If a number is chosen at random from the four-digit numbers formed by using the digits $0, 1, 2, 3, 4, 6$ without repetition,then the probability that it is divisible by $4$ is:

$A$ point is moving on $y=4-2x^2$. The $x$-coordinate of the point is decreasing at the rate of $5 \text{ units/s}$. Then,the rate at which the $y$-coordinate of the point is changing when the point is at $(1,2)$ is

If a line makes angles $\tan ^{-1} \sqrt{7}$ and $\tan ^{-1} \frac{\sqrt{5}}{\sqrt{3}}$ with the $X$-axis and $Y$-axis respectively,then the angle made by it with the $Z$-axis is

The objective and eyepiece of an astronomical telescope are double convex lenses with refractive index $1.5$. When the telescope is adjusted to infinity,the separation between the two lenses is $16 ~cm$. If the space between the lenses is now filled with water (refractive index $4/3$) and the telescope is again adjusted for infinity,then the present separation between the lenses is: (in $~cm$)

The solution of $\frac{dy}{dx} + 1 = e^{x+y}$ is

Which of the following is an incorrect match ?