$ P(X=0 \text{ and } Y=0) = \frac{1}{4}, $
$ P(X=1 \text{ and } Y=0) = \frac{1}{8}, $
$ P(X=0 \text{ and } Y=1) = \frac{1}{2}, $
$ P(X=1 \text{ and } Y=1) = \frac{1}{8}. $

Given $ X = 1 $, the expected value of $ Y $ is

Consider the set $ S $ of points $ (x, y) \in \mathbb{R}^2 $ which minimize the real-valued function $ f(x, y) = (x + y - 1)^2 + (x + y)^2 $. Which of the following statements is true about the set $ S $?

If the function
\[ f(x) = \begin{cases} \sin(2x), & \text{for } x > 0, \\ a + bx, & \text{for } x \leq 0, \end{cases} \]
where $ a $ and $ b $ are constants, is differentiable at $ x = 0 $, then $ a + b $ is equal to:

A 24 cm line AB is vertically standing on a horizontal plane. The station point is located 18 cm above ground and 15 cm in front of the line AB. The picture plane is located in between the line AB and station point perpendicular to the sight line. The distance between the picture plane and the station point is 9 cm. The height of the perspective view of the line AB is ________cm. (rounded off to one decimal place)

If $$ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6, \text{ then } f(1) \text{ is equal to:} $$

The determinant of a 3 × 3 matrix $ M $ is 8. If every element of $ M $ is multiplied by $ -2 $, the determinant of the modified matrix will be:

Which one of the following plots represents $ f(x) = -\frac{|x|}{x} $, where $ x $ is a non-zero real number?
Note: The figures shown are representative.

If $a$, $b$, and $c$ are the roots of $ 2x^3 - 3x^2 + px - 1 = 0 $ and the sum of two roots is 1, the value of $p$ is:

Consider two statements:
Statement 1: $ \lim_{x \to 0} \frac{\tan^{-1} x + \ln \left( \frac{1+x}{1-x} \right) - 2x}{x^5} = \frac{2}{5} $
Statement 2: $ \lim_{x \to 1} x \left( \frac{2}{1-x} \right) = e^2 \; \text{and can be solved by the method} \lim_{x \to 1} \frac{f(x)}{g(x) - 1} $

Find the area of the region defined by the conditions: $ \left\{ (x, y): 0 \leq y \leq \sqrt{9x}, y^2 \geq 3 - 6x \right\} \text{(in square units)} $

Let $ f $ be defined as $ R \setminus \{0\} \to R $ such that $ f(x) = \frac{x}{3} + \frac{3}{x} + 3 $ If $ f(x) $ is strictly increasing in $ (-\infty, \alpha_1) \cup (\alpha_2, \infty) $ and strictly decreasing in $ (\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5) $, then $ \sum_{i=1}^5 (\alpha_i)^2 $ is equal to:

A circle touches the parabola $ y^2 = 9x $ at $ (4, 6) $ and the positive x-axis. Find the radius of the circle.

The value of the following expression: $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 2} + 1}{\tan 2} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 2} - 1}{\tan 2} \right) $