A travelling harmonic wave is represented by the equation $y(x, \,t) = 10^{-3} sin(50t+ 2x),$ where $x$ and $y$ are in
meter and $t$ is in seconds. Which of the following is a correct statement about the wave?
a.
The wave is propagating along the positive $x$ -
axis with speed $25\,m\,s^{-1}$
b.
The wave is propagating along the positive $x$ -
axis with speed $100\,m\,s^{-1}$
c.
The wave is propagating along the negative $x$ -
axis with speed $25\,m\,s^{-1}$
Correct Answer
d.
The wave is propagating along the negative $x$ -
axis with speed $100\,m\,s^{-1}$
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Q2mcq
The galvanometer deflection, when key $K_{1}$ is closed but $K_{2}$ is open, equals $\theta _{0}$ (see figure). On closing $K_{2}$
also and adjusting $R_{2}$ to $5\Omega ,$ the deflection in galvanometer becomes $\frac{\theta _{0}}{5}.$ The resistance of the galvanometer
is, then, given by [Neglect the internal resistance of battery]:
a.
$12\Omega$
b.
$22\Omega$
Correct Answer
c.
$5\Omega$
d.
$25\Omega$
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Q3mcq
Two electric bulbs, rated at $(25\,W, \,220\,V)\,$ and $(100\,W, \,220\,V),$ are connected in series across a $220\,V$
voltage source. If the $25\,W$ and $100\,W$ bulbs draw powers $P_{1}$ and $P_{2}$ respectively, then:
a.
$P_{1} = 4\,W, \,P_{2} = 16\,W$
b.
$P_{1} = 9\,W, \,P_{2} = 16\,W$
c.
$P_{1} = 16\,W, \,P_{2} = 9\,W$
d.
$P_{1} = 16\,W, \,P_{2} = 4\,W$
Correct Answer
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Q4mcq
A proton and an $\alpha$ - particle (with their masses in the ratio of $1 : 4$ and charges in the ratio of $1 : 2$ ) are
accelerated from rest through a potential difference $V.$ If a uniform magnetic field $(B)$ is set up perpendicular
to their velocities, the ratio of the radii $r_{p} :r_{\alpha }$ of the circular paths described by them will be:
a.
$1 : 3$
b.
$1 : \sqrt{2}$
Correct Answer
c.
$1 : 2$
d.
$1 : \sqrt{3}$
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Q5mcq
A satellite of mass $M$ is in a circular orbit of radius $R$ about the center of the earth. A meteorite of the same
mass, falling towards the earth, collides with the satellite completely inelastic. The speeds of the satellite and
the meteorite are the same, just before the collision. The subsequent motion of the combined body will be:
a.
In an elliptical orbit
Correct Answer
b.
Such that it escapes to infinity
c.
In a circular orbit of a different radius
d.
In the same circular orbit of radius $R$
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Q6mcq
The output of the given logic circuit is:
a.
$-$
$AB$
Correct Answer
b.
$-$
$A B$
$-$
c.
$AB+AB$
d.
$- -$
$AB + A B$
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Q7mcq
A $100V$ carrier wave is made to vary between $160\,V$ and $40\,V$ by a modulating signal. What is the modulation
index?
a.
$0.3$
b.
$0.6$
Correct Answer
c.
$0.4$
d.
$0.5$
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Q8mcq
The figure shows a capacitor of capacitance $C$ connected to a battery via a switch, having a total charge $Q$ on
it, in steady-state. When the switch $S$ is turned from position $A$ to position $B$ , the energy dissipated in the
circuit is
a.
$\frac{1}{8} \frac{Q^{2}}{C}$
b.
$\frac{3}{8} \frac{Q^{2}}{C}$
Correct Answer
c.
$\frac{3}{4} \frac{Q^{2}}{C}$
d.
$\frac{5}{8} \frac{Q^{2}}{C}$
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Q9mcq
A simple pendulum, made of a string of length $l$ and $a$ bob of mass $m,$ is released from a small angle $\theta _{0}.$ It
strikes a block of mass $M,$ kept on horizontal surface at its lowest point of oscillations, elastically. It bounces
back and goes up to an angle $\theta _{1}.$ Then $M$ is given by:
A straight rod of length $L$ extends from $x=a$ to $x=L+a.$ The gravitational force it exerts on a point mass
'm' at $x= 0,$ if the mass per unit length of the rod is $A+Bx^{2},$ is given by:
The least count of the main scale of a screw gauge is $1\, mm.$ The minimum number of divisions on its circular
scale required to measure $5\,$ $\mu$ $m$ diameter of a wire is:
a.
$50$
b.
$100$
c.
$500$
d.
$200$
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A cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ . The thermal
conductivity of the material of the inner cylinder is $K_{1}$ and that of the outer cylinder is $K_{2}$ . Assuming no loss of
heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:
a.
$\frac{2K_{1}+3K_{2}}{5}$
b.
$\frac{K_{1}+K_{2}}{2}$
c.
$K_{1}+K_{2}$
d.
$\frac{K+3K}{4}^{1}^{2}$
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In the figure shown, a circuit contains two identical resistors with resistance $R= 5\Omega$ and an inductance with
$L= 2mH.$ An ideal battery of $15V$ is connected in the circuit. What will be the current through the battery
long after the switch is closed?
a.
$5.5\,A$
b.
$6\,A$
c.
$3\,A$
d.
$7.5\,A$
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A particle $A$ of mass $m$ and charge $q$ is accelerated by a potential difference of $50\,V.$ Another particle $B$ of
mass $4m$ and charge $q$ is accelerated by a potential difference of $2500\,V.$ The ratio of de-Broglie wavelengths
$\frac{\lambda _{A}}{\lambda _{B}}$ is close to:
a.
$0.07$
b.
$10.00$
c.
$4.47$
d.
$14.14$
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A light wave is incident normally on a glass slab of refractive index $1.5.$ If $4%$ of light gets reflected and the
amplitude of the electric field of the incident light is $30\,\frac{V}{m} ,$ then the amplitude of the electric field for the wave
propagating in the glass medium will be:
a.
$30 \frac{V}{m}$
b.
$6\frac{V}{m}$
c.
$24 \frac{V}{m}$
d.
$10\frac{V}{m}$
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Two light identical springs of spring constant $k$ are attached horizontally at the two ends of a uniform
horizontal rod $AB$ of length $l$ and mass $m.$ The rod is pivoted at its center 'O' and can rotate freely in
horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is
gently pushed through a small angle and released. The frequency of resulting oscillation is:
a.
$\frac{1}{2\pi} \sqrt{\frac{3k}{m}}$
b.
$\frac{1}{2\pi} \sqrt{\frac{k}{m}}$
c.
$\frac{1}{2\pi} \sqrt{\frac{6k}{m}}$
d.
$\frac{1}{2\pi} \sqrt{\frac{2k}{m}}$
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Let the moment of inertia of a hollow cylinder of length $30\,cm$ (inner radius $10\,cm$ and outer radius $20\,cm$ ),
about its axis be $I$ . The radius of a thin cylinder of the same mass such that its moment of inertia about its axis
is also $I,$ is:
a.
$16\,cm$
b.
$14\,cm$
c.
$12\,cm$
d.
$18\,cm$
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As shown in the figure, two infinitely long, identical wires are bent by $90^{o}$ and placed in such a way that the
segments $LP$ and $QM$ are along the $x$ - axis, while segments $PS$ and $QN$ are parallel to the $y$ - axis. If
$OP =OQ= 4cm,$ and the magnitude of the magnetic field at $O$ is $10^{-4} T,$ and the two wires carry equal
currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at $O$ will
be $(\mu _{0} = 4\pi \times 10^{-7}NA^{-2}):$
a.
$20\,A,$ perpendicular into the page
b.
$20\,A,$ perpendicular out of the page
c.
$40\,A,$ perpendicualr into the page
d.
$40\,A,$ perpendicular out of the page
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A passenger train of length $60\,m$ travels at a speed of $80\, km / hr$ . Another freight train of length $120\,m$ travels
at a speed of $30\, km / hr$ . The ratio of times taken by the passenger train to completely cross the freight train
when: (i) they are moving in the same direction, and (ii) in the opposite directions is:
a.
$\frac{5}{2}$
b.
$\frac{3}{2}$
c.
$\frac{11}{5}$
d.
$\frac{25}{11}$
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An ideal battery of emf $4V$ and resistance $R$ are connected in series in the primary circuit of a potentiometer of
length $1\,m$ and resistance $5\Omega .$ The value of $R,$ to give a potential difference of $5\,mV$ across $10\,cm$ of
potentiometer wire, is:
a.
$480\Omega$
b.
$495\Omega$
c.
$395\Omega$
d.
$490\Omega$
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A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle $60$ with
ground level, but he finds the aeroplane right vertically above his position. If $v$ is the speed of sound, speed of
the plane is:
a.
$\frac{2v}{\sqrt{3}}$
b.
$v$
c.
$\frac{v}{2}$
d.
$\frac{\sqrt{3}}{2}v$
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A particle of mass $m$ moves in a circular orbit in a central potential field $U(r) = \frac{1}{2} kr^{2}.$ If Bohr's quantization
conditions are applied, radii of possible orbitals and energy levels vary with quantum number $n$ as:
There is a uniform spherically symmetric surface charge density at a distance $R_{0}$ from the origin. The charge
distribution is initially at rest and starts expanding because of mutual repulsion. The figure that represents best
the speed $V(R(t))$ of the distribution as a function of its instantaneous radius $R(t)$ is:
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In a meter bridge, the wire of length $1\,m$ has a non-uniform cross-section such that, the variation $\frac{dR}{dl}$ of its
resistance $R$ with length $l$ is $\frac{dR}{dl} \propto \frac{1}{\sqrt{l}}.$ Two equal resistances are connected as shown in the figure. The
galvanometer has zero deflection when the jockey is at point $P.$ What is the length $AP?$
a.
$0.2\,m$
b.
$0.35\,m$
c.
$0.3\,m$
d.
$0.25\,m$
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A point source of light, S is placed at a distance L in front of the center of plane mirror of width d which is
hanging vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror, at a distance
2L as shown below. The distance over which the man can see the image of the light source in the mirror is:
a.
$3d$
b.
$2d$
c.
$d$
d.
$\frac{d}{2}$
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$50 mL$ of $0 .5 M$ oxalic acid is needed to neutralize $25 mL$ of sodium hydroxide solution. What is the amount
of $NaOH$ in $50 mL$ of the given sodium hydroxide solution?
a.
$2 g$
b.
$4 g$
c.
$1 g$
d.
$8 g$
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Decomposition of $X$ exhibits a rate constant of $0 .05$ $\mu$ $g/ year$ . How many years are required for the
decomposition of $5$ $\mu$ $g$ of $X$ into $2 .5$ $\mu$ $g$ ?
a.
$20$
b.
4 $0$
c.
$25$
d.
$50$
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A metal on combustion in excess air forms $X$ . $X$ upon hydrolysis with water yields $H_{2}O_{2}$ and $O_{2}$ along with
another product. The metal is:
a.
$Na$
b.
$Mg$
c.
$Li$
d.
$Rb$
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In the following reaction
$^{HCl}Acetal$
$-Aldehyde + Alcohol \rightarrow \rightarrow$
$Aldehyde\, Alcohol$
$HCHO t -BuOH$
$CH_{3}CHO \, MeOH$
The best combination is:
a.
$HCHO$ and $MeOH$
b.
$CH_{3}CHO$ and $t -BuOH$
c.
$CH_{3}CHO$ and $MeOH$
d.
$HCHO$ and $t- BuOH$
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The standard electrode potential $E^{o}$ and its temperature coefficient $( \frac{dE}{dT})$ for a cell are $2 V$ and
$-5\times 10^{-4} \,V\,K^{-1}$ at $300 K$ , respectively. The reaction is $Zn \,(s) + Cu^{2+} \,(aq) \rightarrow Zn\rightarrow^{2+} \,(aq) + Cu \,(s)$ . The
standard reaction enthalpy $(\Delta _{r}H^{-})$ at $300 K$ in $mol^{-1}$ is
$[$ Use $R = 8\,J\,K^{-1}\, mol^{-1}$ and $F =96,500\, C mol^{-1}]$
a.
$-412.8$
b.
$206.4$
c.
$-384.0$
d.
$192.0$
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Two solids dissociate as follows:
$A\,(s) \rightleftharpoons B\,(g) + C\,(g); K_{P_{1}} = x atm^{2}$
$D\,(s) \rightleftharpoons C\,(g) + E\,(g); K_{P_{2}} = y atm^{2}$
The total pressure when both the solids dissociate simultaneously is:
a.
$\sqrt{x + y}\, atm$
b.
$x^{2} + y^{2}\, atm$
c.
$(x + y)\, atm$
d.
$2(\sqrt{x + y})\, atm$
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The freezing point of a $4%$ aqueous solution of $X$ is equal to the freezing point of a $12%$ aqueous solution of $Y$
. If the molecular weight of $X$ is $A$ , then the molecular weight of $Y$ will be
a.
$2A$
b.
$A$
c.
$4A$
d.
$3A$
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What is the hardness of a water sample $($ in terms of equivalents of $CaCO_{3})$ containing $10^{-3}M CaSO_{4}$ ?
$($ Molar mass of $CaSO_{4} = 136 g mol^{-1})$
a.
$90$ ppm
b.
$50$ ppm
c.
$10$ ppm
d.
$100$ ppm
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The volume of gas $A$ is twice than that of gas $B$ . The compressibility factor of gas $A$ is thrice than that of gas
$B$ at same temperature. What are the pressures of the gases for equal number of moles?
a.
$2P_{A} = 3P_{B}$
b.
$P_{A} = 2P_{B}$
c.
$3P_{A} = 2P_{B}$
d.
$P_{A} = 3P_{B}$
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For a diatomic ideal gas in a closed system, which of the following plots does not correctly describe the
relation between various thermodynamic quantities?
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$o$
What is the work function of the metal if the light of wavelength $4000\,A$ generates photoelectrons of velocity
$6 \times 10^{5} ms^{-1}$ from it?
(Mass of electron $\, = 9 \times 10^{-31}\, kg$ , velocity of light $= 3 \times 10^{8}\, ms^{-1}$ , Planck's constant $= 6 .626 \times 10^{-34}\, Js$ ,
Charge of electron $= 6 .626 \times 10^{-34}\, Js$ )
a.
$0 .9 eV$
b.
$4 .0 eV$
c.
$3 .1 eV$
d.
$2 .1 eV$
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In a chemical reaction, $A + 2B \rightleftharpoons 2C + D^{K}$ , the initial concentration of $B$ was $1.5$ times of the concentration
of $A$ , but the equilibrium concentrations of $A$ and $B$ were found to be equal. The equilibrium constant $(K)$ for
the chemical reaction is:
a.
$1$
b.
$4$
c.
$\frac{1}{4}$
d.
$16$
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The maximum area (in sq. units) of a rectangle having its base on the $x-$ axis and its other two vertices on the
parabola, $y= 12 - x^{2}$ such that the rectangle lies inside the parabola, is :
a.
$20\sqrt{2}$
b.
$32$
c.
$36$
d.
$18\sqrt{3}$
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An ordered pair $(\alpha , \,\beta )$ for which the system of linear equations
$(1 +\alpha )x+\beta y+z= 2$
$\alpha x+ (1 +\beta )y+z= 3$
$\alpha x+\beta y+ 2z= 2$ has a unique solution, is :
a.
$(-3, \,1)$
b.
$(1, \, -3)$
c.
$(2, \,4)$
d.
$(-4, \,2)$
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If the vertices of a hyperbola be at $(-2, \,0)$ and $(2, \,0)$ and one of its foci be at $(-3, \,0)$ , then which one of the
following points does not lie on this hyperbola ?
a.
$(6, \,5\sqrt{2})$
b.
$(-6, \,2\sqrt{10})$
c.
$(2\sqrt{6}, \,5)$
d.
$(4, \,\sqrt{15})$
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If $\lambda$ be the ratio of the roots of the quadratic equation in $x, \,3m^{2}x^{2} +m(m-4)x+ 2 = 0$ , then the least value
of $m$ for which $\lambda + \frac{1}{\lambda} = 1$ , is :
a.
$2 - \sqrt{3}$
b.
$-2 + \sqrt{2}$
c.
$4 -2\sqrt{3}$
d.
$4 -3\sqrt{2}$
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Let $P(4, -4)$ and $Q(9, 6)$ be two points on the parabola, $y^{2} = 4x$ and let $X$ be any point on the arc $POQ$ of
this parabola, where $O$ is the vertex of this parabola, such that the area of $\Delta PXQ$ is maximum. Then this
maximum area (in sq. units) is :
a.
$\frac{625}{4}$
b.
$\frac{75}{2}$
c.
$\frac{125}{4}$
d.
$\frac{125}{2}$
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Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f(x) = min{sinx, \, cosx}$ is not differentiable.
Then $S$ is a subset of which of the following?
Consider three boxes, each containing $10$ balls labelled $1, \,2, \, … . , \,10$ . Suppose one ball is randomly drawn
from each of the boxes. Denote by $n_{i}$ , the label of the ball drawn from the $i^{th}$ box, $(i= 1, \,2, \,3)$ . Then, the
number of ways in which the balls can be chosen such that $n_{1} <n_{2} <n_{3}$ is :
a.
$240$
b.
$82$
c.
$120$
d.
$164$
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The product of three consecutive terms of a $G.P.$ is $512$ . If $4$ is added to each of the first and the second of
these terms, the three terms now form an $A.P.$ , then the sum of the original three terms of the given $G.P.$ is :
a.
$28$
b.
$24$
c.
$32$
d.
$36$
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If a variable line $3x+ 4y- \lambda = 0$ is such that the two circles $x^{2} +y^{2} -2x-2y+ 1 = 0$ and
$x^{2} +y^{2} -18x-2y+ 78 = 0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval :
a.
$[13, \,23]$
b.
$(23, 31)$
c.
$[12, 21]$
d.
$(2, 17)$
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Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx} +y=xlog_{e} x, \,(x> 1)$ . If $2y(2) = log_{e}4 -1$ ,
then $y(e)$ is equal to
a.
$-\frac{e}{2}$
b.
$\frac{e}{4}$
c.
$-\frac{e^{2}}{2}$
d.
$\frac{e^{2}}{4}$
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$1 0 0$
$[ ]$
Let $P = 3 1 0$ and $Q= [q_{ij}]$ be two $3 \times 3\,$ matrices such that $Q- P^{5} =I_{3}$ . Then $\frac{q_{21}+q_{31}}{q_{32}}$ is equal to :
$[ ]$
$9 3 1$
a.
$10$
b.
$9$
c.
$15$
d.
$135$
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Let $C_{1}$ and $C_{2}$ be the centres of the circles $x^{2} +y^{2} -2x-2y-2 = 0$ and $x^{2} +y^{2} -6x-6y+ 14 = 0$
respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the
quadrilateral $PC_{1}QC_{2}$ is :
a.
$6$
b.
$4$
c.
$8$
d.
$9$
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The perpendicular distance from the origin to the plane containing the two lines, $\frac{x\,+\,2}{3} = \frac{y\,-\,2}{5} = \frac{z\,+\,5}{7}$ and
$\frac{x\,-\,1}{1} = \frac{y\,-\,4}{4} = \frac{z\,+\,4}{7}$ , is
a.
$11\sqrt{6}$
b.
$\frac{11}{\sqrt{6}}$
c.
$11$
d.
$6\sqrt{11}$
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A ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of
$10$
$(2\frac{1}{3} + \frac{1}{1}_{2(3)3} )$ is
a.
$1 : 4(16)\frac{1}{3}$
b.
$4(36)\frac{1}{3} : 1$
c.
$2(36) \frac{1}{3} : 1$
d.
$1 : 2(6) \frac{1}{3}$
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In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the
experiment will end in the fifth throw of the die is equal to :
a.
$\frac{150}{6^{5}}$
b.
$\frac{175}{6^{5}}$
c.
$\frac{225}{6^{5}}$
d.
$\frac{200}{6^{5}}$
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