Water is flowing through a horizontal tube having cross-sectional areas of its two ends being $A$ and $A^{\prime}$ such that the ratio $A/A^{\prime}$ is 5. If the pressure difference of water between the two ends is $3\times10^{5} \mathrm{~N\,m}^{-2}$, the velocity of water with which it enters the tube will be (neglect gravity effects)
(1) $5 \mathrm{~m\,s}^{-1}$
(2) $10 \mathrm{~m\,s}^{-1}$
(3) $25 \mathrm{~m\,s}^{-1}$
(4) $50\sqrt{10} \mathrm{~m\,s}^{-1}$

The pressure of an ideal gas varies with volume as $P = \alpha V$, where $\alpha$ is a constant. One mole of the gas is allowed to undergo expansion such that its volume becomes ' $m$ ' times its initial volume. The work done by the gas in the process is
(1) $\frac { \alpha V } { 2 } \left( m ^ { 2 } - 1 \right)$
(2) $\frac { \alpha ^ { 2 } V ^ { 2 } } { 2 } \left( m ^ { 2 } - 1 \right)$
(3) $\frac { \alpha } { 2 } \left( m ^ { 2 } - 1 \right)$
(4) $\frac { \alpha V ^ { 2 } } { 2 } \left( m ^ { 2 } - 1 \right)$

A given ideal gas with $\gamma = \frac{C_{p}}{C_{v}} = 1.5$ at a temperature $T$. If the gas is compressed adiabatically to one-fourth of its initial volume, the final temperature will be
(1) $2\sqrt{2}\,T$
(2) $4T$
(3) $2T$
(4) $8T$

$n$ moles of an ideal gas undergo a process $A \rightarrow B$ as shown in the figure. Maximum temperature of the gas during the process is
(1) $\frac{9P_{0}V_{0}}{nR}$
(2) $\frac{3P_{0}V_{0}}{2nR}$
(3) $\frac{9P_{0}V_{0}}{2nR}$
(4) $\frac{9P_{0}V_{0}}{4nR}$

If a simple pendulum has significant amplitude (up to a factor of $1/e$ of original) only in the period between $t = 0\,\mathrm{s}$ to $t = \tau\,\mathrm{s}$, then $\tau$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with $b$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds:
(1) $\dfrac{0.693}{b}$
(2) $b$
(3) $\dfrac{1}{b}$
(4) $\dfrac{2}{b}$

A ring is suspended from a point $S$ on its rim as shown in the figure. When displaced from equilibrium, it oscillates with time period of 1 second. The radius of the ring is (take $g = \pi ^ { 2 }$ )
(1) 0.15 m
(2) 1.5 m
(3) 1.0 m
(4) 0.5 m

A uniform tube of length 60.5 cm is held vertically with its lower end dipped in water. A sound source of frequency 500 Hz sends sound waves into the tube. When the length of tube above water is 16 cm and again when it is 50 cm , the tube resonates with the source of sound. Two lowest frequencies (in Hz ), to which tube will resonate when it is taken out of water, are (approximately).
(1) 281,562
(2) 281,843
(3) 276,552
(4) 272,544

A wave represented by the equation $y_{1} = a\cos(kx - \omega t)$ is superimposed with another wave to form a stationary wave such that the point $x = 0$ is node. The equation for the other wave is
(1) $a\cos(kx - \omega t + \pi)$
(2) $a\cos(kx + \omega t + \pi)$
(3) $a\cos\left(kx + \omega t + \frac{\pi}{2}\right)$
(4) $a\cos\left(kx - \omega t + \frac{\pi}{2}\right)$

A charge of total amount $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R ( R > r )$ such that the surface charge densities on the two spheres are equal. The electric potential at the common centre is
(1) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R - r ) Q } { \left( R ^ { 2 } + r ^ { 2 } \right) }$
(2) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R + r ) Q } { 2 \left( R ^ { 2 } + r ^ { 2 } \right) }$
(3) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R + r ) Q } { \left( R ^ { 2 } + r ^ { 2 } \right) }$
(4) $\frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { ( R - r ) Q } { 2 \left( R ^ { 2 } + r ^ { 2 } \right) }$

A series combination of $n_{1}$ capacitors, each of capacity $C_{1}$ is charged by source of potential difference 4V. When another parallel combination of $n_{2}$ capacitors each of capacity $C_{2}$ is charged by a source of potential difference $V$, it has the same total energy stored in it as the first combination has. The value of $C_{2}$ in terms of $C_{1}$ is then
(1) $16\frac{n_{2}}{n_{1}}C_{1}$
(2) $\frac{2C_{1}}{n_{1}n_{2}}$
(3) $2\frac{n_{2}}{n_{1}}C_{1}$
(4) $\frac{16C_{1}}{n_{1}n_{2}}$

A resistance $R$ and a capacitance $C$ are connected in series to a battery of negligible internal resistance through a key. The key is closed at $t = 0$. If after $t$ sec the voltage across the capacitance was seven times the voltage across $R$, the value of $t$ is
(1) $3RC\ln 2$
(2) $2RC\ln 2$
(3) $2RC\ln 7$
(4) $3RC\ln 7$

We wish to make a microscope with the help of two positive lenses both with a focal length of 20 mm each and the object is positioned 25 mm from the objective lens. How far apart the lenses should be so that the final image is formed at infinity?
(1) 20 mm
(2) 100 mm
(3) 120 mm
(4) 80 mm

The first diffraction minimum due to the single slit diffraction is seen at $\theta = 30^{\circ}$ for a light of wavelength $5000\,\AA$ falling perpendicularly on the slit. The width of the slit is
(1) $2.5\times10^{-5}\mathrm{~cm}$
(2) $1.25\times10^{-5}\mathrm{~cm}$
(3) $10\times10^{-5}\mathrm{~cm}$
(4) $5\times10^{-5}\mathrm{~cm}$

The maximum number of possible interference maxima for slit separation equal to $1.8\lambda$, where $\lambda$ is the wavelength of light used, in a Young's double slit experiment is
(1) zero
(2) 3
(3) infinite
(4) 5

Hydrogen atom is excited from ground state to another state with principal quantum number equal to 4. Then the number of spectral lines in the emission spectra will be
(1) 2
(2) 3
(3) 5
(4) 6

A diatomic molecule is made of two masses $m_{1}$ and $m_{2}$ which are separated by a distance $r$. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by ($n$ is an integer)
(1) $\dfrac{(m_{1}+m_{2})^{2}n^{2}h^{2}}{2m_{1}^{2}m_{2}^{2}r^{2}}$
(2) $\dfrac{n^{2}h^{2}}{2(m_{1}+m_{2})r^{2}}$
(3) $\dfrac{2n^{2}h^{2}}{(m_{1}+m_{2})r^{2}}$
(4) $\dfrac{(m_{1}+m_{2})n^{2}h^{2}}{2m_{1}m_{2}r^{2}}$

A doubly ionised Li atom is excited from its ground state $(n=1)$ to $n=3$ state. The wavelengths of the spectral lines are given by $\lambda_{32}$, $\lambda_{31}$ and $\lambda_{21}$. The ratio $\lambda_{32}/\lambda_{31}$ and $\lambda_{21}/\lambda_{31}$ are, respectively
(1) $8.1,\;0.67$
(2) $8.1,\;1.2$
(3) $6.4,\;1.2$
(4) $6.4,\;0.67$

$N$ divisions on the main scale of a vernier calliper coincide with $( N + 1 )$ divisions of the vernier scale. If each division of main scale is ' $a$ ' units, then the least count of the instrument is
(1) $a$
(2) $\frac { a } { N }$
(3) $\frac { N } { N + 1 } \times a$
(4) $\frac { a } { N + 1 }$

A 10 kW transmitter emits radio waves of wavelength 500 m. The number of photons emitted per second by the transmitter is of the order of
(1) $10^{37}$
(2) $10^{31}$
(3) $10^{25}$
(4) $10^{43}$

A spectrometer gives the following reading when used to measure the angle of a prism. Main scale reading: 58.5 degree. Vernier scale reading: 09 divisions. Given that 1 division on main scale corresponds to 0.5 degree. Total divisions on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data is
(1) $58.59^{\circ}$
(2) $58.77^{\circ}$
(3) $58.65^{\circ}$
(4) $59^{\circ}$

The logically equivalent preposition of $p \Leftrightarrow q$ is
(1) $(p \Rightarrow q) \wedge (q \Rightarrow p)$
(2) $p \wedge q$
(3) $(p \wedge q) \vee (q \neq p)$
(4) $(p \wedge q) \Rightarrow (q \vee p)$

$\lim _ { x \rightarrow 0 } \left( \frac { x - \sin x } { x } \right) \sin \left( \frac { 1 } { x } \right)$
(1) equals 1
(2) equals 0
(3) does not exist
(4) equals $-1$

The negation of the statement ``If I become a teacher, then I will open a school'' is
(1) I will become a teacher and I will not open a school
(2) Either I will not become a teacher or I will not open a school
(3) Neither I will become a teacher nor I will open a school
(4) I will not become a teacher or I will open a school

If $A = \{x \in z^{+} : x < 10$ and $x$ is a multiple of 3 or 4$\}$, where $z^{+}$ is the set of positive integers, then the total number of symmetric relations on $A$ is
(1) $2^{5}$
(2) $2^{15}$
(3) $2^{10}$
(4) $2^{20}$

Let $p$ and $q$ denote the following statements $p$: The sun is shining $q$: I shall play tennis in the afternoon The negation of the statement ``If the sun is shining then I shall play tennis in the afternoon'', is
(1) $q \Rightarrow \sim p$
(2) $q \wedge \sim p$
(3) $p \wedge \sim q$
(4) $\sim q \Rightarrow \sim p$