List of top Mathematics Questions asked in CUET (PG)

Which partial differential equation represents the Laplace equation in two dimensions?
What is the order of the group of permutations \( S_3 \)?
What are the possible values for the rank of a \(4 \times 3\) matrix?
In a metric space, is every Cauchy sequence necessarily a convergent sequence?
What is the value of the integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \)?
What is the condition for a group \( G \) to be Abelian based on the commutator subgroup?
If \( T: V \to W \) is a linear transformation, what is the relationship between rank(\(T\)), nullity(\(T\)), and dim(\(V\))?
Is the set of all rational numbers \( \mathbb{Q} \) a countable or uncountable set?
What is the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)?
Which theorem states that every bounded sequence in \( \mathbb{R}^n \) has a convergent subsequence?
If \( A \) is a \(3 \times 3\) matrix with eigenvalues \(1, 2, 3\), what is the determinant of \(A^2\)?
How many elements of order 5 are there in a cyclic group of order 25?
What is the value of the limit \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \)?
If a function \( f(x) \) is continuous on a closed interval \( [a,b] \), is it necessarily uniformly continuous?
What is the dimension of the vector space of all \( n \times n \) real symmetric matrices?

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

LIST-ILIST-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:

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)
The value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) is:
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 :
If the matrix \[ A = \begin{pmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{pmatrix} \] satisfies the matrix equation:
The surface area of the plane \(x + 2y + 2z = 12\) cut off by \(x=0, y=0\) and \(x^2+y^2=16\) is
The value of \( \oint_S \vec{F} \cdot d\vec{s} \) where \( \vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k} \) taken over the cylinder \( x^2+y^2=4, z=0 \) and \( z=3 \) is:
The directional derivative of \( \nabla \cdot (\nabla f) \) at the point (1, -2, 1) in the direction of the normal to the surface \( xy^2z = 3x + z^2 \) where \( f = 2x^3y^2z^4 \) and \( \nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \) is
Let \( \vec{F} \) be the vector valued function and f be a scalar function. Let \( \nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \) then,
(A) div (grad f) = \( \nabla^2 f \)
(B) curl curl \( \vec{F} \) = grad curl \( \vec{F} \) - \( \nabla^2 \vec{F} \)
(C) div curl \( \vec{F} \) = \( \vec{0} \)
(D) curl grad f = \( \vec{0} \)
(E) div (\(f\vec{F}\)) = f div \( \vec{F} \) + (grad f) \( \times \vec{F} \)
Choose the correct answer from the options given below:
The plane \(x + y + z = \sqrt{3}\lambda\) touches the sphere \(x^2 + y^2 + z^2 - 2x - 2y - 2z - 6 = 0\) if: