If \(\left(\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right)=\left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)\), then \(\frac{x^2+y^2+z^2}{\gamma}=\)

If \(k\) is one of the roots of the equation \(x^2-25 x+24=0\) such that \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 3 \\ 1 & 1 & k\end{array}\right]\) is a non-singular matrix, then \(A^{-1}=\)

If \(\mathrm{z}_1, \mathrm{z}_2, \mathrm{z}_3\) are the vertices of an equilateral triangle and \(\mathrm{z}\) is its circum centre, then

The coordinates of the point at which \(\mathrm{XZ}\) - plane divides the line joining \(\mathrm{A}(-2,3,4)\) and \(\mathrm{B}(1,2,3)\) is

If \(x=\sin \theta\) and \(y=\cos p \theta\), then \(\left(1-x^2\right) y_2=\)

Let \(f\) be a non-zero real valued continuous function satisfying \(f(x+y)=f(x) \cdot f(y)\) for all \(x, y \in \mathbb{R}\). If \(f(2)=9\), then \(f(6)\) is equal to

If \(x=\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}\), then \(x^2(x-4)^2\) is equal to :

Which of the following produces nitrogen gas after the reaction with nitrous acid?

\(\int_{-1}^2|x| d x=\)

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

If \(f: \mathbf{R} \rightarrow \mathbf{R}\) is defined as \(f(x)=x^2-2 x-3\) then \(f\) is

If the maximum velocity and maximum acceleration of a particle executing simple harmonic motion are respectively \(5 \mathrm{~m} \mathrm{~s}^{-1}\) and \(10 \mathrm{~m} \mathrm{~s}^{-2}\), then the time period of oscillation of the particle is

The pair of elements that form both oxides and nitrides, when burnt in air are