The value of $\cot 70^{\circ} + 4\cos 70^{\circ}$ is

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance $a$ from the centre $P$ (as shown in the figure). Now,the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result,the particles move towards each other on the surface. The magnitude of acceleration,when the separation between them becomes $2x$,is

The mid-point of a chord of the ellipse $x^2+4y^2-2x+20y=0$ is $(2, -4)$. The equation of the chord is

Identify the incorrect statement among the following.

If $u=\sin ^{-1}\left(\frac{x^2+y^2}{x+y}\right)$ then $x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}$ is equal to:

$A$ binary sequence is an array of $0$'s and $1$'s. The number of $n$-digit binary sequences which contain an even number of $0$'s is