A pendulum of length \(2 \; m\) consists of a wooden bob of mass \(50 \; g\). A bullet of mass \(75 \; g\) is fired towards the stationary bob with a speed \(v\). The bullet emerges out of the bob with a speed \(\frac{v}{3}\) and the bob just completes the vertical circle. The value of \(v\) is \(\dots \; ms ^{-1}\). (if \(g =10 \; m / s ^{2}\) )

Two charged conducting spheres of radii \(a\) and \(b\) are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is _______.

If \(V\) is the gravitational potential due to sphere of uniform density on it's surface, then it's value at the center of sphere will be:-

In terms of resistance \(R\) and time \(T\), the dimensions of ratio \(\frac{\mu } {\varepsilon }\) of the permeability \(\mu \) and permittivity \(\varepsilon \)  is

\(\mathrm{A}, \mathrm{B}\) and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about \(P Q\) is \(\frac{X}{15} I\), where \(I\) is the moment of inertia of the disc about its diameter. The value of x is _______.

Two capacitors of capacitances \(C\) and \(2\, C\) are charged to potential differences \(V\) and \(2\, V\), respectively. These are then connected in parallel in such a manner that the positive terminal of one is connected to the negative terminal of the other. The final energy of this configuration is\(.....CV^2\)

The explosive in a Hydrogen bomb is a mixture of \({ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3\) and \({ }_3 \mathrm{Li}^6\) in some condensed form. The chain reaction is given by \( { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \) \( { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1\) During the explosion the energy released is approximately [Given : \(\mathrm{M}(\mathrm{Li})=6.01690\  \mathrm{amu} . \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \  amu.\) \(\mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388\  \mathrm{amu}\),\( and\ \)\(1 \ \mathrm{amu}=931.5\) \(\mathrm{MeV}]\)

Let  \( \lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right. \) \( \left.+\ldots \ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right) \text { be } \frac{\pi}{k},\) using only the principal values of the inverse trigonometric functions. Then \(\mathrm{k}^2\) is equal to ..............

The mean of \(6\) distinct observations is \(6.5\) and their variance is \(10.25\). If \(4\) out of \(6\) observations are \(2,4,5\) and \(7 ,\) then the remaining two observations are:

In a triangle, the sum of lengths of two sides is \(x\) and the product of the lengths of the same  two sides is \(y.\) If \(x^2 - c^2 = y ,\) where \(c\) is the length of the third side of the triangle, then the circumradius of the triangle is

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

A coil is placed in a time varying magnetic field. If the number of turns in the coil were to be halved and the radius of wire doubled, the electrical power dissipated due to the current induced in the coil would be: (Assume the coil to be short circuited.)

If the tangent at \(\left( {1,7} \right)\) to the curve \({x^2} = y - 6\) touches the circle \({x^2} + {y^2} + 16x + 12y + c = 0\) then  the value of \(c\) is: