Statement 1: A function $f: R \rightarrow R$ is continuous at $x_{0}$ if and only if $\lim_{x \rightarrow x_{0}} f(x)$ exists and $\lim_{x \rightarrow x_{0}} f(x) = f(x_{0})$. Statement 2: A function $f: R \rightarrow R$ is discontinuous at $x_{0}$ if and only if, $\lim_{x \rightarrow x_{0}} f(x)$ exists and $\lim_{x \rightarrow x_{0}} f(x) \neq f(x_{0})$.
(1) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
(4) Statement 1 is true, Statement 2 is false.

Two springs of force constants $300 \mathrm {~N} / \mathrm { m }$ (Spring A) and $400 \mathrm {~N} / \mathrm { m }$ (Spring B) are joined together in series. The combination is compressed by 8.75 cm . The ratio of energy stored in A and B is $\frac { E _ { A } } { E _ { B } }$. Then $\frac { E _ { A } } { E _ { B } }$ is equal to:
(1) $\frac { 4 } { 3 }$
(2) $\frac { 16 } { 9 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 9 } { 16 }$

A bullet of mass 10 g and speed $500 \mathrm {~m} / \mathrm { s }$ is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg . It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds into it will be :
(1) $6.25 \mathrm { rad } / \mathrm { sec }$
(2) $0.625 \mathrm { rad } / \mathrm { sec }$
(3) $3.35 \mathrm { rad } / \mathrm { sec }$
(4) $0.335 \mathrm { rad } / \mathrm { sec }$

A uniform sphere of weight $W$ and radius 5 cm is being held by a string as shown in the figure. The tension in the string will be :
(1) $12 \frac { \mathrm {~W} } { 5 }$
(2) $5 \frac { \mathrm {~W} } { 12 }$
(3) $13 \frac { \mathrm {~W} } { 5 }$
(4) $13 \frac { \mathrm {~W} } { 12 }$

A ring of mass $M$ and radius $R$ is rotating about its axis with angular velocity $\omega$. Two identical bodies each of mass $m$ are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be:
(1) $\frac { m ( M + 2 m ) } { M } \omega ^ { 2 } R ^ { 2 }$
(2) $\frac { M m } { ( M + m ) } \omega ^ { 2 } R ^ { 2 }$
(3) $\frac { M m } { ( M + 2 m ) } \omega ^ { 2 } R ^ { 2 }$
(4) $\frac { ( M + m ) M } { ( M + 2 m ) } \omega ^ { 2 } R ^ { 2 }$

The gravitational field, due to the 'left over part' of a uniform sphere (from which a part as shown, has been 'removed out'), at a very far off point, P , located as shown, would be (nearly) :
(1) $\frac { 5 } { 6 } \frac { G M } { x ^ { 2 } }$
(2) $\frac { 8 } { 9 } \frac { G M } { x ^ { 2 } }$
(3) $\frac { 7 } { 8 } \frac { G M } { x ^ { 2 } }$
(4) $\frac { 6 } { 7 } \frac { G M } { x ^ { 2 } }$

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are $a , b$ and $c$ respectively, then the corresponding ratio of increase in their lengths is :
(1) $\frac { 3 c } { 2 a b ^ { 2 } }$
(2) $\frac { 2 a ^ { 2 } c } { b }$
(3) $\frac { 3 a } { 2 b ^ { 2 } c }$
(4) $\frac { 2 a c } { b ^ { 2 } }$

On a linear temperature scale Y , water freezes at $- 160 ^ { \circ } \mathrm { Y }$ and boils at $- 50 ^ { \circ } \mathrm { Y }$. On this Y scale, a temperature of 340 K would be read as : (water freezes at 273 K and boils at 373 K )
(1) $- 73.7 ^ { \circ } \mathrm { Y }$
(2) $- 233.7 ^ { \circ } \mathrm { Y }$
(3) $- 86.3 ^ { \circ } \mathrm { Y }$
(4) $- 106.3 ^ { \circ } \mathrm { Y }$

In the isothermal expansion of 10 g of gas from volume V to 2V the work done by the gas is 575 J. What is the root mean square speed of the molecules of the gas at that temperature?
(1) $398 \mathrm {~m} / \mathrm { s }$
(2) $520 \mathrm {~m} / \mathrm { s }$
(3) $499 \mathrm {~m} / \mathrm { s }$
(4) $532 \mathrm {~m} / \mathrm { s }$

Two simple pendulums of length 1 m and 4 m respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to:
(1) 2
(2) 7
(3) 5
(4) 3

In a transverse wave the distance between a crest and neighbouring trough at the same instant is 4.0 cm and the distance between a crest and trough at the same place is 1.0 cm. The next crest appears at the same place after a time interval of 0.4 s. The maximum speed of the vibrating particles in the medium is:
(1) $\frac { 3\pi } { 2 } \mathrm {~cm} / \mathrm { s }$
(2) $\frac { 5\pi } { 2 } \mathrm {~cm} / \mathrm { s }$
(3) $\frac { \pi } { 2 } \mathrm {~cm} / \mathrm { s }$
(4) $2\pi \mathrm {~cm} / \mathrm { s }$

When two sound waves travel in the same direction in a medium, the displacements of a particle located at ' $x$ ' at time ' $t$ ' is given by : $\begin{aligned} & y _ { 1 } = 0.05 \cos ( 0.50 \pi x - 100 \pi t ) \\ & y _ { 2 } = 0.05 \cos ( 0.46 \pi x - 92 \pi t ) \end{aligned}$ where $y _ { 1 } , y _ { 2 }$ and $x$ are in meters and $t$ in seconds. The speed of sound in the medium is :
(1) $92 \mathrm {~m} / \mathrm { s }$
(2) $200 \mathrm {~m} / \mathrm { s }$
(3) $100 \mathrm {~m} / \mathrm { s }$
(4) $332 \mathrm {~m} / \mathrm { s }$

An engine approaches a hill with a constant speed. When it is at a distance of 0.9 km , it blows a whistle whose echo is heard by the driver after 5 seconds. If the speed of sound in air is $330 \mathrm {~m} / \mathrm { s }$, then the speed of the engine is :
(1) $32 \mathrm {~m} / \mathrm { s }$
(2) $27.5 \mathrm {~m} / \mathrm { s }$
(3) $60 \mathrm {~m} / \mathrm { s }$
(4) $30 \mathrm {~m} / \mathrm { s }$

The surface charge density of a thin charged disc of radius R is $\sigma$. The value of the electric field at the centre of the disc is $\frac { \sigma } { 2\epsilon _ { 0 } }$. With respect to the field at the centre, the electric field along the axis at a distance R from the centre of the disc:
(1) reduces by $70.7\%$
(2) reduces by $29.3\%$
(3) reduces by $9.7\%$
(4) reduces by $14.6\%$

A and B are two sources generating sound waves. A listener is situated at C. The frequency of the source at A is 500 Hz. A, now, moves towards C with a speed $4\mathrm{~m/s}$. The number of beats heard at C is 6. When A moves away from C with speed $4\mathrm{~m/s}$, the number of beats heard at C is 18. The speed of sound is $340\mathrm{~m/s}$. The frequency of the source at B is:
(1) 500 Hz
(2) 506 Hz
(3) 512 Hz
(4) 494 Hz

The gravitational field in a region is given by: $\vec { E } = ( 5 \mathrm{N/kg} ) \hat { i } + ( 12 \mathrm{N/kg} ) \hat { j }$. If the potential at the origin is taken to be zero, then the ratio of the potential at the points $( 12 \mathrm {~m} , 0 )$ and $( 0,5 \mathrm {~m} )$ is:
(1) Zero
(2) 1
(3) $\frac { 144 } { 25 }$
(4) $\frac { 25 } { 144 }$

Two balls of same mass and carrying equal charge are hung from a fixed support of length $l$. At electrostatic equilibrium, assuming that angles made by each thread is small, the separation, $x$ between the balls is proportional to :
(1) $l$
(2) $l ^ { 2 }$
(3) $l ^ { 2 / 3 }$
(4) $l ^ { 1 / 3 }$

A parallel plate capacitor having a separation between the plates d, plate area A and material with dielectric constant K has capacitance $\mathrm { C } _ { 0 }$. Now one-third of the material is replaced by another material with dielectric constant 2K, so that effectively there are two capacitors one with area $\frac { 1 } { 3 } \mathrm {~A}$, dielectric constant 2K and another with area $\frac { 2 } { 3 } \mathrm {~A}$ and dielectric constant K. If the capacitance of this new capacitor is C then $\frac { \mathrm { C } } { \mathrm { C } _ { 0 } }$ is
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

In a metre bridge experiment null point is obtained at 40 cm from one end of the wire when resistance $\mathbf { X }$ is balanced against another resistance Y . If $\mathrm { X } < \mathrm { Y }$, then the new position of the null point from the same end, if one decides to balance a resistance of 3 X against Y , will be close to :
(1) 80 cm
(2) 75 cm
(3) 67 cm
(4) 50 cm

A letter ' $\mathrm { A } ^ { \prime }$ is constructed of a uniform wire with resistance $1.0 \Omega$ per cm , The sides of the letter are 20 cm and the cross piece in the middle is 10 cm long. The apex angle is 60 . The resistance between the ends of the legs is close to:
(1) $50.0 \Omega$
(2) $10 \Omega$
(3) $36.7 \Omega$
(4) $26.7 \Omega$

A shunt of resistance $1 \Omega$ is connected across a galvanometer of $120 \Omega$ resistance. A current of 5.5 ampere gives full scale deflection in the galvanometer. The current that will give full scale deflection in the absence of the shunt is nearly :
(1) 5.5 ampere
(2) 0.5 ampere
(3) 0.004 ampere
(4) 0.045 ampere

An electric current is flowing through a circular coil of radius $R$. The ratio of the magnetic field at the centre of the coil and that at a distance $2 \sqrt { 2 } R$ from the centre of the coil and on its axis is :
(1) $2 \sqrt { 2 }$
(2) 27
(3) 36
(4) 8

A current $i$ is flowing in a straight conductor of length $L$. The magnetic induction at a point on its axis at a distance $\frac{L}{4}$ from its centre will be:
(1) Zero
(2) $\frac{\mu_0 i}{2\pi L}$
(3) $\frac{\mu_0 i}{\sqrt{2}L}$
(4) $\frac{4\mu_0 i}{\sqrt{5}\pi L}$

A series LR circuit is connected to an ac source of frequency $\omega$ and the inductive reactance is equal to $2R$. A capacitance of capacitive reactance equal to $R$ is added in series with $L$ and $R$. The ratio of the new power factor to the old one is:
(1) $\sqrt { \frac { 2 } { 3 } }$
(2) $\sqrt { \frac { 2 } { 5 } }$
(3) $\sqrt { \frac { 3 } { 2 } }$
(4) $\sqrt { \frac { 5 } { 2 } }$

In a series L-C-R circuit, $C = 10^{-11}$ Farad, $L = 10^{-5}$ Henry and $R = 100$ Ohm, when a constant D.C. voltage E is applied to the circuit, the capacitor acquires a charge $10^{-9}\mathrm{~C}$. The D.C. source is replaced by a sinusoidal voltage source in which the peak voltage $E_0$ is equal to the constant D.C. voltage E. At resonance the peak value of the charge acquired by the capacitor will be:
(1) $10^{-15}\mathrm{~C}$
(2) $10^{-6}\mathrm{~C}$
(3) $10^{-10}\mathrm{~C}$
(4) $10^{-8}\mathrm{~C}$