A charge \(Q\) is placed at a distance \(a/2\) above the centre of the square surface of edge \(a\) as shown in the figure. The electric flux through the square surface is

\(n\) polarizing sheets are arranged such that each makes an angle \(45^{\circ}\) with the proceeding sheet. An unpolarized light of intensity \(I\) is incident into this arrangement. The output intensity is found to be \(\frac{I}{64}\). The value of \(n\) will be:

Consider a frame that is made up of two thin massless rods \(AB\) and \(AC\) as shown in the figure. \(A\) vertical force \(\overrightarrow{ P }\) of magnitude \(100 \;N\) is applied at point \(A\) of the frame. Suppose the force is \(\overrightarrow{ P }\) resolved parallel to the arms \(AB\) and \(AC\) of the frame. The magnitude of the resolved component along the arm \(AC\) is \(xN\). The value of \(x\), to the nearest integer, is ............ [Given : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)  

A mass \(m = 1.0\,kg\) is put on a flat pan attached to a vertical spring fixed on the ground. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes simple harmonic motion. The spring constant is \(500\,N/m.\) What is the amplitude \(A\) of the motion, so that the mass \(m\) tends to get detached from the pan ? (Take \(g = 10\,m/s^2\) ). The spring is stiff enough so that it does not get distorted during the motion.

If two vectors \(\vec{P}=\hat{i}+2 m \hat{j}+m \hat{k}\) and \(\vec{Q}=4 \hat{i}-2 \hat{j}+ mk\) are perpendicular to each other. Then, the value of \(m\) will be :

In a hydraulic lift, the surface area of the input piston is \(6 \mathrm{~cm}^2\) and that of the output piston is \(1500 \mathrm{~cm}^2\). If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is _______ kJ.

A metallic conductor of length \(1\; m\) rotates in a vertical plane parallel to east-west direction about one of its end with angular velocity \(5 \;rad / s\). If the horizontal component of earth's magnetic field is \(0.2 \times 10^{-4} \;T\), then emf induced between the two ends of the conductor is .............

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is \(R_0= 10\, m\). What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity \(v = 20\, m/s\), on a horizontal surface ......... \(^o.\) \((g = 10\, m/s^2)\)

The height of any point \(P\) above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point \(P\) will be : (Given \(g=\) acceleration due to gravity at the surface of earth)

An electron in a hydrogen atom revolves around its nucleus with a speed of \(6.76 \times 10^6\,ms ^{-1}\) in an orbit of radius \(0.52\,\mathring A\). The magnetic field produced at the nucleus of the hydrogen atom is \(......T\).