A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta ,\) where \(\theta \) is the angle by which it has rotated, is given as \(k\theta ^2.\) If its moment of inertia is \(I\) then the angular acceleration of the disc is

A transformer operating at primary voltage \(8\,kV\) and secondary voltage \(160\,V\) serves a load of \(80\,kW\). Assuming the transformer to be ideal with purely resistive load and working on unity power factor, the loads in the primary and secondary circuit would be.

A transparent film of refractive index, 2.0 is coated on a glass slab of refractive index, 1.45. What is the minimum thickness of transparent film to be coated for the maximum transmission of Green light of wavelength 550 nm . [Assume that the light is incident nearly perpendicular to the glass surface.]

In Young's double slit experiment, monochromatic light of wavelength \(5000\) \(A\) is used. The slits are \(1.0 \mathrm{~mm}\) apart and screen is placed at \(1.0 \mathrm{~m}\) away from slits. The distance from the centre of the screen where intensity becomes half of the maximum intensity for the first time is _______ \(10^{-6} \mathrm{~m}\).

In a long glass tube, mixture of two liquids A and B with refractive indices 1.3 and 1.4 respectively, forms a convex refractive meniscus towards \(A\). If an object placed at 13 cm from the vertex of the meniscus in A forms an image with a magnification of ' \(-2^{\prime}\) then the radius of curvature of meniscus is :

Identify the correct statements from the following: \((A)\) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative. \((B)\) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative. \((C)\) Work done by friction on a body sliding down an inclined plane is positive. \((D)\) Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity in zero. \((E)\) Work done by the air resistance on an oscillating pendulum in negative. Choose the correct answer from the options given below:

The mean free path of molecules of a certain gas at \(STP\) is \(1500\,d\), where \(d\) is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at \(373\,K\) is approximately \(..........\,d\)

For the hyperbola \(H : x ^{2}- y ^{2}=1\) and the ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b>0\), let the \((1)\) eccentricity of \(E\) be reciprocal of the eccentricity of \(H\), and \((2)\) the line \(y=\sqrt{\frac{5}{2}} x+K\) be a common tangent of \(E\) and \(H\) Then \(4\left(a^{2}+b^{2}\right)\) is equal to

If \(A\, = \,\left[ {\begin{array}{*{20}{c}}
1&2&x\\
3&{ - 1}&2
\end{array}} \right]\) and \(B\, = \,\left[ {\begin{array}{*{20}{c}}
y\\
x\\
1
\end{array}} \right]\) be such that \(AB\, = \,\left[ {\begin{array}{*{20}{c}}
6\\
8
\end{array}} \right],\) then

A rubber ball is released from a height of \(5\, m\) above the floor. It bounces back repeatedly, always rising to \(\frac{81}{100}\) of the height through which it falls. Find the average speed of the ball. (Take \(g =10 ms ^{-2}\) ) (in \(ms ^{-1}\))

Let \(S\) and \(S\,'\) be the foci of an ellipse and \(B\) be any one of the extremities of its minor axis. If \(\Delta S\,'BS\) is a right angled triangle with right angle at \(B\) and area \((\Delta S\,'BS) = 8\,sq.\) units, then the length of a latus rectum of the ellipse is

Let \(A(6,8), B(10 \cos \alpha,-10 \sin \alpha)\) and \(C(-10 \sin \alpha, 10 \cos \alpha)\), be the vertices of a triangle. If \(L(a, 9)\) and \(G(h, k)\) be its orthocenter and centroid respectively, then \((5 a-3 h+6 k+100 \sin 2 \alpha)\) is equal to ______ -.

Consider the function \(\mathrm{f}:(0,2) \rightarrow \mathrm{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function\( g(x)\) defined by \(g(x)=\left\{\begin{array}{cc}\min \{f(t)\}, & 0 < t \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x< 2\end{array}\right.\). Then