The angular frequency of the damped oscillator is given by, $\omega = \sqrt{\left(\frac{\mathrm{k}}{\mathrm{m}} - \frac{\mathrm{r}^2}{4\mathrm{~m}^2}\right)}$ where k is the spring constant, m is the mass of the oscillator and $r$ is the damping constant. If the ratio $\frac{r^2}{\mathrm{mk}}$ is $8\%$, the change in time period compared to the undamped oscillator is approximately as follows:
(1) increases by $1\%$
(2) increases by $8\%$
(3) decreases by $1\%$
(4) decreases by $8\%$

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $x$ is the displacement and $\mathrm{a},\mathrm{b},\mathrm{c}$ are positive constants?
(1) $a+bx-cx^{2}$
(2) $bx^{2}$
(3) $a-bx+cx^{2}$
(4) $-bx$

An object is located in a fixed position in front of a screen. Sharp image is obtained on the screen for two positions of a thin lens separated by 10 cm. The size of the images in two situations are in the ratio $3:3$. What is the distance between the screen and the object?
(1) 124.5 cm
(2) 144.5 cm
(3) 65.0 cm
(4) 99.0 cm

In a compound microscope the focal length of objective lens is 1.2 cm and focal length of eye piece is 3.0 cm. When object is kept at 1.25 cm in front of objective, final image is formed at infinity. Magnifying power of the compound microscope should be:
(1) 200
(2) 100
(3) 400
(4) 150

Two monochromatic light beams of intensity 16 and 9 units are interfering. The ratio of intensities of bright and dark parts of the resultant pattern is:
(1) $\frac{16}{9}$
(2) $\frac{4}{3}$
(3) $\frac{7}{1}$
(4) $\frac{49}{1}$

In an experiment of single slit diffraction pattern, first minimum for red light coincides with first maximum of some other wavelength. If wavelength of red light is $6600\,\AA$, then wavelength of first maximum will be:
(1) $3300\,\AA$
(2) $4400\,\AA$
(3) $5500\,\AA$
(4) $6600\,\AA$

A photon of wavelength $\lambda$ is scattered from an electron, which was at rest. The wavelength shift $\Delta\lambda$ is three times of $\lambda$ and the angle of scattering $\theta$ is $60^\circ$. The angle at which the electron recoiled is $\phi$. The value of $\tan\phi$ is: (electron speed is much smaller than the speed of light)
(1) 0.16
(2) 0.22
(3) 0.25
(4) 0.28

A beam of light has two wavelengths of $4972\,\AA$ and $6216\,\AA$ with a total intensity $3.6\times10^{-3}\,\mathrm{Wm}^{-2}$ equally distributed among the two wavelengths. The beam falls normally on an area of $1\,\mathrm{cm}^{2}$ of a clean metallic surface of work function 2.3 eV. Assume that there is no loss of light by reflection and that each capable photon ejects one electron. The number of photoelectrons liberated in 2 s is approximately:
(1) $6\times10^{11}$
(2) $9\times10^{11}$
(3) $11\times10^{11}$
(4) $15\times10^{11}$

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 vector $\vec { A }$ is rotated by a small angle $\Delta \theta$ radians ( $\Delta \theta \ll 1$ ) to get a new vector $\vec { B }$. In that case $| \vec { B } - \vec { A } |$ is :
(1) $| \vec { A } | \left[ 1 - \frac { ( \Delta \theta ) ^ { 2 } } { 2 } \right]$
(2) 0
(3) $| \vec { A } | \Delta \theta$
(4) $| \vec { B } | \Delta \theta - | \vec { A } |$

From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of $48 \mathrm {~m} / \mathrm { s }$. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $g = 32 \mathrm {~m} / \mathrm { s } ^ { 2 }$, is:
(1) 112
(2) 88
(3) 128
(4) 100

A large number ( $n$ ) of identical beads, each of mass $m$ and radius $r$ are strung on a thin smooth rigid horizontal rod of length $L ( L \gg r )$ and are at rest at random positions. The rod is mounted between two rigid supports. If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic):
(1) $\frac { m v ^ { 2 } } { L - n r }$
(2) $\frac { m v ^ { 2 } } { L - 2 n r }$
(3) $\frac { m v ^ { 2 } } { 2 ( L - n r ) }$
(4) Zero

A particle is moving in a circle of radius $r$ under the action of a force $F = \alpha r ^ { 2 }$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy $= 0$ for $r = 0$ ):
(1) $\frac { 5 } { 6 } \alpha r ^ { 3 }$
(2) $\alpha r ^ { 3 }$
(3) $\frac { 1 } { 2 } \alpha r ^ { 3 }$
(4) $\frac { 4 } { 3 } \alpha r ^ { 3 }$