List of top Mathematics Questions asked in CUET (PG)

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 \(m, n \in \mathbb{N}\) 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

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 value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) is:

The function \( f(z) = |z|^2 \) is differentiable, at

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 the positively oriented circle represented by \( |z|=2 \), then \( \int_C \frac{e^{2z}}{z-4} dz \) is:

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)

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 statement is true:

Let [x] be the greatest integer function, where x is a real number, then \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} ([x] + [y] + [z]) \, dx \, dy \, dz = \)

If \( S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right) ... \left(1-\frac{1}{n^2}\right) \), then S is equal to:

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(T\textsuperscript{2}), then which of the following are correct?
A. N(T) = R(T)
B. N(T) = N(T\textsuperscript{2})
C. N(T) \(\cap\) R(T) = \{0\
D. R(T) = R(T\textsuperscript{2})}

Which of the following subsets form subgroups of the group <ℤ, +>?

(A). H₁ = {0}
(B). H₂ = {n+1 : n ∈ ℤ}
(C). H₃ = {2n : n ∈ ℤ}
(D). H₄ = {2n+1 : n ∈ ℤ}

Choose the correct answer from the options given below:

What is the key principle behind Monte Carlo simulation?

What is the fundamental assumption behind a Markov model?

If a subset B is a basis of a vector space V, then
(A). B generates V.
(B). B contains zero vector.
(C). B is linearly independent.
(D). B is the only basis of V.
Choose the correct answer from the options given below:

The series \( \sum_{n=1}^{\infty} \frac{1}{n} \)

The equation of a straight line passes through the point (4,-5) and is perpendicular to the straight line 3x + 4y + 5 = 0.

Let an unbiased die be thrown and the random variable X be the number appears on its top. Then the expectation of X is

The integral \( \int_{0}^{\pi/2} \sin^5 x \cos^7 x \,dx = \)

The solution of \(y = xp + \frac{m}{p}\) where \(p = \frac{dy}{dx}\) is