A long straight wire of radius a carries a steady current I. The current is uniformly distributed across its cross section. The ratio of the magnetic field at $$\frac{a}{2}$$ and $$2 a$$ from axis of the wire is :

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is :

The position of the image formed by the combination of lenses is :

If $$\epsilon_{\mathrm{o}}$$ is the permittivity of free space and $$\mathrm{E}$$ is the electric field, then $$\epsilon_{\mathrm{o}} \mathrm{E}^2$$ has the dimensions :

Two satellite A and B go round a planet in circular orbits having radii 4R and R respectively. If the speed of $$\mathrm{A}$$ is $$3 v$$, the speed of $$\mathrm{B}$$ will be :

Water boils in an electric kettle in 20 minutes after being switched on. Using the same main supply, the length of the heating element should be _________ to __________ times of its initial length if the water is to be boiled in 15 minutes.

A diatomic gas $$(\gamma=1.4)$$ does $$100 \mathrm{~J}$$ of work in an isobaric expansion. The heat given to the gas is :

If $$M_0$$ is the mass of isotope $${ }_5^{12} B, M_p$$ and $$M_n$$ are the masses of proton and neutron, then nuclear binding energy of isotope is:

A given object takes $$\mathrm{n}$$ times the time to slide down $$45^{\circ}$$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $$45^{\circ}$$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is :

There are 100 divisions on the circular scale of a screw gauge of pitch $$1 \mathrm{~mm}$$. With no measuring quantity in between the jaws, the zero of the circular scale lies 5 divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found that 4 linear scale divisions are clearly visible while 60 divisions on circular scale coincide with the reference line. The diameter of the wire is :

A capacitor has air as dielectric medium and two conducting plates of area $$12 \mathrm{~cm}^2$$ and they are $$0.6 \mathrm{~cm}$$ apart. When a slab of dielectric having area $$12 \mathrm{~cm}^2$$ and $$0.6 \mathrm{~cm}$$ thickness is inserted between the plates, one of the conducting plates has to be moved by $$0.2 \mathrm{~cm}$$ to keep the capacitance same as in previous case. The dielectric constant of the slab is : (Given $$\epsilon_0=8.834 \times 10^{-12} \mathrm{~F} / \mathrm{m}$$)

Least count of a vernier caliper is $$\frac{1}{20 \mathrm{~N}} \mathrm{~cm}$$. The value of one division on the main scale is $$1 \mathrm{~mm}$$. Then the number of divisions of main scale that coincide with $$\mathrm{N}$$ divisions of vernier scale is :

A cube of ice floats partly in water and partly in kerosene oil. The ratio of volume of ice immersed in water to that in kerosene oil (specific gravity of Kerosene oil = 0.8, specific gravity of ice = 0.9):

Given below are two statements :

Statement (I) : The mean free path of gas molecules is inversely proportional to square of molecular diameter.

Statement (II) : Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.

In the light of the above statements, choose the correct answer from the options given below :

The angle of projection for a projectile to have same horizontal range and maximum height is :

A proton and an electron have the same de Broglie wavelength. If $$\mathrm{K}_{\mathrm{p}}$$ and $$\mathrm{K}_{\mathrm{e}}$$ be the kinetic energies of proton and electron respectively, then choose the correct relation :

In a hypothetical fission reaction

$${ }_{92} X^{236} \rightarrow{ }_{56} \mathrm{Y}^{141}+{ }_{36} Z^{92}+3 R$$

The identity of emitted particles (R) is :

A thin circular disc of mass $$\mathrm{M}$$ and radius $$\mathrm{R}$$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $$\omega$$. If another disc of same dimensions but of mass $$\mathrm{M} / 2$$ is placed gently on the first disc co-axially, then the new angular velocity of the system is :

A coil of negligible resistance is connected in series with $$90 \Omega$$ resistor across $$120 \mathrm{~V}, 60 \mathrm{~Hz}$$ supply. A voltmeter reads $$36 \mathrm{~V}$$ across resistance. Inductance of the coil is :

A plane progressive wave is given by $$y=2 \cos 2 \pi(330 \mathrm{t}-x) \mathrm{m}$$. The frequency of the wave is :

An alternating emf $$\mathrm{E}=110 \sqrt{2} \sin 100 \mathrm{t}$$ volt is applied to a capacitor of $$2 \mu \mathrm{F}$$, the rms value of current in the circuit is ________ $$\mathrm{mA}$$.

If the net electric field at point $$\mathrm{P}$$ along $$\mathrm{Y}$$ axis is zero, then the ratio of $$\left|\frac{q_2}{q_3}\right|$$ is $$\frac{8}{5 \sqrt{x}}$$, where $$x=$$ ________.

Two slits are $$1 \mathrm{~mm}$$ apart and the screen is located $$1 \mathrm{~m}$$ away from the slits. A light of wavelength $$500 \mathrm{~nm}$$ is used. The width of each slit to obtain 10 maxima of the double slit pattern within the central maximum of the single slit pattern is __________ $$\times 10^{-4} \mathrm{~m}$$.

A potential divider circuit is connected with a dc source of $$20 \mathrm{~V}$$, a light emitting diode of glow in voltage $$1.8 \mathrm{~V}$$ and a zener diode of breakdown voltage of $$3.2 \mathrm{~V}$$. The length (PR) of the resistive wire is $$20 \mathrm{~cm}$$. The minimum length of PQ to just glow the LED is _________ $$\mathrm{cm}$$.

An object of mass $$0.2 \mathrm{~kg}$$ executes simple harmonic motion along $$x$$ axis with frequency of $$\left(\frac{25}{\pi}\right) \mathrm{Hz}$$. At the position $$x=0.04 \mathrm{~m}$$ the object has kinetic energy $$0.5 \mathrm{~J}$$ and potential energy $$0.4 \mathrm{~J}$$. The amplitude of oscillation is ________ $$\mathrm{cm}$$.

A circular table is rotating with an angular velocity of $$\omega \mathrm{~rad} / \mathrm{s}$$ about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of $$1 \mathrm{~m}$$ on the groove. All the surfaces are smooth. If the radius of the table is $$3 \mathrm{~m}$$, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $$x \sqrt{2} \omega \mathrm{~m} / \mathrm{s}$$, where the value of $$x$$ is _________.

A heater is designed to operate with a power of $$1000 \mathrm{~W}$$ in a $$100 \mathrm{~V}$$ line. It is connected in combination with a resistance of $$10 \Omega$$ and a resistance $$R$$, to a $$100 \mathrm{~V}$$ mains as shown in figure. For the heater to operate at $$62.5 \mathrm{~W}$$, the value of $$\mathrm{R}$$ should be _______ $$\Omega$$.

Small water droplets of radius $$0.01 \mathrm{~mm}$$ are formed in the upper atmosphere and falling with a terminal velocity of $$10 \mathrm{~cm} / \mathrm{s}$$. Due to condensation, if 8 such droplets are coalesced and formed a larger drop, the new terminal velocity will be ________ $$\mathrm{cm} / \mathrm{s}$$.

The coercivity of a magnet is $$5 \times 10^3 \mathrm{~A} / \mathrm{m}$$. The amount of current required to be passed in a solenoid of length $$30 \mathrm{~cm}$$ and the number of turns 150, so that the magnet gets demagnetised when inside the solenoid is ________ A.

A body of mass M thrown horizontally with velocity v from the top of the tower of height H touches the ground at a distance of $$100 \mathrm{~m}$$ from the foot of the tower. A body of mass $$2 \mathrm{~M}$$ thrown at a velocity $$\frac{v}{2}$$ from the top of the tower of height $$4 \mathrm{H}$$ will touch the ground at a distance of _______ m.