Two distinct numbers $a$ and $b$ are selected at random from $1,2,3, \ldots, 50$. The probability, that their product $a b$ is divisible by 3 , is

The number of solutions of $\tan ^{-1} 4 x+\tan ^{-1} 6 x=\frac{\pi}{6}$, where $-\frac{1}{2 \sqrt{6}} < x < \frac{1}{2 \sqrt{6}}$, is equal to :

If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^2-9 y^2=9$, then a possible value of $\alpha$ is :

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is