A direct current of \(4\,A\) and an alternating current of peak value \(4\,A\) flow through resistance of \(3\,\Omega\) and \(2\,\Omega\) respectively. The ratio of heat produced in the two resistances in same interval of time will be.

Given below are two statements: one is labelled as Assertion \(\mathbf{A}\) and the other is labelled as Reason \(\mathbf{R}\)
Assertion A: In a central force field, the work done is independent of the path chosen.
Reason R: Every force encountered in mechanics does not have an associated potential energy.
In the light of the above statements, choose the most appropriate answer from the options given below

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A :\) An electron microscope can achieve better resolving power than an optical microscope. Reason \(R\) : The de Broglie's wavelength of the electrons emitted from an electron gun is much less than wavelength of visible light. In the light of the above statements, choose the correct answer from the options given below

A particle performs simple harmonic motion with a period of \(2\) second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is \(\frac{1}{ a } s .\) The value of \('a'\) to the nearest integer is.........

The force required to stretch a wire of crosssection \(1 cm ^{2}\) to double its length will be ........ \( \times 10^{7}\,N\) (Given Yong's modulus of the wire \(=2 \times 10^{11}\,N / m ^{2}\) )

A solid sphere (\(A\)) of mass \(5m\) and a spherical shell (\(B\)) of mass \(m\), both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of \(A\) and \(B\), they start rolling without slipping with an acceleration of \(a_A\) and \(a_B\), respectively. The ratio of \(a_A\) and \(a_B\) is __________.

An object is placed at a distance of \(12\,cm\) in front of a plane mirror. The virtual and erect image is formed by the mirror. Now the mirror is moved by \(4\,cm\) towards the stationary object. The distance by which the position of image would be shifted, will be:

When a diode is forward biased, it has a voltage drop of \(0.5\, V.\) The safe limit of current through the diode is \(10\, mA\). If a battery of emf \(1.5\, V\) is used in the circuit, the value of minimum resistance to be connected in series with the diode so that the current does not exceed the safe limit is\(.....\Omega\)

The gravitational potential at a point above the surface of earth is \(-5.12 \times 10^7 \mathrm{~J} / \mathrm{kg}\) and the acceleration due to gravity at that point is \(6.4 \mathrm{~m} / \mathrm{s}^2\). Assume that the mean radius of earth to be \(6400 \mathrm{~km}\). The height of this point above the earth's surface is _______.

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in\(P-V\) diagram. The relation between the ratio \(\frac{V_a}{V_d}\) and the ratio \(\frac{V_b}{V_c}\) is _______.

Match List \(I\) with List \(II\) :

List \(I\) (Physical Quantity)	List \(II\) (Dimensional Formula)
\((A)\) Pressure gradient	\((I)\) \(\left[ M ^0 L ^2 T ^{-2}\right]\)
\((B)\) Energy density	\((II)\) \(\left[ M ^1 L ^{-1} T ^{-2}\right]\)
\((C)\) Electric Field	\((III)\) \(\left[ M ^1 L ^{-2} T ^{-2}\right]\)
\((D)\) Latent heat	\((IV)\) \(\left[ M ^1 L ^1 T ^{-3} A ^{-1}\right]\)

Choose the correct answer from the options given below:

The total charge on the system of capacitance \(C _{1}=1\,\mu F , C _{2}=2\,\mu F , C _{3}=4\,\mu F\) and \(C _{4}=3\,\mu F\) connected in parallel is \(......\mu C\) (Assume a battery of \(20\,V\) is connected to the combination)

A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of \(2 \times 10^5 \mathrm{~ms}^{-1}\). When the electric field is switched off, the proton moves along a circular path of radius 2 cm . The magnitude of electric field is \(x \times 10^4 \mathrm{~N} / \mathrm{C}\). The value of \(x\) is _______ Take the mass of the proton \(=1.6 \times 10^{-27} \mathrm{~kg}\).