$\int_{-\pi / 2}^{\pi / 2} \sin ^4 x \cos ^6 x \, dx$ is equal to

If the pair of straight lines given by $A x^2+2 H x y+B y^2=0$ $(H^2>A B)$ forms an equilateral triangle with the line $a x+b y+c=0$,then $(A+3 B)(3 A+B)$ is equal to :

Two identical charged spherical drops, each of capacitance $C$, merge to form a single drop. The resultant capacitance is:

$\mathop {\lim }\limits_{x \to 0} \frac{{x\tan 2x - 2x\tan x}}{{{{(1 - \cos 2x)}^2}}}$ is

The circuit shown contains a $12 \ V$ battery,a $4 \ \Omega$ resistor,and two branches in parallel,each containing an ideal diode and a resistor. Branch $1$ has diode $D_1$ and a $3 \ \Omega$ resistor,while branch $2$ has diode $D_2$ and a $2 \ \Omega$ resistor. Determine the total current flowing in the circuit. (in $A$)

The angle of elevation of an object from a point $P$ on the level ground is $\alpha$. Moving $d$ metres on the ground towards the object,the angle of elevation is found to be $\beta$. Then the height (in metres) of the object is