In a non-leap year, the probability of getting 53 Sunday or 53 Tuesday or 53 Thursday is

\(\int_0^{\pi / 2} \frac{\pi \sin x}{1+\cos ^2 x} d x\) is equal to

If $\theta =\frac{\pi }{12}$ and $x=\log \left(\cot \left(\frac{\pi }{4}+\theta \right)\right)$, then $\cosh x=$

Consider the system of equations
\[
\begin{aligned}
& a x+b y+c z=2 \\
& b x+c y+a z=2 \\
& c x+a y+b z=2
\end{aligned}
\]
where \(a, b, c\) are real numbers such that \(a+b+c=0\). Then, the system

\(A B C D\) is a parallelogram and \(P\) is a point on the segment \(\mathbf{A D}\) dividing it internally in the ratio \(3: 1\). If the line \(\mathbf{B P}\) meets the diagonal \(A C\) in \(Q\), then \(A Q: Q C\) equals

If \(\frac{27 x^2+32 x+16}{\beta x+2)^2(1-x)}=\frac{A}{3 x+2}+\frac{B}{\beta x+2)^2}\) \(+\frac{C}{1-x}\), then \(A B+B C+C A=\)

If \(3 x+6 y+2=0, x+y+1=0,2 x-y+3=0\) are three given lines then the point \(\left(\frac{-4}{3}, \frac{1}{3}\right)\) is

A man weighing \(75 \mathrm{~kg}\) is standing in a lift. The weight of the man standing on a weighing machine kept in the lift when the lift is moving downwards freely under gravity is

Match the following table. \begin{array}{|c|c|c|c|} \hline & List-I & & List-II \ \hline A. & \begin{array}{l}\text { Michelson-Morley } \ \text { experiment }\end{array} & I. & \begin{array}{l}\text { The existence of } \ \text { anti-matter }\end{array} \ \hline B. & \begin{array}{l}\text { Stern-Gerlach } \ \text { experiment }\end{array} & II. & \begin{array}{l}\text { The existence of } \ \text { de-Broglie matter waves }\end{array} \ \hline C. & \begin{array}{l}\text { Davisson-Germer } \ \text { experiment }\end{array} & III. & Electrons have spins \ \hline D. & \begin{array}{l}\text { Anderson discovery } \ \text { of positron }\end{array} & \mathrm{IV}. & \begin{array}{l}\text { The non-existence of } \ \text { ether }\end{array} \ \hline \end{array} \(\begin{array}{llll}A & B & C & D\end{array}\)

The sum of the least positive arguments of the distinct cube roots of the complex number \((1-i \sqrt{3})\) is

If \(a, b, c\) are distinct real numbers and \(P, Q, R\) are three points whose position vectors are respectively \(a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}, b \hat{\mathbf{i}}+c \hat{\mathbf{j}}+a \hat{\mathbf{k}}\) and \(c \hat{\mathbf{i}}+a \hat{\mathbf{j}}+b \hat{\mathbf{k}}\), then \(\angle Q P R=\)

For a reaction $2\mathrm{A}\rightleftharpoons \mathrm{B}+\mathrm{C}, {\mathrm{K}}_{\mathrm{c}}$ is $2\times {10}^{-3}.$ At a given time, the reaction mixture has $\left[\mathrm{A}\right]=\left[\mathrm{B}\right]=\left[\mathrm{C}\right]=3\times {10}^{-4}\mathrm{M}.$ Which of the following options is correct?

In any triangle, \(A B C=\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2}=\)