Let $g(x)$ be an antiderivative for $f(x)$. Then $\ln(1 + (g(x))^2)$ is an antiderivative for

In the following reaction,$X$ and $Y$ respectively are:
$C_2H_5OH \xrightarrow{KMnO_4 / H^{+}} X$
$X \xrightarrow[H_2SO_4 / \Delta]{Y} CH_3COOC_2H_5$

Helium gas undergoes a cycle $ABCDA$ (consisting of two isochoric and two isobaric lines) as shown in the figure. The efficiency of this cycle is nearly ..... $\%$ (Assume the gas to be close to an ideal gas).

$A$ tangent is drawn to the ellipse $\frac{x^2}{27} + y^2 = 1$ at the point $(3\sqrt{3} \cos \theta, \sin \theta)$,where $\theta \in (0, \pi/2)$. The value of $\theta$ for which the sum of the intercepts on the axes made by this tangent is minimum,is:

$A$ bag $X$ contains $2$ white and $3$ black balls and another bag $Y$ contains $4$ white and $2$ black balls. One bag is selected at random and a ball is drawn from it. Then the probability for the ball chosen to be white is

$A$ magnetic needle lying parallel to a magnetic field is turned through $60^{\circ}$. The work done on it is $W$. The torque required to maintain the magnetic needle in the position mentioned above is