If $f(x) = \frac{1}{1+\frac{1}{x}}$ and $g(x) = \frac{1}{1+\frac{1}{f(x)}}$,then $g^{\prime}(2)$ is equal to

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar unit vectors such that $\bar{a} \times (\bar{b} \times \bar{c}) = \frac{\bar{b} + \bar{c}}{\sqrt{2}}$,then the angle between $\bar{a}$ and $\bar{b}$ is:

The solution of the differential equation $\frac{dy}{dx} + \frac{3x^2}{1+x^3}y = \frac{\sin^2 x}{1+x^3}$ is

The blocks shown in the figure move with a constant velocity of $10 \, m/s$ towards the right. All surfaces in contact are rough. The friction force applied by block $B$ on block $A$ is ........... $N$.

The number of divisors of $7!$ is

Let $\alpha, \beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is