List of top Mathematics Questions asked in CUET (PG)

Match List-I with List-II and choose the correct option:

LIST-I (Set)	LIST-II (Supremum/Infimum)
(A) \( S = \{2, 3, 5, 10\} \)	(III) Sup S = 10, Inf S = 2
(B) \( S = (1, 2] \cup [3, 8) \)	(IV) Sup S = 8, Inf S = 1
(C) \( S = \{2, 2^2, 2^3, \dots, 2^n, \dots\} \)	(II) Sup S = 5, Inf S = -5
(D) \( S = \{x \in \mathbb{Z} : x^2 \le 25\} \)	(I) Inf S = 2

Choose the correct answer from the options given below:

Which of the following are correct?

A. A set \( S = \{(x, y) \mid xy \leq 1 : x, y \in \mathbb{R}\} \) is a convex set.
B. A set \( S = \{(x, y) \mid x^2 + 4y^2 \leq 12 : x, y \in \mathbb{R}\} \) is a convex set.
C. A set \( S = \{(x, y) \mid y^2 - 4x \leq 0 : x, y \in \mathbb{R}\} \) is a convex set.
D. A set \( S = \{(x, y) \mid x^2 + 4y^2 \geq 12 : x, y \in \mathbb{R}\} \) is a convex set.

Match List-I with List-II and choose the correct option:

LIST-I (Function)	LIST-II (Value)
(A) \( \int_{\gamma} \frac{1}{z-a} \, dz \), where \( \gamma: \|z-a\|=r, r > 0 \)	(III) \( 2i\pi \)
(B) \( \int_{\gamma} \frac{z+2}{z} \, dz \), where \( \gamma: z = 2e^{it}, 0 \le t \le \pi \)	(IV) \( i\pi \)
(C) \( \int_{\gamma} \frac{e^{2z}}{(z-1)(z-2)} \, dz \), where \( \gamma: \|z\|=3 \)	(II) \( 2i\pi(e^4 - e^2) \)
(D) \( \int_{\gamma} \frac{z^2 - z + 1}{2(z-1)} \, dz \), where \( \gamma: \|z\|=2 \)	(I) \( -4 + 2i\pi \)

Choose the correct answer from the options given below:

Let \( R \) be the planar region bounded by the lines \( x = 0 \), \( y = 0 \) and the curve \( x^2 + y^2 = 4 \) in the first quadrant. Let \( C \) be the boundary of \( R \), oriented counter clockwise. Then, the value of:

\[ \oint_C x(1 - y) \, dx + (x^2 - y^2) \, dy \] is equal to:

Match List-I with List-II and choose the correct option:

LIST-I	LIST-II
(A) The solution of an ordinary differential equation of order 'n' has	(III) 'n' arbitrary constants
(B) The solution of a differential equation which contains no arbitrary constant is	(IV) particular solution
(C) The solution of a differential equation which is not obtained from the general solution is	(I) singular solution
(D) The solution of a differential equation containing as many arbitrary constants as the order of a differential equation is	(II) complete primitive

Choose the correct answer from the options given below:

If p is a prime number and a group G is of the order p2, then G is:

The value of \( v_3 \) for which the vector \( \vec{v} = e^y \sin x \hat{i} + e^y \cos x \hat{j} + v_3 \hat{k} \) is solenoidal, is:

Match List-I with List-II and choose the correct option:

\[ \begin{array}{|l|l|} \hline \textbf{LIST-I (Function)} & \textbf{LIST-II (Expansion)} \\ \hline A. \log(1-x) & I. 1 + \frac{1}{3} + \frac{1}{6} + \frac{3}{40} + \frac{15}{336} + \dots \\ \hline B. \sin^{-1} x & II. 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \\ \hline C. \log 2 & III. x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} + \dots, -1 < x \le 1 \\ \hline D. \frac{\pi}{2} & IV. -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots, -1 \le x < 1 \\ \hline \end{array} \]

The value of the integral \( \iint_R (x+y) \,dy\,dx \) in the region R bounded by \( x=0, x=2, y=x, y=x+2 \), is

For any Linear Programming Problem (LPP), choose the correct statement:
A. There exists only finite number of basic feasible solutions to LPP
B. Any convex combination of k - different optimum solution to a LPP is again an optimum solution to the problem
C. If a LPP has feasible solution, then it also has a basic feasible solution
D. A basic solution to AX = b is degenerate if one or more of the basic variables vanish

Which of the following are correct:
A. Every infinite bounded set of real number has a limit point
B. The set \( S = \{x : 0<x \le 1, x \in \mathbb{R}\} \) is a closed set
C. The set of whole real numbers is open as well closed set
D. The set \( S = \{1, -1, \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3}, ... \} \) is neither open set nor closed set

Let \( f(x) = |x| + |x-1| + |x+1| \) be a function defined on \( \mathbb{R} \), then \( f(x) \) is:

Let \(f(x)\) be a real valued function defined for all \(x \in \mathbb{R}\), such that \(|f(x)-f(y)| \le (x-y)^2\), for all \(x,y \in \mathbb{R}\), then

If, \( u=y^3-3x^2y \) be a harmonic function then its corresponding analytic function \( f(z) \), where \( z=x+iy \), is:

Which of the following forms a linear transformation:

The value of the integral \( \int_0^\infty \int_0^x x e^{-x^2/y} dy \, dx \) is:

If U and W are distinct 4-dimensional subspaces of a vector space V of dimension 6, then the possible dimensions of \( U \cap W \) is:

The value of \( \iiint_E \frac{dx\,dy\,dz}{x^2+y^2+z^2} \), where E: \( x^2+y^2+z^2 \le a^2 \), is

The given series \( 1 - \frac{1}{2^p} + \frac{1}{3^p} - \frac{1}{4^p} + ... (p>0) \) is conditionally convergent, if 'p' lies in the interval:

A complete solution of \( y'' + a_1 y' + a_2 y = 0 \) is \( y = b_1 e^{-x} + b_2 e^{-3x} \), where \( a_1, a_2, b_1 \) and \( b_2 \) are constants, then the respective values of \( a_1 \) and \( a_2 \) are: