A uniform thin rod AB of length $L$ has linear mass density $\mu ( x ) = a + \frac { b x } { L }$, where $x$ is measured from A. If the CM of the rod lies at a distance of $\left( \frac { 7 } { 12 } L \right)$ from A, then $a$ and $b$ are related as:
(1) $2 a = b$
(2) $a = 2 b$
(3) $a = b$
(4) $3 a = 2 b$

A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is $12 \mathrm { rad } \mathrm { s } ^ { - 1 }$, the magnitude of its angular momentum about a point on the ground right under the center of the circle is:
(1) $14.4 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
(2) $11.52 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
(3) $20.16 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
(4) $8.64 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$

A pendulum with the time period of 1 s is losing energy due to damping. At a certain time, its energy is 45 J. If after completing 15 oscillations its energy has become 15 J, then its damping constant (in $\mathrm { s } ^ { - 1 }$) will be
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 15 } \ln 3$
(3) $\frac { 1 } { 30 } \ln 3$
(4) 2

A cylindrical block of wood (density $= 650 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$), of base area $30 \mathrm {~cm} ^ { 2 }$ and height 54 cm, floats in a liquid of density $900 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly):
(1) 52 cm
(2) 26 cm
(3) 39 cm
(4) 65 cm

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) $s \vee ( r \vee \sim s )$

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 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 increases four times, then its side is doubled.
(3) if the area of a square does not increase four times, then its side is not doubled.
(4) if the side of a square is not doubled, then its area does not increase four times.

The angle of elevation of the top of a vertical tower from a point A , due east of it is $45 ^ { \circ }$. The angle of elevation of the top of the same tower from a point B , due south of A is $30 ^ { \circ }$. If the distance between A and B is $54 \sqrt { 2 } m$, then the height of the tower (in meters), is:
(1) 108
(2) $36 \sqrt { 3 }$
(3) $54 \sqrt { 3 }$
(4) 54

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) $32 \%$

A car is standing 200 m behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and the car has acceleration $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The car will catch up with the bus after time :
(1) $\sqrt { 120 } \mathrm {~s}$
(2) 15 s
(3) $\sqrt { 110 } \mathrm {~s}$
(4) $10 \sqrt { 2 } \mathrm {~s}$

A conical pendulum of length $l$ makes an angle $\theta = 45 ^ { \circ }$ with respect to $Z$-axis and moves in a circle in the $X Y$ plane. The radius of the circle is 0.4 m and its center is vertically below $O$. The speed of the pendulum, in its circular path, will be - (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
(1) $0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

The machine as shown has 2 rods of length 1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :
(1) Speed which is $\frac { 3 } { 4 }$ th of that of the roller when the weight is 0.4 m above the ground
(2) Constant speed
(3) Decreasing speed
(4) Increasing speed

Two particles $A$ and $B$ of equal mass $M$ are moving with the same speed $v$ as shown in figure. They collide completely inelastic and move as a single particle $C$. The angle $\theta$ that the path of $C$ makes with the $X$-axis is given by-
(1) $\tan \theta = \frac { \sqrt { 3 } - \sqrt { 2 } } { 1 - \sqrt { 2 } }$
(2) $\tan \theta = \frac { 1 - \sqrt { 2 } } { \sqrt { 2 } ( 1 + \sqrt { 3 } ) }$
(3) $\tan \theta = \frac { 1 - \sqrt { 3 } } { 1 + \sqrt { 2 } }$
(4) $\tan \theta = \frac { \sqrt { 3 } + \sqrt { 2 } } { 1 - \sqrt { 2 } }$

A circular hole of radius $\frac { R } { 4 }$ is made in a thin uniform disc having mass $M$ and radius $R$, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point $O$ and perpendicular to the plane of the disc is-
(1) $\frac { 219 M R ^ { 2 } } { 256 }$
(2) $\frac { 237 M R ^ { 2 } } { 512 }$
(3) $\frac { 197 M R ^ { 2 } } { 256 }$
(4) $\frac { 19 M R ^ { 2 } } { 512 }$

The mass density of a spherical body is given by $\rho ( r ) = \frac { k } { r }$ for $r \leq R$ and $\rho ( r ) = 0$ for $r > R$, where $r$ is the distance from the center. The correct graph that describes qualitatively the acceleration, $a$ of a test particle as a function of $r$ is: (options given as graphs (1), (2), (3), (4))

$N$ moles of diatomic gas in a cylinder is at a temperature $T$. Heat is supplied to the cylinder such that the temperature remains constant but $n$ moles of the diatomic gas get converted into monoatomic gas. The change in the total kinetic energy of the gas is
(1) 0
(2) $\frac { 5 } { 2 } n R T$
(3) $\frac { 3 } { 2 } n R T$
(4) $\frac { 1 } { 2 } n R T$

A block of mass 0.1 kg is connected to an elastic spring of spring constant $640 \mathrm {~N} \mathrm {~m} ^ { - 1 }$ and oscillates in a damping medium of damping constant $10 ^ { - 2 } \mathrm {~kg} \mathrm {~s} ^ { - 1 }$. The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to-
(1) 2 s
(2) 5 s
(3) 3 s
(4) 7 s

In an experiment to determine the period of a simple pendulum of length 1 m, it is attached to different spherical bobs of radii $r _ { 1 }$ and $r _ { 2 }$. The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be $5 \times 10 ^ { - 4 } \mathrm {~s}$, the difference in radii, $\left| r _ { 1 } - r _ { 2 } \right|$ is best-given by
(1) 0.01 cm
(2) 0.05 cm
(3) 0.5 cm
(4) 1 cm

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by, $y ( x , t ) = 0.5 \sin \left( \frac { 5 \pi } { 4 } x \right) \cos ( 200 \pi t )$. What is the speed of the travelling wave moving in the positive $x$ direction? ($x$ and $t$ are in meter and second, respectively)
(1) $120 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $160 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $180 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

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