If the internal energy of an ideal gas decreases by the same amount as the work done by the system,the process is

The value of $\cos ^{-1} \left[ \cot \left\{ \sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}} + \cos ^{-1} \frac{\sqrt{12}}{4} + \sec ^{-1} \sqrt{2} \right\} \right]$ is

$\log _4 2 - \log _8 2 + \log _{16} 2 - \dots$ is equal to

$A$ six-faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be $7$. The probability that the number $3$ has appeared at least once is:

The equation of the lines passing through the origin and having slopes $3$ and $-\frac{1}{3}$ is

The time period of a particle in simple harmonic motion is $8 \ s$. At $t=0$,it is at the mean position. The ratio of the distances travelled by it in the first and second seconds is