The proposition $( \sim p ) \vee ( p \wedge \sim q )$ is equivalent to
(1) $p \rightarrow \sim q$
(2) $p \wedge \sim q$
(3) $q \rightarrow p$
(4) none

Let a vertical tower $AB$ have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that $AP = 2AB$. If $\angle BPC = \beta$, then $\tan\beta$ is equal to:
(1) $\dfrac{6}{7}$
(2) $\dfrac{1}{4}$
(3) $\dfrac{2}{9}$
(4) $\dfrac{4}{9}$

A body of mass $m$ starts moving from rest along $x$-axis so that its velocity varies as $v = a \sqrt { s }$ where $a$ is a constant and $s$ is the distance covered by the body. The total work done by all the forces acting on the body in the first $t$ second after the start of the motion is
(1) $8 m a ^ { 4 } t ^ { 2 }$
(2) $\frac { 1 } { 4 } m a ^ { 4 } t ^ { 2 }$
(3) $4 m a ^ { 4 } t ^ { 2 }$
(4) $\frac { 1 } { 8 } m a ^ { 4 } t ^ { 2 }$

As shown in the figure, forces of $10 ^ { 5 } \mathrm {~N}$ each are applied in opposite directions, on the upper and lower faces of a cube of side 10 cm , shifting the upper face parallel to itself by 0.5 cm . If the side of another cube of the same material is, 20 cm then under similar conditions as above, the displacement will be: [Figure]
(1) 1.00 cm
(2) 0.25 cm
(3) 0.37 cm
(4) 0.75 cm

A thin circular disk is in the $x y$ plane as shown in the figure. The ratio of its moment of inertia about $z$ and $z'$ axes will be:
(1) $1 : 4$
(2) $1 : 5$
(3) $1 : 3$
(4) $1 : 2$

A disc rotates about its axis of symmetry in a horizontal plane at a steady rate of 3.5 revolutions per second. A coin placed at a distance of 1.25 cm from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is $\left( \mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$
(1) 0.5
(2) 0.7
(3) 0.3
(4) 0.6

A proton of mass $m$ collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of $90 ^ { \circ }$ with respect to each other. The mass of unknown particle is:
(1) $\frac { \mathrm { m } } { \sqrt { 3 } }$
(2) $\frac { \mathrm { m } } { 2 }$
(3) 2 m
(4) m

A thin rod MN , free to rotate in the vertical plane about the fixed end N , is held horizontal. When the end M is released the speed of this end, when the rod makes an angle $\alpha$ with the horizontal, will be proportional to: [Figure]
(1) $\sqrt { \cos \alpha }$
(2) $\cos \alpha$
(3) $\sin \alpha$
(4) $\sqrt { \sin \alpha }$

A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m moving in the same horizontal plane from opposite sides of the bar with speeds 2 v and $v$ respectively. The masses stick to the bar after collision at a distance $\frac { \mathrm { L } } { 3 }$ and $\frac { \mathrm { L } } { 6 }$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be: [Figure]
(1) $\frac { \mathrm { v } } { 6 \mathrm {~L} }$
(2) $\frac { 6 \mathrm { v } } { 5 \mathrm {~L} }$
(3) $\frac { 3 \mathrm { v } } { 5 \mathrm {~L} }$
(4) $\frac { \mathrm { v } } { 5 \mathrm {~L} }$

When an air bubble of radius $r$ rises from the bottom to the surface of a lake, its radius becomes $\frac { 5r } { 4 }$. Taking the atmospheric pressure to be equal to 10 m height of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature):
(1) 10.5 m
(2) 8.7 m
(3) 11.2 m
(4) 9.5 m

A body takes 10 minutes to cool from $60 ^ { \circ } \mathrm { C }$ to $50 ^ { \circ } \mathrm { C }$. The temperature of surroundings is constant at $25 ^ { \circ } \mathrm { C }$. Then, the temperature of the body after next 10 minutes will be approximately
(1) $43 ^ { \circ } \mathrm { C }$
(2) $47 ^ { \circ } \mathrm { C }$
(3) $41 ^ { \circ } \mathrm { C }$
(4) $45 ^ { \circ } \mathrm { C }$

An oscillator of mass $M$ is at rest in its equilibrium position in a potential, $V = \frac { 1 } { 2 } k ( x - X ) ^ { 2 }$. A particle of mass $m$ comes from the right with speed $u$ and collides completely inelastic with $M$ and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is: ( $M = 10 , m = 5 , u = 1 , k = 1$ )
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) $\sqrt { \frac { 3 } { 5 } }$
(4) $\frac { 1 } { 2 }$

Two Carnot engines A and B are operated in series. Engine A receives heat from a reservoir at 600 K and rejects heat to a reservoir at temperature T . Engine B receives heat rejected by engine A and in turn rejects it to a reservoir at 100 K . If the efficiencies of the two engines A and B are represented by $\eta _ { A }$ and $\eta _ { B }$ respectively, then what is the value of $\frac { \eta _ { \mathrm { A } } } { \eta _ { \mathrm { B } } }$
(1) $\frac { 12 } { 7 }$
(2) $\frac { 12 } { 5 }$
(3) $\frac { 5 } { 12 }$
(4) $\frac { 7 } { 12 }$

A particle executes simple harmonic motion and it is located at $x = a , b$ and $c$ at time $t _ { 0 } , 2 t _ { 0 }$ and $3 t _ { 0 }$ respectively. The frequency of the oscillation is:
(1) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { a + c } { 2 b } \right)$
(2) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { a + 2 b } { 3 c } \right)$
(3) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { a + b } { 2 c } \right)$
(4) $\frac { 1 } { 2 \pi t _ { 0 } } \cos ^ { - 1 } \left( \frac { 2 a + 3 c } { b } \right)$

The value closest to the thermal velocity of a Helium atom at room temperature (300 K) in $\mathrm { ms } ^ { - 1 }$ is: $\left[ \mathrm { k } _ { \mathrm { B } } = 1.4 \times 10 ^ { - 23 } \mathrm {~J} / \mathrm { K } ; \mathrm { m } _ { \mathrm { He } } = 7 \times 10 ^ { - 27 } \mathrm {~kg} \right]$
(1) $1.3 \times 10 ^ { 4 }$
(2) $1.3 \times 10 ^ { 5 }$
(3) $1.3 \times 10 ^ { 2 }$
(4) $1.3 \times 10 ^ { 3 }$

5 beats/ second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m . The frequency of the fork will be:
(1) 195 Hz
(2) 251 Hz
(3) 150 Hz
(4) 300 Hz

A parallel plate capacitor with area $200 \mathrm {~cm} ^ { 2 }$ and separation between the plates 1.5 cm , is connected across a battery of emf V . If the force of attraction between the plates is $25 \times 10 ^ { - 6 } \mathrm {~N}$, the value of $V$ is approximately: $$\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \frac { \mathrm { C } ^ { 2 } } { \mathrm {~N} . \mathrm { m } } \right) ^ { 2 } \right)$$ (1) 150 V
(2) 100 V
(3) 250 V
(4) 300 V

A capacitor $\mathrm { C } _ { 1 }$ is charged up to a voltage $\mathrm { V } = 60 \mathrm {~V}$ by connecting it to battery B through switch (1). Now $\mathrm { C } _ { 1 }$ is disconnected from battery and connected to a circuit consisting of two uncharged capacitors $\mathrm { C } _ { 2 } = 3.0 \mu \mathrm {~F}$ and $\mathrm { C } _ { 3 } = 6.0 \mu \mathrm {~F}$ through a switch (2) as shown in the figure. The sum of final charges on $\mathrm { C } _ { 2 }$ and $\mathrm { C } _ { 3 }$ is: [Figure]
(1) $36 \mu \mathrm { C }$
(2) $20 \mu \mathrm { C }$
(3) $54 \mu \mathrm { C }$
(4) $40 \mu \mathrm { C }$

A constant voltage is applied between two ends of a metallic wire. If the length is halved and the radius of the wire is doubled, the rate of heat developed in the wire will be:
(1) Increased 8 times
(2) Doubled
(3) Halved
(4) Unchanged

A current of 1 A is flowing on the sides of an equilateral triangle of side $4.5 \times 10 ^ { - 2 } \mathrm {~m}$. The magnetic field at the centre of the triangle will be:
(1) $4 \times 10 ^ { - 5 } \mathrm {~Wb} / \mathrm { m } ^ { 2 }$
(2) Zero
(3) $2 \times 10 ^ { - 5 } \mathrm {~Wb} / \mathrm { m } ^ { 2 }$
(4) $8 \times 10 ^ { - 5 } \mathrm {~Wb} / \mathrm { m } ^ { 2 }$

A copper rod of mass m slides under gravity on two smooth parallel rails, with separation $l$ and set at an angle of $\theta$ with the horizontal. At the bottom, rails are joined by a resistance $R$. There is a uniform magnetic field $B$ normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is: [Figure]
(1) $\frac { \mathrm { mgR } \cos \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$
(2) $\frac { \mathrm { mgR } \sin \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$
(3) $\frac { \mathrm { mgR } \tan \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$
(4) $\frac { \mathrm { mgR } \cot \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$

At the centre of a fixed large circular coil of radius R , a much smaller circular coil of radius $r$ is placed. The two coils are concentric and are in the same plane. The larger coil carries a current I. The smaller coil is set to rotate with a constant angular velocity $\omega$ about an axis along their common diameter. Calculate the emf induced in the smaller coil after a time $t$ of its start of rotation.
(1) $\frac { \mu _ { 0 } I } { 2 R } \omega r ^ { 2 } \sin \omega t$
(2) $\frac { \mu _ { 0 } I } { 4 R } \omega \pi r ^ { 2 } \sin \omega t$
(3) $\frac { \mu _ { 0 } I } { 2 R } \omega \pi r ^ { 2 } \sin \omega t$
(4) $\frac { \mu _ { 0 } \mathrm { I } } { 4 \mathrm { R } } \omega \mathrm { r } ^ { 2 } \sin \omega \mathrm { t }$

A plane polarized monochromatic EM wave is travelling in a vacuum along $z$ direction such that at $\mathrm { t } = \mathrm { t } _ { 1 }$ it is found that the electric field is zero at a spatial point $z _ { 1 }$. The next zero that occurs in its neighbourhood is at $z _ { 2 }$. The frequency of the electromagnetic wave is:
(1) $\frac { 3 \times 10 ^ { 8 } } { \left| z _ { 2 } - z _ { 1 } \right| }$
(2) $\frac { 6 \times 10 ^ { 8 } } { \left| z _ { 2 } - z _ { 1 } \right| }$
(3) $\frac { 1.5 \times 10 ^ { 8 } } { \left| z _ { 2 } - z _ { 1 } \right| }$
(4) $\frac { 1 } { t _ { 1 } + \frac { \left| z _ { 2 } - z _ { 1 } \right| } { 3 \times 10 ^ { 8 } } }$

A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of 10 cm . The separation between the two lenses is 2 cm . The focal lengths of the component lenses are:
(1) $18 \mathrm {~cm} , 20 \mathrm {~cm}$
(2) $10 \mathrm {~cm} , 12 \mathrm {~cm}$
(3) $12 \mathrm {~cm} , 14 \mathrm {~cm}$
(4) $16 \mathrm {~cm} , 18 \mathrm {~cm}$

If the de Broglie wavelengths associated with a proton and an $\alpha$-particle are equal, then the ratio of velocities of the proton and the $\alpha$-particle will be:
(1) $1 : 4$
(2) $1 : 2$
(3) $4 : 1$
(4) $2 : 1$