The equation of the tangent at $(-4, -4)$ on the curve $x^2 = -4y$ is

Beats are produced by two waves given by $y_1 = a \sin(2000 \pi t)$ and $y_2 = a \sin(2008 \pi t)$. The number of beats heard per second is

The locus of the centre of the circle,which cuts the circle $x^2+y^2-20x+4=0$ orthogonally and touches the line $x=2$,is

$A$ number $n$ is chosen at random from $\{1, 2, 3, 4, \ldots, 1000\}$. The probability that $n$ is a number that leaves remainder $1$ when divided by $7$ is:

Which one of the following is a secondary alcohol?

$A$ thin converging lens of focal length $f=25 \ cm$ forms the image of an object on a screen placed at a distance of $75 \ cm$ from the lens. The screen is moved closer to the lens by a distance of $25 \ cm$. The distance through which the object has to be shifted,so that its image on the screen is sharp again,is (in $cm$)