A radioactive nuclei with decay constant $0.5/\mathrm{s}$ is being produced at a constant rate of 100 nuclei/s. If at $t = 0$ there were no nuclei, the time when there are 50 nuclei is:
(1) 1 s
(2) $2\ln\left(\frac{4}{3}\right) \mathrm{s}$
(3) $\ln 2 \mathrm{~s}$
(4) $\ln\left(\frac{4}{3}\right) \mathrm{s}$

A piece of bone of an animal from a ruin is found to have ${}^{14}\mathrm{C}$ activity of 12 disintegrations per minute per gm of its carbon content. The ${}^{14}\mathrm{C}$ activity of a living animal is 16 disintegrations per minute per gm. How long ago nearly did the animal die? (Given half life of ${}^{14}\mathrm{C}$ is $\mathrm{t}_{1/2}=5760$ years)
(1) 1672 years
(2) 2391 years
(3) 3291 years
(4) 4453 years

If $\lambda_0$ and $\lambda$ be threshold wavelength and wavelength of incident light, the velocity of photoelectron ejected from the metal surface is:
(1) $\sqrt{\frac{2h}{m}\left(\lambda_0 - \lambda\right)}$
(2) $\sqrt{\frac{2hc}{m}\left(\lambda_\mathrm{o} - \lambda\right)}$
(3) $\sqrt{\frac{2hc}{m}\left(\frac{\lambda_0 - \lambda}{\lambda\lambda_0}\right)}$
(4) $\sqrt{\frac{2h}{m}\left(\frac{1}{\lambda_\mathrm{o}} - \frac{1}{\lambda}\right)}$

If $m$ and $e$ are the mass and charge of the revolving electron in the orbit of radius $r$ for hydrogen atom, the total energy of the revolving electron will be:
(1) $\frac{1}{2}\frac{e^{2}}{r}$
(2) $-\frac{e^{2}}{r}$
(3) $\frac{me^{2}}{r}$
(4) $-\frac{1}{2}\frac{e^{2}}{r}$

Based on the equation: $$\Delta\mathrm{E} = -2.0 \times 10^{-18} \mathrm{~J} \left(\frac{1}{\mathrm{n}_2^2} - \frac{1}{\mathrm{n}_1^2}\right)$$ the wavelength of the light that must be absorbed to excite hydrogen electron from level $n = 1$ to level $\mathrm{n} = 2$ will be: ($\mathrm{h} = 6.625 \times 10^{-34} \mathrm{Js}$, $\mathrm{C} = 3 \times 10^8 \mathrm{~ms}^{-1}$)
(1) $1.325 \times 10^{-7} \mathrm{~m}$
(2) $1.325 \times 10^{-10} \mathrm{~m}$
(3) $2.650 \times 10^{-7} \mathrm{~m}$
(4) $5.300 \times 10^{-10} \mathrm{~m}$

The de-Broglie wavelength of a particle of mass 6.63 g moving with a velocity of $100\,\mathrm{ms}^{-1}$ is:
(1) $10^{-33}\,\mathrm{m}$
(2) $10^{-35}\,\mathrm{m}$
(3) $10^{-31}\,\mathrm{m}$
(4) $10^{-25}\,\mathrm{m}$

Excited hydrogen atom emits light in the ultraviolet region at $2.47\times10^{15}\,\mathrm{Hz}$. With this frequency, the energy of a single photon is: ($\mathrm{h}=6.63\times10^{-34}\,\mathrm{Js}$)
(1) $8.041\times10^{-40}\,\mathrm{J}$
(2) $2.680\times10^{-19}\,\mathrm{J}$
(3) $1.640\times10^{-18}\,\mathrm{J}$
(4) $6.111\times10^{-17}\,\mathrm{J}$

The rate coefficient ($k$) for a particular reaction is $1.3\times10^{-4}\,\mathrm{M}^{-1}\mathrm{s}^{-1}$ at $100^{\circ}\mathrm{C}$, and $1.3\times10^{-3}\,\mathrm{M}^{-1}\mathrm{s}^{-1}$ at $150^{\circ}\mathrm{C}$. What is the energy of activation ($\mathrm{E}_{\mathrm{A}}$) (in kJ) for this reaction? ($\mathrm{R}=$ molar gas constant $=8.314\,\mathrm{JK}^{-1}\mathrm{mol}^{-1}$)
(1) 16
(2) 60
(3) 99
(4) 132

The contrapositive of the statement "if I am not feeling well, then I will go to the doctor" is
(1) if I will go to the doctor, then I am not feeling well.
(2) if I am feeling well, then I will not go to the doctor.
(3) if I will not go to the doctor, then I am feeling well.
(4) if I will go to the doctor, then I am feeling well.

The statement $\sim ( p \leftrightarrow \sim q )$ is
(1) A tautology
(2) A fallacy
(3) Equivalent to $p \leftrightarrow q$
(4) Equivalent to $\sim p \leftrightarrow q$

The contrapositive of the statement "I go to school if it does not rain" is
(1) If it rains, I go to school.
(2) If it rains, I do not go to school.
(3) If I go to school, it rains.
(4) If I do not go to school, it rains.

If $X = \left\{ 4 ^ { n } - 3 n - 1 : n \in N \right\}$ and $Y = \{ 9 ( n - 1 ) : n \in N \}$, where $N$ is the set of natural numbers, then $X \cup Y$ is equal to
(1) $X$
(2) $Y$
(3) $N$
(4) $Y - X$

Let $P$ be the relation defined on the set of all real numbers such that $P = \left\{ ( a , b ) : \sec ^ { 2 } a - \tan ^ { 2 } b = 1 \right\}$. Then, $P$ is
(1) reflexive and symmetric but not transitive
(2) symmetric and transitive but not reflexive
(3) reflexive and transitive but not symmetric
(4) an equivalence relation

A source of sound emits sound waves at frequency $f _ { 0 }$. It is moving towards an observer with fixed speed $v _ { s } \left( v _ { s } < v \right)$, where $v$ is the speed of sound in air. If the observer were to move towards the source with speed $v _ { 0 }$, one of the following two graphs (A and B) will give the correct variation of the frequency $f$ heard by the observer as $v _ { 0 }$ is changed.
The variation of $f$ with $v _ { 0 }$ is given correctly by:
(1) Graph A with slope $= \frac { f _ { 0 } } { \left( v - v _ { s } \right) }$
(2) Graph A with slope $= \frac { f _ { 0 } } { \left( v + v _ { s } \right) }$
(3) Graph B with slope $= \frac { f _ { 0 } } { \left( v - v _ { s } \right) }$
(4) Graph B with slope $= \frac { f _ { 0 } } { \left( v + v _ { s } \right) }$

In a Young's double slit experiment with light of wavelength $\lambda$, the separation of slits is $d$ and distance of screen is $D$ such that $D \gg d \gg \lambda$. If the Fringe width is $\beta$, the distance from point of maximum intensity to the point where intensity falls to half of the maximum intensity on either side is:
(1) $\frac { \beta } { 4 }$
(2) $\frac { \beta } { 3 }$
(3) $\frac { \beta } { 6 }$
(4) $\frac { \beta } { 2 }$

The de-Broglie wavelength associated with the electron in the $n = 4$ level is:
(1) Half of the de-Broglie wavelength of the electron in the ground state
(2) Four times the de-Broglie wavelength of the electron in the ground state
(3) $\frac { 1 } { 4 }$ th of the de-Broglie wavelength of the electron in the ground state
(4) Two times the de-Broglie wavelength of the electron in the ground state

Let $N _ { \beta }$ be the number of $\beta$ particles emitted by 1 gram of $\mathrm { Na } ^ { 24 }$ radioactive nuclei having a half life of 15 h. In 7.5 h, the number $N _ { \beta }$ is close to $\left[ N _ { \mathrm { A } } = 6.023 \times 10 ^ { 23 } \mathrm {~mole} ^ { - 1 } \right]$
(1) $1.75 \times 10 ^ { 22 }$
(2) $6.2 \times 10 ^ { 21 }$
(3) $7.5 \times 10 ^ { 21 }$
(4) $1.25 \times 10 ^ { 22 }$

For the equilibrium, $\mathrm { A } ( \mathrm { g } ) \rightleftharpoons \mathrm { B } ( \mathrm { g } ) , \Delta \mathrm { H }$ is $- 40 \mathrm {~kJ} / \mathrm { mol }$. If the ratio of the activation energies of the forward $\left( \mathrm { E } _ { \mathrm { f } } \right)$ and reverse $\left( \mathrm { E } _ { \mathrm { b } } \right)$ reactions is $\frac { 2 } { 3 }$ then:
(1) $\mathrm { E } _ { \mathrm { f } } = 30 \mathrm {~kJ} / \mathrm { mol } , \mathrm { E } _ { \mathrm { b } } = 70 \mathrm {~kJ} / \mathrm { mol }$
(2) $\mathrm { E } _ { \mathrm { f } } = 70 \mathrm {~kJ} / \mathrm { mol } , \mathrm { E } _ { \mathrm { b } } = 30 \mathrm {~kJ} / \mathrm { mol }$
(3) $\mathrm { E } _ { \mathrm { f } } = 80 \mathrm {~kJ} / \mathrm { mol } , \mathrm { E } _ { \mathrm { b } } = 120 \mathrm {~kJ} / \mathrm { mol }$
(4) $\mathrm { E } _ { \mathrm { f } } = 60 \mathrm {~kJ} / \mathrm { mol } , \mathrm { E } _ { \mathrm { b } } = 100 \mathrm {~kJ} / \mathrm { mol }$

The negation of $\sim s \vee ( \sim r \wedge s )$ is equivalent to
(1) $s \wedge r$
(2) $s \wedge \sim r$
(3) $s \wedge ( r \wedge \sim s )$
(4) $s \vee ( r \vee \sim s )$

The negation of $\sim s \vee (\sim r \wedge s)$ is equivalent to:
(1) $s \wedge \sim r$
(2) $s \wedge (r \wedge \sim s)$
(3) $s \vee (r \vee \sim s)$
(4) $s \wedge r$

If $\mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d }$ are inputs to a gate and x is its output, then, as per the following time graph, the gate is:
(1) OR (2) NAND (3) NOT (4) AND

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to:
(1) $p \wedge q$
(2) $p \vee q$
(3) $p \vee \sim q$
(4) $\sim p \wedge q$

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to: (1) $\sim p \wedge q$ (2) $p \wedge q$ (3) $p \vee q$ (4) $p \vee \sim q$

The contrapositive of the following statement, ``If the side of a square doubles, then its area increases four times'', is: (1) If the area of a square increases four times, then its side is not doubled. (2) If the area of a square does not increase four times, then its side is not doubled. (3) If the area of a square does not increase four times, then its side is doubled. (4) If the side of a square is not doubled, then its area does not increase four times.

A physical quantity $P$ is described by the relation $P = a ^ { \frac { 1 } { 2 } } b ^ { 2 } c ^ { 3 } d ^ { - 4 }$. If the relative errors in the measurement of $a , b , c$ and $d$ respectively, are $2 \% , 1 \% , 3 \%$ and $5 \%$. Then the relative error in $P$ will be:
(1) $12 \%$
(2) $8 \%$
(3) $25 \%$
(4) $32 \%$