Two fair dice (with faces labeled $1, 2, 3, 4, 5,$ and $6$) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is (rounded off to two decimal places).

Consider the vectors $\boldsymbol{a} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \boldsymbol{b} = \begin{bmatrix} 0 \\ 3\sqrt{2} \end{bmatrix}.$ For real-valued scalar variable $x$, the value of $$\min{x} \lVert \boldsymbol{a}x - \boldsymbol{b} \rVert2$$ is (rounded off to two decimal places). $\lVert \cdot \rVert2$ denotes the Euclidean norm, i.e., for $\boldsymbol{y} = \begin{bmatrix} y1 \\ y2 \end{bmatrix}$, $\lVert \boldsymbol{y} \rVert2 = \sqrt{y1^2 + y2^2}$.

Consider the matrix $A$ below: $A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}$ For which of the following combinations of $\alpha, \beta, \text{ and } \gamma$, is the rank of $A$ at least three? (i) $\alpha = 0$ and $\beta = \gamma \neq 0$ (ii) $\alpha = \beta = \gamma = 0$ (iii) $\beta = \gamma = 0$ and $\alpha \neq 0$ (iv) $\alpha = \beta = \gamma \neq 0$

Consider the function $f: \mathbb{R} \to \mathbb{R}$, defined as $f(x) = 2x^3 - 3x^2 - 12x + 1.$ Which of the following statements is/are correct? (Here, $\mathbb{R}$ is the set of real numbers.)

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $2, 8$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval. Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality $\beta \leq \int{2}^{8} f(x) dx \leq \alpha$ is guaranteed to hold.

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

The function $y(t)$ satisfies $$t^2 y''(t) - 2ty'(t) + 2y(t) = 0,$$ where $y'(t)$ and $y''(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y'(0) = 1$ and $y'(1) = -1$, the maximum value of $y(t)$ over $0, 1$ is \\\\\ (rounded off to two decimal places).

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is \\\\\\.

Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors $\{2 \quad -3 \quad \alpha, \quad 3 \quad -1 \quad 3, \quad 1 \quad -5 \quad 7\}$ does not form a basis of $\mathbb{R}^3$ is .

Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $0,1$. The probability that $X \geq Y$ is .

Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the $60^\circ\text{N}$ latitude circle, that contains the north pole in its interior is .

Consider the matrix $\begin{bmatrix} 1 & k \\ 2 & 1 \end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?