Let $A,B,C,D$ be square real matrices such that $C^T=DAB, D^T=ABC, S=ABCD$, then $S^2$ is equal to

If \(\alpha, \beta\) are the eccentric angles of the extremities of a focal chord (other than the major axis) of the ellipse \(x^2+4 y^2=4\) then \(\sqrt{3} \cos \frac{\alpha+\beta}{2}=\)

If \(\mathrm{A}(0,0,0), \mathrm{B}(3,4,0), \mathrm{C}(0,12,5)\) are the vertices of a triangle ABC, then the \(x\)-coordinate of its incentre is

Two dice are thrown and the sum of the numbers appeared on the dice is noted. If \(A\) is the event of getting a prime number as their sum and \(B\) is the event of getting a number greater than 8 as their sum, then \(\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=\)

Let \(\mathrm{X}\) - axis be the transverse axis and \(\mathrm{Y}\) - axis be the conjugate axis of a hyperbola \(\mathrm{H}\). Let the eccentricity of \(\mathrm{H}\) be the reciprocal of the eccentricity of the ellipse \(\frac{x^2}{4}+\frac{y^2}{2}=1\). If \((5,4)\) is a point on \(H\), then the length of the transverse axis of \(\mathrm{H}\) is

The variance of the random variable \(X\) having the following distribution

If \(a, b, c\) are distinct positive real numbers, then the value of the determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is

Match the following.

\(\begin{aligned} & \int \frac{x d x}{\sqrt[15]{\left(1+x^2\right)^{12}\left(2+x^2\right)^{18}}}=\alpha\left(\frac{1+x^2}{2+x^2}\right)^{1 / n}+C \Rightarrow \\ & \frac{n}{\alpha}=\end{aligned}\)

Match List-I with List-II
\(\begin{array}{|c|l|l|l|}\hline & \text{List-I (Reaction)} & & \text{List-II (Enzyme)} \\ \hline \text{A.} & \begin{array}{l} \text{Hydrolysis of starch to} \\ \text{maltose}\end{array} & \text{I} & \text{Diastase} \\ \hline \text{B.} & \begin{array}{l} \text{Conversion of proteins} \\ \text{to peptides} \end{array} & \text{II} & \text{Pepsin} \\ \hline \text{C.} & \begin{array}{l} \text{Hydrolysis of sucrose} \\ \text{to glucose and fructose}\end{array} & \text{III} & \text{Invertase} \\ \hline \text{D.} & \text{Glucose to ethanol} & \text{IV} & \text{Zymase} \\ \hline\end{array}\)
The correct answer is

A body is projected at an angle other than \(90^{\circ}\) with the horizontal with same velocity. If the time of ascent of the body is ls, then the maximum height it can reach is (Take, \(g=10 \mathrm{~ms}^{-2}\) )

If \(Z_r=\cos \left(\frac{\pi}{2^r}\right)+i \sin \left(\frac{\pi}{2^r}\right)\) for \(r=1,2,3, \ldots\). , then \(Z_1 Z_2 Z_3 \ldots \infty\) equals to

The energy of second Bohr orbit of hydrogen atom is -3.4 eV. The energy of the fourth Bohr orbit of the \(\mathrm{He}^{+}\)ion will be