If $\tan ^{-1} x = y$,then . . . . . . .

If the coefficients $a$ and $b$ of a quadratic expression $x^2+ax+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+ax+b=0$ has real roots is

Consider the following statements $A$ and $B$ and identify the correct answer:
$A$. When a person walks on a rough surface, the direction of the frictional force exerted by the surface on the person is in the direction of his motion.
$B$. When a cycle is in motion, the force of friction exerted by the ground on the front wheel is in the backward direction.

If $\log (1+x) - \frac{2x}{2+x}$ is increasing, then

Assertion $A$: An astronaut inside a massive spaceship orbiting around the Earth will experience a finite but small gravitational force.
Reason $R$: The centripetal force necessary to keep the spaceship in orbit around the Earth is provided by the gravitational force between the Earth and the spaceship.

If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$,then the locus of $P(h, k)$ is a conic $C$ whose eccentricity equals