Given: $\mathrm { E } _ { \mathrm { Fe } ^ { 3 + } / \mathrm { Fe } } ^ { \circ } = -0.036 \mathrm {~V} , \quad \mathrm { E } _ { \mathrm { Fe } ^ { 2 + } / \mathrm { Fe } } ^ { \circ } = -0.439 \mathrm {~V}$. The value of standard electrode potential for the change, $\mathrm { Fe } _ { ( \mathrm { aq } ) } ^ { 3 + } + \mathrm { e } ^ { - } \rightarrow \mathrm { Fe } ^ { 2 + } ( \mathrm { aq } )$ will be:
(1) $-0.072$ V
(2) $0.385$ V
(3) $0.770$ V
(4) $-0.270$ V

The half life period of a first order chemical reaction is 6.93 minutes. The time required for the completion of $99\%$ of the chemical reaction will be $( \log 2 = 0.301 )$:
(1) 230.3 minutes
(2) 23.03 minutes
(3) 46.06 minutes
(4) 460.6 minutes

A small particle of mass $m$ is projected at an angle $\theta$ with the x-axis with an initial velocity $\mathrm { v } _ { 0 }$ in the $\mathrm { x } - \mathrm { y }$ plane as shown in the figure. At a time $t < \frac { v _ { 0 } \sin \theta } { g }$, the angular momentum of the particle is where $\hat { \mathrm { i } } , \hat { \mathrm { j } }$ and $\hat { \mathrm { k } }$ are unit vectors along $\mathrm { x } , \mathrm { y }$ and z-axis respectively.
(1) $- \mathrm { mgv } _ { 0 } \mathrm { t } ^ { 2 } \cos \theta \hat { \mathrm { j } }$
(2) $\mathrm { mgv } _ { 0 } t \cos \theta \hat { \mathrm { k } }$
(3) $- \frac { 1 } { 2 } m g v _ { 0 } t ^ { 2 } \cos \theta \hat { k }$
(4) $\frac { 1 } { 2 } m g v _ { 0 } t ^ { 2 } \cos \theta \hat { i }$

Two fixed frictionless inclined plane making an angle $30 ^ { \circ }$ and $60 ^ { \circ }$ with the vertical are shown in the figure. Two block $A$ and $B$ are placed on the two planes. What is the relative vertical acceleration of $A$ with respect to $B$?
(1) $4.9 \mathrm {~ms} ^ { - 2 }$ in horizontal direction
(2) $9.8 \mathrm {~ms} ^ { - 2 }$ in vertical direction
(3) zero
(4) $4.9 \mathrm {~ms} ^ { - 2 }$ in vertical direction

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U ( x ) = \frac { a } { x ^ { 12 } } - \frac { b } { x ^ { 6 } }$, where a and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is $D = \left[ U ( x = \infty ) - U _ { \text {at equilibrium} } \right]$, $D$ is
(1) $\frac { b ^ { 2 } } { 2 a }$
(2) $\frac { b ^ { 2 } } { 12 a }$
(3) $\frac { b ^ { 2 } } { 4 a }$
(4) $\frac { b ^ { 2 } } { 6 a }$

The figure shows the position-time $(x-t)$ graph of one-dimensional motion of a body of mass 0.4 kg. The magnitude of each impulse is
(1) 0.4 Ns
(2) 0.8 Ns
(3) 1.6 Ns
(4) 0.2 Ns

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length $s = t ^ { 3 } + 5$, where $s$ is in metres and $t$ is in seconds. The radius of the path is 20 m. The acceleration of 'P' when $t = 2 \mathrm {~s}$ is nearly
(1) $13 \mathrm {~m} / \mathrm { s } ^ { 2 }$
(2) $12 \mathrm {~m} / \mathrm { s } ^ { 2 }$
(3) $7.2 \mathrm {~m} / \mathrm { s } ^ { 2 }$
(4) $14 \mathrm {~m} / \mathrm { s } ^ { 2 }$

For a particle in uniform circular motion the acceleration $\vec { a }$ at a point $P ( R , \theta )$ on the circle of radius R is (here $\theta$ is measured from the $x$-axis)
(1) $- \frac { v ^ { 2 } } { R } \cos \theta \hat { i } + \frac { v ^ { 2 } } { R } \sin \theta \hat { j }$
(2) $- \frac { v ^ { 2 } } { R } \sin \theta \hat { i } + \frac { v ^ { 2 } } { R } \cos \theta \hat { j }$
(3) $- \frac { v ^ { 2 } } { R } \cos \theta \hat { i } - \frac { v ^ { 2 } } { R } \sin \theta \hat { j }$
(4) $\frac { v ^ { 2 } } { R } \hat { i } + \frac { v ^ { 2 } } { R } \hat { j }$

Let $C$ be the capacitance of a capacitor discharging through a resistor R. Suppose $t _ { 1 }$ is the time taken for the energy stored in the capacitor to reduce to half its initial value and $t _ { 2 }$ is the time taken for the charge to reduce to one-fourth its initial value. Then the ratio $t _ { 1 } / t _ { 2 }$ will be
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 4 }$
(4) 2

A rectangular loop has a sliding connector PQ of length $\ell$ and resistance $\mathrm { R } \Omega$ and it is moving with a speed $v$ as shown. The set-up is placed in a uniform magnetic field going into the plane of the paper. The three currents $I _ { 1 } , I _ { 2 }$ and $I$ are
(1) $\mathrm { I } _ { 1 } = - \mathrm { I } _ { 2 } = \frac { \mathrm { B } \ell \mathrm { v } } { \mathrm { R } } , \mathrm { I } = \frac { 2 \mathrm {~B} \ell \mathrm { v } } { \mathrm { R } }$
(2) $\mathrm { I } _ { 1 } = \mathrm { I } _ { 2 } = \frac { \mathrm { B } \ell \mathrm { v } } { 3 \mathrm { R } } , \mathrm { I } = \frac { 2 \mathrm {~B} \ell \mathrm { v } } { 3 \mathrm { R } }$
(3) $I _ { 1 } = I _ { 2 } = I = \frac { B \ell v } { R }$
(4) $I _ { 1 } = I _ { 2 } = \frac { B \ell v } { 6 R } , I = \frac { B \ell v } { 3 R }$

In a series LCR circuit $R = 200 \Omega$ and the voltage and the frequency of the main supply is 220 V and 50 Hz respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $30 ^ { \circ }$. On taking out the inductor from the circuit the current leads the voltage by $30 ^ { \circ }$. The power dissipated in the LCR circuit is
(1) 305 W
(2) 210 W
(3) Zero W
(4) 242 W

A nucleus of mass $M + \Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac { M } { 2 }$ each. Speed of light is $c$. The binding energy per nucleon for the parent nucleus is $E _ { 1 }$ and that for the daughter nuclei is $E _ { 2 }$. Then
(1) $\mathrm { E } _ { 2 } = 2 \mathrm { E } _ { 1 }$
(2) $E _ { 1 } > E _ { 2 }$
(3) $E _ { 2 } > E _ { 1 }$
(4) $\mathrm { E } _ { 1 } = 2 \mathrm { E } _ { 2 }$

A nucleus of mass $M + \Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac { M } { 2 }$ each. Speed of light is $C$. The speed of daughter nuclei is
(1) $c \frac { \Delta m } { M + \Delta m }$
(2) $c \sqrt { \frac { 2 \Delta m } { M } }$
(3) $c \sqrt { \frac { \Delta \mathrm {~m} } { \mathrm { M } } }$
(4) $c \sqrt { \frac { \Delta m } { M + \Delta m } }$

A radioactive nucleus (initial mass number $A$ and atomic number Z) emits $3\alpha$-particles and 2 positrons. The ratio of number of neutrons to that of protons in the final nucleus will be
(1) $\frac { A - Z - 8 } { Z - 4 }$
(2) $\frac { A - Z - 4 } { Z - 8 }$
(3) $\frac { A - Z - 12 } { Z - 4 }$
(4) $\frac { A - Z - 4 } { Z - 2 }$

An object, moving with a speed of $6.25 \mathrm{~m}/\mathrm{s}$, is decelerated at a rate given by :
$$\frac{\mathrm{dv}}{\mathrm{dt}} = -2.5\sqrt{\mathrm{v}}$$
where $v$ is the instantaneous speed. The time taken by the object, to come to rest, would be:
(1) 2 s
(2) 4 s
(3) 8 s
(4) 1 s

A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountain is $v$, the total area around the fountain that gets wet is:
(1) $\pi \frac{v^{4}}{g^{2}}$
(2) $\frac{\pi}{2}\frac{v^{4}}{g^{2}}$
(3) $\pi \frac{v^{2}}{g^{2}}$
(4) $\pi \frac{v^{4}}{g}$

A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the pulley, is
(1) g
(2) $\frac{2}{3}\mathrm{~g}$
(3) $\frac{g}{3}$
(4) $\frac{3}{2}g$

A pulley of radius 2 m is rotated about its axis by a force $\mathrm{F} = \left(20\mathrm{t} - 5\mathrm{t}^{2}\right)$ Newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10\,\mathrm{kg\,m}^2$, the number of rotations made by the pulley before its direction of motion is reversed, is:
(1) more than 3 but less than 6
(2) more than 6 but less than 9
(3) more than 9
(4) less than 3

Two bodies of masses m and 4m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is:
(1) $-\frac{4\mathrm{Gm}}{\mathrm{r}}$
(2) $-\frac{6\mathrm{Gm}}{\mathrm{r}}$
(3) $-\frac{9\mathrm{Gm}}{r}$
(4) zero

A Carnot engine operating between temperatures $T_{1}$ and $T_{2}$ has efficiency $\frac{1}{6}$. When $T_{2}$ is lowered by 62 K, its efficiency increases to $\frac{1}{3}$. Then $T_{1}$ and $T_{2}$ are, respectively:
(1) 372 K and 330 K
(2) 330 K and 268 K
(3) 310 K and 248 K
(4) 372 K and 310 K

A thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by:
(1) $\frac{(\gamma-1)}{2\gamma\mathrm{R}}\mathrm{Mv}^{2}\mathrm{~K}$
(2) $\frac{\gamma Mv^{2}}{2\mathrm{R}}\mathrm{K}$
(3) $\frac{(\gamma-1)}{2R}Mv^{2}K$
(4) $\frac{(\gamma-1)}{2(\gamma+1)R}\mathrm{Mv}^{2}\mathrm{~K}$

Three perfect gases at absolute temperatures $T_{1}, T_{2}$ and $T_{3}$ are mixed. The masses of molecules are $m_{1}, m_{2}$ and $m_{3}$ and the number of molecules are $n_{1}, n_{2}$ and $n_{3}$ respectively. Assuming no loss of energy, the final temperature of the mixture is:
(1) $\frac{n_{1}T_{1}+n_{2}T_{2}+n_{3}T_{3}}{n_{1}+n_{2}+n_{3}}$
(2) $\frac{n_{1}T_{1}+n_{2}T_{2}^{2}+n_{3}T_{3}^{2}}{n_{1}T_{1}+n_{2}T_{2}+n_{3}T_{3}}$
(3) $\frac{n_{1}^{2}T_{1}^{2}+n_{2}^{2}T_{2}^{2}+n_{3}^{2}T_{3}^{2}}{n_{1}T_{1}+n_{2}T_{2}+n_{3}T_{3}}$
(4) $\frac{\left(T_{1}+T_{2}+T_{3}\right)}{3}$

Two particles are executing simple harmonic motion of the same amplitude A and frequency $\omega$ along the $x$ axis. Their mean position is separated by distance $X_{0}\left(X_{0}>A\right)$. If the maximum separation between them is $\left(X_{0}+A\right)$, the phase difference between their motion is:
(1) $\frac{\pi}{3}$
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{6}$
(4) $\frac{\pi}{2}$

A mass $M$, attached to a horizontal spring, executes S.H.M. with amplitude $A_{1}$. When the mass $M$ passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude $A_{2}$. The ratio of $\left(\frac{A_{1}}{A_{2}}\right)$ is:
(1) $\frac{M+m}{M}$
(2) $\left(\frac{M}{M+m}\right)^{1/2}$
(3) $\left(\frac{M+m}{M}\right)^{1/2}$
(4) $\frac{M}{M+m}$

The transverse displacement $y(x,t)$ of a wave on a string is given by $y(x,t) = e^{-\left(ax^{2}+bt^{2}+2\sqrt{ab}xt\right)}$. This represents a
(1) wave moving in $-x$ direction with speed $\sqrt{\frac{b}{a}}$
(2) standing wave of frequency $\sqrt{b}$
(3) standing wave of frequency $\frac{1}{\sqrt{b}}$
(4) wave moving in $+x$ direction with speed $\sqrt{\frac{a}{b}}$