A mass of \(10 \,kg\) is suspended vertically by a rope of length \(5 \,m\) from the roof. A force of \(30 \,N\) is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is \(\theta=\tan ^{-1}\left(x \times 10^{-1}\right)\). The value of \(x\) is ................ \(\text { (Given } g =10 \,m / s ^{2} \text { ) }\)

A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio \(2:2:3\). The two pieces of equal masses fly off perpendicular to each other with a speed of \(18~m/s\) each. The velocity of the heavier fragment is ___________ \(m/s\).

A small metallic sphere of diameter 2 mm and density \(10.5 g / cm ^3\) is dropped in glycerine having viscosity 10 Poise and density \(1.5 g / cm ^3\) respectively. The terminal velocity attained by the sphere is ___________ \(cm / s\).
\(\left(\pi=\frac{22}{7}\right.\) and \(\left.g=10 m / s ^2\right)\)

A point object in air is in front of the curved surface of a plano-convex lens. The radius of curvature of the curved surface is \(30 \;\mathrm{cm}\) and the refractive index of the lens material is \(1.5 .\) then the focal length of the lens (in \(\mathrm{cm}\) ) is

Which of the following Maxwell's equations is valid for time varying conditions but not valid for static conditions

A liquid of density \(600\) kg/m\(^3\) flowing steadily in a tube of varying cross-section. The cross-section at a point \(A\) is \(1.0\) cm\(^2\) and that at \(B\) is \(20\) mm\(^2\). Both the points \(A\) and \(B\) are in same horizontal plane, the speed of the liquid at \(A\) is \(10\) cm/s. The difference in pressures at \(A\) and \(B\) points is ________ Pa.

The \(X-Y\) plane be taken as the boundary between two transparent media \(M _{1}\) and \(M _{2} . M _{1}\) in \(Z \geq 0\) has a refractive index of \(\sqrt{2}\) and \(M _{2}\) with \(Z <0\) has a refractive index of \(\sqrt{3}\). A ray of light travelling in \(M_{1}\) along the direction given by the vector \(\overrightarrow{ A }=4 \sqrt{3 i }-3 \sqrt{3 \hat{j}}-5 \hat{ k }\), is incident on the plane of separation. The value of difference between the angle of incident in \(M _{1}\) and the angle of refraction in \(M _{2}\) will be \(....\) degree.

Two different wires having lengths \(L _{1}\) and \(L _{2}\) and respective temperature coefficient of linear expansion \(\alpha_{1}\) and \(\alpha_{2},\) are joined end-to-end. Then the effective temperature coefficient of linear expansion is

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at \(47^{\circ} \mathrm{C}\) ?

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 .............

A body moves on a frictionless plane starting from rest. If \(\mathrm{S}_{\mathrm{n}}\) is distance moved between \(\mathrm{t}=\mathrm{n}-1\) and \(\mathrm{t}\) \(=\mathrm{n}\) and \(\mathrm{S}_{\mathrm{n}-1}\) is distance moved between \(\mathrm{t}=\mathrm{n}-2\) and \(t=n-1\), then the ratio \(\frac{S_{n-1}}{S_n}\) is \(\left(1-\frac{2}{x}\right)\) for \(n\) \(=10\). The value of \(x\) is _______.