List of top Mathematics Questions

Which one of the following subsets of $\mathbb{R}$ has a non-empty interior?

For every $ n \in \mathbb{N} $, let $ f_n : \mathbb{R} \to \mathbb{R} $ be a function. From the given choices, pick the statement that is the negation of \[ \text{“For every } x \in \mathbb{R} \text{ and for every real number } \varepsilon>0, \text{ there exists an integer } N>0 \text{ such that } \sum_{i=1}^p |f_{N+i}(x)|<\varepsilon \text{ for every integer } p>0.” \]

For an integer $k \ge 0$, let $P_k$ denote the vector space of all real polynomials in one variable of degree less than or equal to $k$. Define a linear transformation $T : P_2 \to P_3$ by \[ T(f(x)) = f''(x) + x f(x). \] Which one of the following polynomials is not in the range of $T$?

Consider the surface \[ S = \{(x,y,xy) \in \mathbb{R}^3 : x^2 + y^2 \le 1\}. \] Let $\vec{F} = y\hat{i} + x\hat{j} + \hat{k}$. If $\hat{n}$ is the continuous unit normal field to the surface $S$ with positive $z$-component, then \[ \iint_S \vec{F} \cdot \hat{n} \, dS \] equals

Let $y$ be the solution of \[ (1+x)y''(x) + y'(x) - \frac{1}{1+x} y(x) = 0, \quad x \in (-1,\infty), \] with initial conditions $y(0) = 1, \ y'(0) = 0.$ Then

Let $n>1$ be an integer. Consider the following two statements for an arbitrary $n \times n$ matrix $A$ with complex entries. I. If $A^k = I_n$ for some integer $k \ge 1$, then all the eigenvalues of $A$ are $k^{\text{th}}$ roots of unity.
II. If, for some integer $k \ge 1$, all the eigenvalues of $A$ are $k^{\text{th}}$ roots of unity, then $A^k = I_n.$
Then

Let $ M_n(\mathbb{R}) $ be the real vector space of all $ n \times n $ matrices with real entries, $ n \ge 2 $. Let $ A \in M_n(\mathbb{R}) $. Consider the subspace $ W $ of $ M_n(\mathbb{R}) $ spanned by $\{I_n, A, A^2, A^3, \ldots\}$. Then the dimension of $ W $ over $\mathbb{R}$ is necessarily

Let $p$ and $t$ be positive real numbers. Let $D_t$ be the closed disc of radius $t$ centered at $(0,0)$, i.e., \[ D_t = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le t^2 \}. \] Define \[ I(p,t) = \iint_{D_t} \frac{dx\,dy}{(p^2 + x^2 + y^2)^2}. \] Then $\lim_{t \to \infty} I(p,t)$ is finite

Let $ P : \mathbb{R} \to \mathbb{R} $ be a continuous function such that $ P(x)>0 $ for all $x \in \mathbb{R}$. Let $y$ be a twice differentiable function on $\mathbb{R}$ satisfying \[ y''(x) + P(x)y'(x) - y(x) = 0 \] for all $x \in \mathbb{R}$. Suppose that there exist two real numbers $a,b$ ($a<b$) such that $y(a) = y(b) = 0$. Then

Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function satisfying $ f(x) = f(x+1) $ for all $x \in \mathbb{R}$. Then

Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function such that for all $x \in \mathbb{R}$, \[ \int_0^1 f(xt) \, dt = 0. \quad \text{(*)} \] Then

How many elements of the group $\mathbb{Z}_{50}$ have order 10?

Let $ 0<\alpha<1 $ be a real number. The number of differentiable functions $ y : [0,1] \to [0,\infty) $, having continuous derivative on $[0,1]$ and satisfying \[ y'(t) = (y(t))^{\alpha}, \ t \in [0,1], \quad y(0) = 0, \] is

Let $X_1, X_2, X_3, X_4$ be independent random variables following the standard normal distribution. Let $Y$ be defined as \[ Y = (X_1 + X_2)^2 + (X_3 + X_4)^2. \] Then the variance of $Y$ equals __________. (in integer)

For $\beta>0$, let the variables $x_1$ and $x_3$ be the optimal basic feasible solution of the linear programming problem \[ \text{maximize } \; z = x_1 + 2x_2 + 3x_3 \]

If the optimal value is 7, then $\beta$ equals ____________. (in integer)

The optimal value of the constrained optimization problem \[ \text{minimize } 2xy \quad \text{subject to } 9x^2 + 4y^2 \leq 36 \] is _____________. (in integer)

Let $y(x) > 0$ be a solution of the differential equation \[ \frac{dy}{dx} + y = y^2. \] If $y(\ln 2) = \dfrac{1}{3}$, where $\ln$ denotes the natural logarithmic function, then $y(\ln 3)$ equals ___________. (round off to 2 decimal places)

Let $X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$ be two normally distributed random variables, where $\mu_1 = 2, \mu_2 = 3$ and $\sigma_1^2 = 4, \sigma_2^2 = 9$. The correlation coefficient between them is 0.5. The variance of the random variable $(X_1 + X_2)$ is ___________. (in integer)

Consider the first order difference equation \[ x_n = \left( \frac{n + 1}{n} \right) x_{n - 1}, \quad n = 1, 2, 3, \ldots \] If $x_0 = 2$, then $x_{100} - x_{50}$ equals __________. (in integer)

The value of the integral \[ \int_{0}^{9} \frac{x - 1}{1 + \sqrt{x}} \, dx \] is _____________. (in integer)

The amount of money a gambler can win in a casino is determined by three independent rolls of a six-faced fair dice. The gambler wins Rs. 800 if he gets three sixes, Rs. 400 if he gets two sixes, and Rs. 100 in the event of getting only one six. The gambler does not win or lose any money in all other possible outcomes. The probability that a gambler will win at least Rs. 400 is ___________. (round off to 2 decimal places)

Let $k \in \mathbb{R}$. Which of the following statements is/are correct for the roots of the quadratic equation \[ x^2 + 2(k + 1)x + 9k - 5 = 0 \] ?

Let $x, y \in \mathbb{R}$ and the matrix \[ M = \begin{bmatrix} x + y & x - y \\ x - y & x + y \end{bmatrix}. \] Also, let $\text{adj}(M)$ be the adjoint and $\det(M)$ be the determinant of the matrix $M$. If \[ M \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}, \] then

For any two sets $S_1, S_2 \subseteq \mathbb{R}$, define the set $S_1 - S_2 = \{x \in S_1, x \notin S_2\}$. Let \[ P = \{x \in \mathbb{R} : x^2 - 2x - 3 \le 0\} \quad \text{and} \quad Q = \{x \in \mathbb{R} : \log_5(1 + x^2) \le 1\}. \] Then,

Let $X$ and $Y$ be two random variables with the joint probability density function \[ f_{X,Y}(x,y) = \begin{cases} 6xy, & 0 < y \le \sqrt{x} \le 1, \\ 0, & \text{otherwise.} \end{cases} \] Then, the conditional probability $P(Y \ge \tfrac{1}{3} \mid X = \tfrac{2}{3})$ is