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

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) $\frac { b ^ { 2 } } { 6 a }$

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) $\frac { A - Z - 4 } { Z - 2 }$

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) $\frac{\pi}{2}$

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 $\sqrt{\frac{a}{b}}$

A fully charged capacitor $C$ with initial charge $q_{0}$ is connected to a coil of self inductance $L$ at $t=0$. The time at which the energy is stored equally between the electric and the magnetic field is:
(1) $\frac{\pi}{4}\sqrt{LC}$
(2) $2\pi\sqrt{LC}$
(3) $\sqrt{LC}$
(4) $\pi\sqrt{LC}$

Consider the following statements $P$: Suman is brilliant $Q$: Suman is rich $R$: Suman is honest The negation of the statement ``Suman is brilliant and dishonest if and only if Suman is rich'' can be expressed as
(1) $\sim(\mathrm{Q}\leftrightarrow(\mathrm{P}\wedge\sim\mathrm{R}))$
(2) $\sim Q\leftrightarrow\sim P\wedge R$
(3) $\sim(P\wedge\sim R)\leftrightarrow Q$
(4) $\sim P\wedge(Q\leftrightarrow\sim R)$

Let $R$ be the set of real numbers. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: $A=\{(x,y)\in R\times R: y-x \text{ is an integer}\}$ is an equivalence relation on $R$. Statement-2: $B=\{(x,y)\in R\times R: x=\alpha y \text{ for some rational number }\alpha\}$ is an equivalence relation on $R$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

The amount of heat produced in an electric circuit depends upon the current ( $I$ ), resistance ( $R$ ) and time ( $t$ ). If the error made in the measurements of the above quantities are $2 \% , 1 \%$ and $1 \%$ respectively then the maximum possible error in the total heat produced will be
(1) $1 \%$
(2) $2 \%$
(3) $6 \%$
(4) $3 \%$

A particle of mass m is at rest at the origin at time $\mathrm{t} = 0$. It is subjected to a force $\mathrm{F}(\mathrm{t}) = \mathrm{F}_{0}\mathrm{e}^{-\mathrm{bt}}$ in the $x$ direction. Its speed $v(t)$ is depicted by which of the following curves?

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements. If two springs $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ of force constants $k_{1}$ and $k_{2}$, respectively, are stretched by the same force, it is found that more work is done on spring $\mathrm{S}_{1}$ than on spring $\mathrm{S}_{2}$. Statement 1: If stretched by the same amount, work done on $\mathrm{S}_{1}$ will be more than that on $\mathrm{S}_{2}$. Statement 2: $k_{1} < k_{2}$
(1) Statement 1 is false, Statement 2 is true
(2) Statement 1 is true, Statement 2 is false
(3) Statement 1 is true, Statement 2 is the correct explanation for statement 1
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for statement 1.

Two point masses of mass $m _ { 1 } = f M$ and $m _ { 2 } = ( 1 - f ) M ( f < 1 )$ are in outer space (far from gravitational influence of other objects) at a distance $R$ from each other. They move in circular orbits about their centre of mass with angular velocities $\omega _ { 1 }$ for $m _ { 1 }$ and $\omega _ { 2 }$ for $m _ { 2 }$. In that case
(1) $( 1 - f ) \omega _ { 1 } = f \omega$
(2) $\omega _ { 1 } = \omega _ { 2 }$ and independent of $f$
(3) $f \omega _ { 1 } = ( 1 - f ) \omega _ { 2 }$
(4) $\omega _ { 1 } = \omega _ { 2 }$ and depend on $f$

Two cars of masses $m_{1}$ and $m_{2}$ are moving in circles of radii $r_{1}$ and $r_{2}$, respectively. Their speeds are such that they make complete circles in the same time $t$. The ratio of their centripetal acceleration is
(1) $m_{1}r_{1} : m_{2}r_{2}$
(2) $m_{1} : m_{2}$
(3) $r_{1} : r_{2}$
(4) $1 : 1$

The mass of a spaceship is 1000 kg. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10\,\mathrm{m/s}^{2}$ and 6400 km respectively. The required energy for this work will be:
(1) $6.4 \times 10^{11}$ Joules
(2) $6.4 \times 10^{8}$ Joules
(3) $6.4 \times 10^{9}$ Joules
(4) $6.4 \times 10^{10}$ Joules

This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: When moment of inertia $I$ of a body rotating about an axis with angular speed $\omega$ increases, its angular momentum $L$ is unchanged but the kinetic energy $K$ increases if there is no torque applied on it. Statement 2: $L = I\omega$, kinetic energy of rotation $= \frac{1}{2}I\omega^{2}$
(1) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of the Statement 1.
(4) Statement 1 is true, Statement 2 is false.

Assuming the earth to be a sphere of uniform density, the acceleration due to gravity inside the earth at a distance of $r$ from the centre is proportional to
(1) $r$
(2) $r^{-1}$
(3) $r^{2}$
(4) $r^{-2}$