List of top Mathematics Questions asked in CUET (PG)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: Every solution $\phi$ of the differential equation $y'' + \omega^2 y = A \cos \omega x$; $A$ and $\omega$ are positive constants, satisfies $|\phi(x)| \to \infty$ as $x \to \infty$.
Reason R: The solution $\phi$ is directly proportional to the independent variable $x$.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Every solution of the differential equation $y'' + 2y' + 2y = 0$ tends to zero as $x \to \infty$.
Reason R : The real part of the roots of the polynomial $\lambda^2 + 2\lambda + 2$ are negative.
In the light of the above statements, choose the correct answer from the options given below

Which of the following statements are correct:
A. Permutation group $S_n$ has order $n!$, $n > 1$.
B. Alternating group $A_n$ of degree $n$ has order $\frac{n!}{2}$, $n > 1$. C. Set of all even permutations forms a subgroup of $S_n$.
D. Every permutation in $S_n$, $n > 1$ is a product of 2-cycles.
E. $A_4$ has a subgroup of order 6.
Choose the correct answer from the options given below:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The map $f : \mathbb{C} \to \mathbb{C}$ defined by $f(z) = \sin z$ is bounded. Reason R : The function $f(z) = \sin z$ is an entire map.
In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The function $f : \mathbb{C} \to \mathbb{C}$ defined by $f(z) = z - \bar{z}$ is analytic at every point in $\mathbb{C}$ Reason R : $f$ does not satisfy the Cauchy - Riemann equations at any point.
In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $\langle x_n \rangle$ be a sequence in $\mathbb{R}$, where $x_n = \sin\left(\frac{1}{n}\right)$, then $\langle x_n \rangle$ is convergent in $\mathbb{R}$. Reason R : $\langle x_n \rangle$ is bounded.
In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The set of rationals $\mathbb{Q}$ is closed in the real line $\mathbb{R}$. Reason R : The closure of $\mathbb{Q}$ in $\mathbb{R}$ is $\mathbb{R}$. In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $P_0(t) = 1, P_1(t) = t, P_2(t) = t^2, P_3(t) = t^3$, then $\{P_0, P_1, P_2, P_3\}$ is linearly independent. Reason R : $C_0 P_0 + C_1 P_1 + C_2 P_2 + C_3 P_3 = 0 \implies C_0 = 0, C_1 = 0, C_2 = 0, C_3 = 0$.
In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Set $\{(1,0,0), (0,1,0), (0,0,1)\}$ forms a basis of $V_3(\mathbb{R})$. Reason R : Let $V$ be a vector space over a field $F$ and $S$ be a non-empty subset of $V$, then $S$ is a basis of $V$ if
(i) $S$ is linearly dependent
(ii) $\text{span}(S) = V$. In the light of the above statements, choose the correct answer from the options given below

Orthogonal projection of $\vec{v}$ on $\vec{a}$ is

Consider the linear programming problem
\[ Z_{\max} = 2x_1 + 5x_2 \quad \text{subject to the restrictions:} \] \[ x_1 \leq 4 \] \[ x_2 \leq 3 \] \[ 2x_1 + 3x_2 \leq 14 \] and $x_1 \geq 0, x_2 \geq 0$. Then the optimal solution is.

If $f(x, y)$ is a real-valued function of two variables, then $\vec{\nabla} f(x, y)$ is:

Let $c$ be boundary of $[0, 1] \times [0, 1]$ oriented counter clockwise then $\int_c (y^4 + x^3) dx + 2 x^6 dy$ is

Let $\vec{F} = y e^z \hat{i} + x e^z \hat{j} + x y e^z \hat{k}$, then integral of $\vec{F}$ around the boundary of oriented surface $S$ is

Let $\vec{F}(x, y, z) = x \hat{i} + x y \hat{j} + \hat{k}$, then $\text{curl}(\vec{F})$ is

The value of $\vec{\nabla}\left(\frac{f}{g}\right)$ at points where $g(x) \neq 0$ is given by

Let $\vec{F}(x, y, z) = x^2 y \hat{i} + z \hat{j} + x y z \hat{k}$, then $\text{div} \vec{F}$ is given by

The set of linearly independent solutions of the differential equation $(D^4 - D^3)y = 0; D = \frac{d}{dx}$ is

The solution of the differential equation $(2 \cos y) y' + \sin y = x^2 \csc y, y \neq 0$ is:

The integrating factor of the differential equation $(e^x - \sin y) dx + \cos y \, dy = 0$ is

Let $u, v$ and $w$ be the non-zero solutions of the differential equation
\[ (D^3 - 6D^2 + 11D - 6)y = 0; \quad D = \frac{d}{dx} \] Then the Wronskian of $u, v$ and $w$ is

A solution of the differential equation $(D^2 - 1)y = 2^x + e^{-x}; D = \frac{d}{dx}$ is

A solution curve of the equation $x y' = 2y$ passing through $(1, 4)$, also passes through

Consider the differential equation
\[ y'' - 4y' + 20y = 0 \] with $y\left(\frac{\pi}{2}\right) = 0$, and $y'\left(\frac{\pi}{2}\right) = 1$, then the value of $y\left(\frac{\pi}{8}\right)$ is

The value of $\iiint_B (x + 2y + 3z)^2 \, dx \, dy \, dz$ is, where $B$ is the box $[0, 1] \times \left[-\frac{1}{2}, 0\right] \times \left[0, \frac{1}{3}\right]$