The CUET PG Mathematics exam in 2025 is scheduled between 13th May and 3rd June, with access to the question paper, answer key, and solution PDF available post-exam. This test evaluates a candidate’s command over calculus, linear algebra, real and complex analysis, differential equations, number theory, and applied mathematics.
Candidates will face 75 questions in 60 minutes, totaling 300 marks. 4 marks are awarded per correct answer, and 1 mark is deducted for incorrect ones.
CUET PG Mathematics 2025 Question Paper with Answer Key PDF
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CUET PG Mathematics 2025 Question Paper with Solutions
If p is a prime number and a group G is of the order p2, then G is:
If \( S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right) \dots \left(1-\frac{1}{n^2}\right) \), then S is equal to:
Let R be the planar region bounded by the lines x = 0, y = 0 and the curve x2 + y2 = 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:
Let [x] be the greatest integer function, where x is a real number, then \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} ([x] + [y] + [z]) \, dx \, dy \, dz = \)
Let V(F) be a finite dimensional vector space and T: V \(\to\) V be a linear transformation. Let R(T) denote the range of T and N(T) denote the null space of T. If rank(T) = rank(T2), then which of the following are correct?
A. N(T) = R(T)
B. N(T) = N(T2)
C. N(T) \(\cap\) R(T) = 0
D. R(T) = R(T2)
Match List-I with List-II and choose the correct option:
For the function \( f(x, y) = x^3 + y^3 - 3x - 12y + 12 \), which of the following are correct:
A. minima at (1,2)
B. maxima at (-1,-2)
C. neither a maxima nor a minima at (1,-2) and (-1,2)
D. the saddle points are (-1,2) and (1,-2)
Which of the following statement is true:
Match List-I with List-II and choose the correct option:
Consider the following: Let f(z) be a complex valued function defined on a subset \( S \subset \mathbb{C} \) of complex numbers. Then which of the following are correct?
A. The order of a zero of a polynomial equals to the order of its first non-vanishing derivative at that zero of the polynomial
B. Zeros of non-zero analytic function are isolated
C. Zeros of f(z) are obtained by equating the numerator to zero if there is no common factor in the numerator and the denominator of f(z)
D. Limit points of zeros of an analytic function is an isolated essential singularity
Which of the following are correct?
A. A set \( S = \{(x, y) | xy \le 1 : x, y \in \mathbb{R}\} \) is a convex set
B. A set \( S = \{(x, y) | x^2 + 4y^2 \le 12 : x, y \in \mathbb{R}\} \) is a convex set
C. A set \( S = \{(x, y) | y^2 - 4x \le 0 : x, y \in \mathbb{R}\} \) is a convex set
D. A set \( S = \{(x, y) | x^2 + 4y^2 \ge 12 : x, y \in \mathbb{R}\} \) is a convex set
The value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) is:
Match List-I with List-II and choose the correct option:
Match List-I with List-II and choose the correct option:
The function \( f(z) = |z|^2 \) is differentiable, at
If C is the positively oriented circle represented by \( |z|=2 \), then \( \int_C \frac{e^{2z}}{z-4} dz \) is:
Let f be a continuous function on \(\mathbb{R}\) and \( F(x) = \int_{x-2}^{x+2} f(t) dt \), then F'(x) is
If C is a triangle with vertices (0,0), (1,0) and (1,1) which are oriented counter clockwise, then \( \oint_C 2xydx + (x^2+2x)dy \) is equal to:
The integral domain of which cardinality is not possible:
A. 5
B. 6
C. 7
D. 10
Let \(m, n \in \mathbb{N}\) such that \(m < n\) and \(P_{m \times n}(\mathbb{R})\) and \(Q_{n \times m}(\mathbb{R})\) are matrices over real numbers and let \(\rho(V)\) denotes the rank of the matrix V. Then, which of the following are NOT possible.
A. \( \rho(PQ) = n \)
B. \( \rho(QP) = m \)
C. \( \rho(PQ) = m \)
D. \( \rho(QP) = \lfloor(m+n)/2\rfloor \), where \(\lfloor \rfloor\) is the greatest integer function
Which of the following are subspaces of vector space \(\mathbb{R}^3\):
A. \( \{(x,y,z) : x+y=0\} \)
B. \( \{(x,y,z) : x-y=0\} \)
C. \( \{(x,y,z) : x+y=1\} \)
D. \( \{(x,y,z) : x-y=1\} \)
Maximize \( Z = 2x+3y \), subject to the constraints:
\( x+y \le 2 \)
\( 2x+y \le 3 \)
\( x,y \ge 0 \)
Which one of the following mathematical structure forms a group?
If \( A = \begin{pmatrix} 2 & 4 & 1
0 & 2 & -1
0 & 0 & 1 \end{pmatrix} \) satisfies \( A^3 + \mu A^2 + \lambda A - 4I_3 = 0 \), then the respective values of \( \lambda \) and \( \mu \) are:
Let A and B be two symmetric matrices of same order, then which of the following statement are correct:
A. AB is symmetric
B. A+B is symmetric
C. \( A^T B = AB^T \)
D. \( BA = (AB)^T \)
Match List-I with List-II and choose the correct option:
Consider the function \( f(x, y) = x^2 + xy^2 + y^4 \), then which of the following statement is correct:
If \( f(z) = (x^2-y^2-2xy) + i(x^2-y^2+2xy) \) and \( f'(z)=cz \), where c is a complex constant, then \( |c| \) is equals to:
Match List-I with List-II and choose the correct option:
The solution of the differential equation \( \frac{xdy - ydx}{xdx + ydy} = \sqrt{x^2+y^2} \) is:
If, \( I_n = \int_{-\pi}^{\pi} \frac{\cos(nx)}{1+2^x} dx, n=0, 1, 2, \dots \), then which of the following are correct:
A. \( I_n = I_{n+2} \), for all \( n=0, 1, 2, \dots \)
B. \( I_n = 0 \), for all \( n=0, 1, 2, \dots \)
C. \( \sum_{n=1}^{10} I_n = 2^{10} \)
D. \( \sum_{n=1}^{10} I_n = 0 \)
The number of maximum basic feasible solution of the system of equations AX = b, where A is m \( \times \) n matrix, b is n \( \times \) 1 column matrix and rank of A is \(\rho(A) = m\), is:
If \( \vec{F} = x^2 \hat{i} + z \hat{j} + yz \hat{k} \), for \( (x, y, z) \in \mathbb{R}^3 \), then \( \oiint _{S} \vec{F} \cdot d\vec{S} \), where S is the surface of the cube formed by \( x = \pm 1, y = \pm 1, z = \pm 1 \), is
Find the residue of \( (67 + 89 + 90 + 87) \pmod{11} \)
The solution of the differential equation \( (xy^3 + y)dx + (2x^2y^2 + 2x + 2y^4)dy = 0 \) is:
If G is a cyclic group of order 12, then the order of Aut(G) is:
Which of the following function is discontinuous at every point of \(\mathbb{R}\)?
If the vectors \( \begin{pmatrix} 1
-1
3 \end{pmatrix}, \begin{pmatrix} 1
2
-3 \end{pmatrix}, \begin{pmatrix} p
0
1 \end{pmatrix} \) are linearly dependent, then the value of p is:
If \( \vec{F} \) is a vector point function and \( \phi \) is a scalar point function, then match List-I with List-II and choose the correct option:
The value of integral \( \oint_C \frac{z^3-z}{(z-z_0)^3} dz \), where \( z_0 \) is outside the closed curve C described in the positive sense, is
The solution of the differential equation \( (x^2 - 4xy - 2y^2)dx + (y^2 - 4xy - 2x^2)dy = 0 \), is
If \( x \in \mathbb{R} \) and a particular integral (P.I.) of \( (D^2 - 2D + 4)y = e^x \sin x \) is \( \frac{1}{2} e^x f(x) \), then \( f(x) \) is:
The value of \( \int_{0}^{\pi} \frac{x \sin x}{1+\cos^2 x} dx \) is:
The orthogonal trajectory of the cardioid \( r = a(1 - \cos\theta) \), where 'a' is an arbitrary constant is:
If \( \vec{F} \) be the force and C is a non-closed arc, then \( \int_C \vec{F} \cdot d\vec{r} \) represents:
In Green's theorem, \( \oint_C (x^2y dx + x^2 dy) = \iint_R f(x,y) dx dy \), where C is the boundary described counter clockwise of the triangle with vertices (0, 0), (1, 0), (1, 1) and R is the region bounded by a simple closed curve C in the x-y plane, then \( f(x,y) \) is equal to:
The value of \( \int_0^{1+i} (x^2 - iy) dz \), along the path \( y = x^2 \) is:
Let \( f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be defined as \( f(x,y) = \begin{cases} \frac{x}{\sqrt{x^2+y^2}} & ; (x,y) \neq (0,0)
1 & ; (x,y) = (0,0) \end{cases} \), then which of the following statement is true?
The system of linear equations \( x+y+z=6 \), \( x+2y+5z=10 \), \( 2x+3y+\lambda z=\mu \) has a unique solution, if
For the given linear programming problem,
Minimum Z = 6x + 10y
subject to the constraints
\( x \ge 6 \); \( y \ge 2 \); \( 2x+y \ge 10 \); \( x,y \ge 0 \),
the redundant constraints are:
Which of the following statements are true for group of permutations?
A. Every permutation of a finite set can be written as a cycle or a product of disjoint cycles
B. The order of a permutation of a finite set written in a disjoint cycle form is the least common multiple of the lengths of the cycles
C. If \(A_n\) is a group of even permutation of n-symbol (\(n > 1\)), then the order of \(A_n\) is n!
D. The pair of disjoint cycles commute
A ring (R, +, .), where all elements are idempotent is always:
If \( v = \sin^{-1} \left( \frac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}} \right) \), then \( x\frac{\partial v}{\partial x} + y\frac{\partial v}{\partial y} \) is equal to :
Let D be the region bounded by a closed cylinder \( x^2+y^2=16 \), \( z=0 \) and \( z=4 \), then the value of \( \iiint_D (\nabla \cdot \vec{v}) dV \), where \( \vec{v} = 3x^2\hat{i} + 6y^2\hat{j} + z\hat{k} \), is:
The value of the double integral \( \iint_R e^{x^2} dxdy \), where R is a region given by \( 2y \le x \le 2 \) and \( 0 \le y \le 1 \), is:
Let A be a \( 2 \times 2 \) matrix with \( \det(A) = 4 \) and \( trace(A) = 5 \). Then the value of \( trace(A^2) \) is:
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:
If 'a' is an imaginary cube root of unity, then \( (1-a+a^2)^5 + (1+a-a^2)^5 \) is equal to:
The solution of the differential equation \( xdy - ydx = (x^2+y^2)dx \), 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 \( 1 - \frac{1}{2^p} + \frac{1}{3^p} - \frac{1}{4^p} + \dots (p>0) \) is conditionally convergent, if 'p' lies in the interval:
Which of the following set of vectors forms the basis for \( \mathbb{R}^3 \)?
If p is a prime number and O(G) denotes the order of a group G and p divides O(G), then group G has an element of order p. Then, this is a statement of
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:
Which of the following forms a linear transformation:
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
The value of the integral \( \int_0^\infty \int_0^x x e^{-x^2/y} dy \, dx \) is:
If, \( u=y^3-3x^2y \) be a harmonic function then its corresponding analytic function \( f(z) \), where \( z=x+iy \), 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:
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 \( S = \{1, -1, \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3}, \dots \} \) is neither open set nor closed set
Match List-I with List-II and choose the correct option:
The locus of point z which satisfies \( \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3} \) is:
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