ has a vibrational mode.
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Q9mcq
A particle of mass $m$ is moving with speed $2v$ and collides with a mass $2m$ moving with speed $v$ in the same
direction. After the collision, the first mass is stopped completely while the second one splits into two particles
each of mass $m,$ which move at an angle $45^{o}$ with respect to the original direction. The speed of each of the
moving particle will be
a.
$\sqrt{2}v$
b.
$\frac{v}{\sqrt{2}}$
c.
$2\sqrt{2}v$
Correct Answer
d.
$\frac{v}{(2\sqrt{2})}$
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Q10mcq
$||$
A particle $'P'$ is formed due to a completely inelastic collision of particles $'x'$ and $'y'$ having de-Broglie
wavelengths $'$ $\lambda$ $_{x}'$ and $'$ $\lambda$ $_{y}'$ respectively. If $x$ and $y$ were moving in opposite directions, then the de-Broglie
wavelength of $'P'$ is:
A moving coil galvanometer has a coil with $175$ turns and area $1\,cm^{2}.$ It uses a torsion band of torsion constant
$10^{-6}\,N\,m/rad$ The coil is placed in a magnetic field B parallel to its plane. The coil deflects by $1^{o}$ for a
current of $1\,mA$ The value of B (in tesla) is approximately:
a.
$10^{-2}$
b.
$10^{-4}$
c.
$10^{-3}$
d.
$10^{-1}$
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In a conductor, if the number of conduction electrons per unit volume is $8.5 \times 10^{28} m^{-3}$ and mean free time is
$25fs$ (femto second), it’s approximate resistivity is: $(m_{e} = 9.1 \times 10^{-31} kg)$
a.
$10^{-7} \Omega \,m$
b.
$10^{-6} \Omega \,m$
c.
$10^{-5} \Omega \,m$
d.
$10^{-8} \Omega \,m$
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The position of a particle as a function of time $t,$ is given by $x(t) =at+bt^{2} - ct^{3}$ where $a, \,b$ and $c$ are
constants. When the particles zero acceleration, then its velocity will be:
a.
$a+ \frac{b^{2}}{3c}$
b.
$a+ \frac{b^{2}}{2c}$
c.
$a+ \frac{b^{2}}{c}$
d.
$a+ \frac{b^{2}}{4c}$
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Two coils $'P'$ and $'Q'$ are separated by some distance. When a current of $3\,A$ flows through coil $'P',$ a
magnetic flux of $10^{-3}\,Wb$ passes through $'Q'.$ No current is passed through $'Q'.$ When no current passes
through $'P'$ and a current of $2\,A$ passes through $'Q',$ the flux through $'P'$ is:
a.
$3.67 \times 10^{-3} Wb$
b.
$6.67 \times 10^{-4} Wb$
c.
$3.67 \times 10^{-4} Wb$
d.
$6.67 \times 10^{-3} Wb$
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Two cars A and B are moving away from each other in opposite directions. Both the cars are moving with
speed of $20\,m\,s^{-1}$ with respect to the ground. If an observer in car A detects a frequency $2000\,Hz$ of the sound
coming from car B, what is the natural frequency of the sound source in car B?
(speed of sound in air $= 340\,m\,s^{-1}$ )
a.
$2150\,Hz$
b.
$2300\,Hz$
c.
$2250\,Hz$
d.
$2060\,Hz$
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A very long solenoid of radius $R$ is carrying current $I(t) = kte^{-$\alpha$t}(k > 0),$ as a function of time $(t \geq 0).$
Counterclockwise current is taken to be positive. A circular conducting coil of radius $2R$ is placed in the
equitorial plane of the solenoid and concentric with the solenoid. The current induced in the outer coil is
correctly depicted, as a function of time, by:
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Two materials having coefficients of thermal conductivity $3K$ and $K$ and thickness $d$ and $3d$ respectively, are
joined to form a slab as shown in the figure. The temperatures of the outer surfaces are $\theta$ $_{2}$ and $\theta$ $_{1}$ respectively,
$($ $\theta$ $_{2} >$ $\theta$ $_{1}).$ The temperature at the interface is
The position vector of a particle changes with time according to the relation $\vec{r}(t) = 15t^{2}ˆi+(4 -20t^{2})ˆj.$ What
is the magnitude of the acceleration at $t= 1?$
a.
$40$
b.
$25$
c.
$100$
d.
$50$
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Moment of inertia of a body about a given axis is $1.5\,kg\,m^{2}.$ Initially the body is at rest. In order to produce a
rotational kinetic energy of $1200\,J,$ the angular acceleration of $20\,rad/s^{2}$ must be applied about the axis for a
duration of:
a.
$3\,s$
b.
$2\,s$
c.
$2.5\,s$
d.
$5\,s$
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Sunlight of intensity $50\,W\,m^{-2}$ is incident normally on the surface of a solar panel. Some part of incident
energy ( $25%$ ) is reflected from the surface and the rest is absorbed. The force exerted on $1\,m^{2}$ surface area will
be close to $(c= 3 \times 10^{8}\,m\,s^{-1})$
a.
$15 \times 10^{-8} N$
b.
$20 \times 10^{-8} N$
c.
$10 \times 10^{-8} N$
d.
$35 \times 10^{-8} N$
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Diameter of the objective lens of a telescope is $250\,cm.$ For light of wavelength $600\,nm$ coming from a distant
object, the limit of resolution of the telescope is close to:
a.
$1.5 \times 10^{-7} rad$
b.
$3.0 \times 10^{-7} rad$
c.
$2.0 \times 10^{-7} rad$
d.
$4.5 \times 10^{-7} rad$
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A convex lens of focal length $20\,cm$ produces images of the same magnification 2 when an object is kept at
two distances $x_{1}$ and $x_{2}(x_{1} >x_{2})$ from the lens. The ratio of $x_{1}$ and $x_{2}$ is:
a.
$2 : 1$
b.
$5 : 3$
c.
$4 : 3$
d.
$3 : 1$
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A wooden block floating in a bucket of water has $\frac{4}{5}$ of its volume submerged. When certain amount of an oil is
poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water
and half in oil. The density of oil relative to that of water is:
a.
$0.8$
b.
$0.7$
c.
$0.5$
d.
$0.6$
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Four point charges $-q, \, + q, \, + q$ and $-q$ are placed on y-axis at $y = -2d, \,y = -d,$ and $y = +2d,$
respectively. The magnitude of the electric field $E$ at a point on the x-axis at $x = D,$ with $D \gg d,$ will behave
as:
a.
$E \propto \frac{1}{D^{4}}$
b.
$E \propto \frac{1}{D}$
c.
$E \propto \frac{1}{D^{3}}$
d.
$E \propto \frac{1}{D^{2}}$
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A thin convex lens $L$ (refractive index $= 1.5$ ) is placed on a plane mirror $M$ . When a pin is placed at $A$ , such
that $OA = 18\,cm,$ its real inverted image is formed at $A$ itself, as shown in figure. When liquid of refractive
index $\mu _{l}$ is put between the lens and the mirror, the pin has to be moved to $A',$ such that $OA' = 27 cm,$ to get
its inverted real image at $A'$ itself. The value of $\mu _{l}$ will be
a.
$\frac{4}{3}$
b.
$\sqrt{3}$
c.
$\frac{3}{2}$
d.
$\sqrt{2}$
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A massless spring $(k= 800 N/m),$ attached with a mass $(500 g)$ is completely immersed in $1\,kg$ of water. The
spring is stretched by $2\,cm$ and released so that it starts vibrating. What would be the order of magnitude of the
change in the temperature of water when the vibrations stop completely? (Assume that the water container and
spring receive negligible heat and specific heat of mass $= 400\,J/kg\,K,$ specific heat of water $= 4184 J/kg K$ )
a.
$10^{-5}\,K$
b.
$10^{-1}\,K$
c.
$10^{-3}\,K$
d.
$10^{-4}\,K$
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At a given temperature $T,$ gases $Ne, \,Ar, \,Xe$ and $Kr$ are found to deviate from ideal gas behaviour. Their
equation of state is given as $p = \frac{RT}{V-b}$ at $T.$ Here, $b$ is the van der Waals constant. Which gas will exhibit
steepest increase in the plot of $Z$ (compression factor) vs $p?$
a.
$Ne$
b.
$Kr$
c.
$Ar$
d.
$Xe$
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The correct statements among I to III are:
(I) Valence bond theory cannot explain the color exhibited by transition metal complexes.
(II) Valence bond theory can predict quantitatively the magnetic properties of transition metal complexes.
(III) Valence bond theory cannot distinguish ligands as weak and strong field ones.
a.
(I) and (II) only
b.
(II) and (III) only
c.
(I) and (III) only
d.
(I), (II) and (III)
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$10\,mL$ of $1\,mM$ surfactant solution forms a monolayer covering $0.24\,cm^{2}$ on a polar substrate. If the polar
head is approximated as a cube, what is its edge length?
a.
$1.0\,pm$
b.
$0.1\,nm$
c.
$2.0\,pm$
d.
$2.0\,nm$
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Consider the given plot of enthalpy of the following reaction between $A$ and $B.$
$A + B \rightarrow C + D.\rightarrow$ Identify the incorrect statement.
a.
Formation of $A$ and $B$ from $C$ has highest enthalpy of activation.
b.
$D$ is kinetically stable product.
c.
$C$ is the thermodynamically stable product.
d.
Activation enthalpy to form $C$ is $5\,kJ\,mol^{-1}$ less than that to form $D.$
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A solution of $Ni(NO_{3})_{2}$ is electrolyzed between platinum electrode $0.1$ Faraday electricity. How many mole of
$Ni$ will be deposited at the cathode?
a.
$0.10$
b.
$0.15$
c.
$0.05$
d.
$0.20$
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The correct statements among I to III regrading group 13 element oxides are,
(I) Boron trioxide is acidic.
(II) Oxides of aluminum and gallium are amphoteric.
(III) Oxides of indium and thallium are basic.
a.
(I) and (III) only
b.
(I) and (II) only
c.
(I), (II) and (III)
d.
(II) and (III) only
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During compression of a spring the work done is $10\,kJ$ and $2\,kJ$ escaped to the surroundings as heat. The
change in internal energy, $\Delta U(in kJ)$ is:
a.
$-8$
b.
$8$
c.
$12$
d.
$-12$
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Molal depression constant for a solvent is $4.0\,K\,kg\,mol^{-1}.$ The depression in the freezing point of the solvent
for $0.03 mol kg^{-1}$ solution of $K_{2}SO_{4}$ is:
(Assume complete dissociation of the electrolyte)
a.
$0.36\,K$
b.
$0.24\,K$
c.
$0.18\,K$
d.
$0.12\,K$
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In an acid-base titration, $0.1\,M\,HCl$ solution was added to the $NaOH$ solution of unknown strength. Which of
the following correctly shown the change of $pH$ of the titration mixture in this experiment?
a.
$A$
b.
$B$
c.
$D$
d.
$C$
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If $m$ is chosen in the quadratic equation $(m^{2} + 1)x^{2} -3x+(m^{2} + 1)^{2} = 0$ such that the sum of its roots is
greatest, then the absolute difference of the cubes of its roots is:
a.
$4\sqrt{3}$
b.
$10\sqrt{5}$
c.
$8\sqrt{3}$
d.
$8\sqrt{5}$
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Two poles standing on a horizontal ground are of heights $5\,m$ and $10\,m$ respectively. The line joining their
tops makes an angle of $15^{\circ}$ with the ground. Then the distance $($ in $m)$ between the poles, is
a.
$10(\sqrt{3}-1)$
b.
$\frac{5}{2}(2 + \sqrt{3})$
c.
$5(2 + \sqrt{3})$
d.
$5(\sqrt{3}+ 1)$
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The vertices $B$ and $C$ of a $\Delta ABC$ lie on the line, $\frac{x+2}{3} = \frac{y-1}{0} = \frac{z}{4}$ such that $BC = 5$ units. Then the area (in
sq. units) of this triangle, given the point $A(1, -1, 2),$ is
a.
$6$
b.
$2\sqrt{34}$
c.
$\sqrt{34}$
d.
$5\sqrt{17}$
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If some three consecutive coefficients in the binomial expansion of $(x+ 1)^{n}$ in powers of $x$ are in the ratio
$2 : 15 : 70,$ then the average of these three coefficients is:
a.
$227$
b.
$964$
c.
$625$
d.
$232$
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The mean and the median of the following ten numbers in increasing order
$10, \,22, \,26, \,29, \,34, \,x, \,42, \,67, \,70, \,y$ are $42$ and $35$ respectively, then $\frac{y}{x}$ is equal to:
a.
$\frac{9}{4}$
b.
$\frac{7}{3}$
c.
$\frac{7}{2}$
d.
$\frac{8}{3}$
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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the
second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls
used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side
contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of
balls used to form the equilateral triangle is
a.
$262$
b.
$190$
c.
$225$
d.
$157$
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If the system of equations $2x+ 3y- z= 0, x+ky-2z= 0$ and $2x- y+z= 0$ has a non-trivial solution
$(x, \,y, \,z),$ then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} +k$ is equal to
a.
$-\frac{1}{4}$
b.
$\frac{1}{2}$
c.
$-4$
d.
$\frac{3}{4}$
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If the two lines $x+ (a-1)y= 1$ and $2x+a^{2}y= 1, (a\in R-{0,1})$ are perpendicular, then the distance of
their point of intersection from the origin is
a.
$\frac{2}{\sqrt{5}}$
b.
$\frac{\sqrt{2}}{5}$
c.
$\frac{2}{5}$
d.
$\sqrt{\frac{2}{5}}$
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Two newspapers $A$ and $B$ are published in a city. It is known that $25%$ of the city population reads $A$ and $20%$
reads $B$ while $8%$ reads both $A$ and $B.$ Further, $30%$ of those who read $A$ but not $B$ look into advertisements
and $40%$ of those who read $B$ but not $A$ also look into advertisements, while $50%$ of those who read both $A$
and $B$ look into advertisements. Then the percentage of the population who look into advertisements is:
a.
$13.5$
b.
$12.8$
c.
$13.9$
d.
$13$
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Let $P$ be the plane, which contains the line of intersection of the planes, $x+y+z-6 = 0$ and
$2x+ 3y+z+ 5 = 0$ and it is perpendicular to the $xy$ -plane. Then the distance of the point $(0, \,0, \,256)$ from $P$
is equal to:
a.
$205\sqrt{5}$ units
b.
$\frac{17}{\sqrt{5}}$ units
c.
$\frac{11}{\sqrt{5}}$ units
d.
$63\sqrt{5}$ units
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If $f :R\rightarrow \rightarrow R$ is a differentiable function and $f(2) = 6,$ then $_{x}lim_{\rightarrow 2\rightarrow } \int _{6}^{f(x)} \frac{2tdt}{(x-2)}$ is:
a.
$0$
b.
$2f'(2)$
c.
$24f'(2)$
d.
$12f'(2)$
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If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $ˆi, \, \frac{$\pi$}{4}$ with $ˆj$ and $\theta \in (0,\pi )$ with $ˆk,$ then a value of $\theta$ is:
a.
$\frac{5\pi}{6}$
b.
$\frac{5\pi}{12}$
c.
$\frac{\pi}{4}$
d.
$\frac{2\pi}{3}$
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If the tangent to the parabola $y^{2} =x$ at a point $(\alpha ,\beta ), (\beta > 0)$ is also a tangent to the ellipse, $x^{2} + 2y^{2} = 1$
then $\alpha$ is equal to:
a.
$\sqrt{2}-1$
b.
$2 + 1\sqrt{2}$
c.
$\sqrt{2}+ 1$
d.
$2 -1\sqrt{2}$
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A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $tan^{-1}(\frac{1}{2}).$ Water is
poured into it at a constant rate of $5\,cubic m/min.$ Then the rate $($ in $m/min),$ at which the level of water is
rising at the instant when the depth of water in the tank is $10\,m;$ is:
a.
$\frac{1}{10$\pi$}$
b.
$\frac{1}{15$\pi$}$
c.
$\frac{1}{5$\pi$}$
d.
$\frac{2}{$\pi$}$
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A rectangle is inscribed in a circle with a diameter lying along the line $3y=x+ 7.$ If the two adjacent vertices
of the rectangle are $(-8, \,5)$ and $(6, \,5),$ then the area of the rectangle $($ in sq. units $)$ is:
a.
$72$
b.
$98$
c.
$56$
d.
$84$
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