Inverse of the matrix \(\left[\begin{array}{cc}\cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta\end{array}\right]\) is

If \(|\mathrm{a}|=3,|\mathrm{~b}|=4\), then a value of \(\lambda\) for which \(\mathrm{a}+\lambda \mathrm{b}\) is perpendicular to \(\mathrm{a}-\lambda \mathrm{b}\) is

If \(a=\cos 2 \alpha+i \sin 2 \alpha, b=\cos 2 \beta+i \sin 2 \beta\), then \(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\)

Let L be the foot of the perpendicular drawn from the point \(P(5, 3k-7, -4)\) to the YZ-plane. If the distance of point L from the origin is \(\sqrt{41}\) units, then the possible value of '\(k\)' is:

\(8^{3 \log _{8} 5}\) is equal to

The circle \(x^2+y^2+4 x-7 y+12=0\) cuts an intercept on \(Y\)-axis of length

If the direction ratios of two lines are given by \(3 l m-4 l n+m n=0\) and \(l+2 m+3 n=0\), then the angle between the lines is

The magnifying power of a telescope is 9 . When it is adjusted for parallel rays, the distance between the objective and eyepiece is \(20 \mathrm{~cm}\). The focal length of lenses are

A proton is moving along negative X axis. If a uniform magnetic field is applied parallel to the positive Z axis, then choose the correct answer from the following options;
1. The proton will experience magnetic force along positive Y direction and will move along a circular path in the XY plane
2. The proton will experience magnetic force along negative Y direction and will move along a circular path in the XY plane
3. The proton will experience magnetic force along positive Y direction and will move along a circular path in the XZ plane
4. The proton will experience magnetic force along negative Y direction and will move along a circular path in the XZ plane

\(S \equiv x^2+y^2+2 x+3 y+1=0\) and
\(S^{\prime} \equiv x^2+y^2+4 x+3 y+2=0\) are two circles.
The point \((-3,-2)\) lies

A long spring, when stretched by a distance \(x\), has potential energy \(U\). On increasing the stretching to \(n x\), the potential energy of the spring will be

Mass of \(0.1\) mole of methane is

Denatured alcohol is