The p.d.f. of a continuous random variable $X$ is $f(x)$.
\[ f(x)= \begin{cases} \dfrac{x^2}{18}, & -3 \le x \le 3 \\ 0, & \text{otherwise} \end{cases} \] Then find $P(|X|<2)$.

Which of the following statements has the truth value $T$?
A. Cube roots of unity are in Geometric Progression and their sum is $1$
B. \[ 4 + 7 > 10 \iff 2 + 8 < 10 \]
C. \[ \exists x \in \mathbb{N} \text{ such that } x^2 - 3x + 2 = 0 \text{ and } \exists n \in \mathbb{N} \text{ such that } n \text{ is an odd number} \]
D. \[ 3+i \text{ is a complex number } \; \text{or} \; \sqrt{2}+\sqrt{3}=\sqrt{5} \]

If the function $f(x)$ is continuous in $[0, \pi]$ then $a - b =$

If the pair of straight lines $xy - x + y - 1 = 0$ and the line $x + ky - 3 = 0$ are concurrent, then the value of $k$ is equal to

If $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$ then

The angle between the lines $3x = 2y = -z$ and $-x = 6y = -4z$ is

For a real number $x$, $[x]$ denotes the greatest integer less than or equal to $x$. Then the value of $\left[ \frac{1}{2} \right] + \left[ \frac{1}{2} + \frac{1}{100} \right] + \left[ \frac{1}{2} + \frac{2}{100} \right] + \dots + \left[ \frac{1}{2} + \frac{99}{100} \right] =$

If the sum of the squares of the distances of a point $P(x, y, z)$ from the three co-ordinate axes is $324$, then the distance of point $P$ from the origin is ....

The degree of the differential equation $\frac{d^2y}{dx^2} + 3 \left( \frac{dy}{dx} \right)^2 = x^2 \log \left( \frac{d^2y}{dx^2} \right)$ is

A coin is tossed until one head appears or a tail appears $4$ times in succession. The probability distribution of the number of tosses is

In a triangle $ ABC $ with usual notations, if \[ \tan \left( \frac{B-C}{2} \right) = x \cot \left( \frac{A}{2} \right), \] then $ x = \, ? $