A body of mass ' \({m}\) ' dropped from a height ' \({h}\) ' reaches the ground with a speed of \(0.8 \sqrt{{gh}}\). The value of workdone by the air-friction is \(.....\,{mgh}\)

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

A reversible engine has an efficiency of \(\frac{1}{4}\). If the temperature of the sink is reduced by \(58^{\circ} {C}\), its efficiency becomes double. Calculate the temperature of the sink. (In \(^{\circ} {C}\))

One end of a metal wire is fixed to a ceiling and a load of \(2 \mathrm{~kg}\) hangs from the other end. A similar wire is attached to the bottom of the load and another load of \(1 \mathrm{~kg}\) hangs from this lower wire. Then the ratio of longitudinal strain of upper wire to that of the lower wire will be _______. [Area of cross section of wire \(=0.005 \mathrm{~cm}^2\), \(\mathrm{Y}=2 \times 10^{11}\  \mathrm{Nm}^{-2}\) and \(\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right]\)

An electron rotates in a circle around a nucleus having positive charge \(\mathrm{Ze}\). Correct relation between total energy ( \(E\) ) of electron to its potential energy \((U)\) is _______.

The de Broglie wavelength of a proton and \(\alpha\) -particle are equal. The ratio of their velocities is ...... .

A satellite is in an elliptical orbit around a planet \(P.\) It is observed that the velocity of the satellite when it is farthest from the planet is \(6\) time less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is\(......\)

\(B _{ X }\) and \(B _{ Y }\) are the magnetic field at the centre of two coils of two coils \(X\) and \(Y\) respectively, each carrying equal current. If coil \(X\) has \(200\) turns and \(20 cm\) radius and coil \(Y\) has \(400\) turns and \(20 cm\) radius, the ratio of \(B _{ X }\) and \(B _{ Y }\) is

The point of intersection of the normals to the parabola \(y^ 2\, = 4x\) at the ends of its latus rectum is 

Let \(S=\{1,2,3,4,5,6\} .\) Then the probability that a randomly chosen onto function \(\mathrm{g}\) from \(\mathrm{S}\) to \(\mathrm{S}\) satisfies \(g(3)=2 g(1)\) is :

The magnetic field in a plane electromagnetic wave is \(\mathrm{B}_y=\left(3.5 \times 10^{-7}\right) \sin \left(1.5 \times 10^3 \mathrm{x}+0.5\right.\) \(\left.\times 10^{11} \mathrm{t}\right) \mathrm{T}\). The corresponding electric field will be _______.

A helicopter flying horizontally with a speed of \(360 \mathrm{~km} / \mathrm{h}\) at an altitude of 2 km , drops an object at an instant. The object hits the ground at a point O , 20 s after it is dropped. Displacement of ' O ' from the position of helicopter where the object was released is :
(use acceleration due to gravity \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) and neglect air resistance)

A concave-convex lens of refractive index 1.5 and the radii of curvature of its surfaces are 30 cm and 20 cm , respectively. The concave surface is upwards and is filled with a liquid of refractive index 1.3. The focal length of the liquid-glass combination will be _______.