List of top Questions asked in CUET (PG)

Given below are two statements
Statement I Every homogeneous equation of second degree in x y and z represents a cone whose vertex is at the origin
Statement II If two equations representing the guiding curve are such that the one equation is of the first degree then the required cone with vertex at the origin is obtained by making the other equation homogeneous with the help of the first equation
Choose the most appropriate answer from the options given below
Solve the following differential equation by the method of Laplace transform: \[ y''' + 2y'' - y' - 2y = 0 \] given that $y(0) = 0$, $y'(0) = 0$, and $y''(0) = 6$.
Choose the correct answer from the options below:
Given below are two statements:
{Statement (I):} Two families of curves such that every member of either family cuts each member of the other family at right angles are called orthogonal trajectories of each other.
{Statement (II):} The orthogonal trajectories of the curve \( xy = c \) is \( y = \frac{1}{x} \).
Choose the most appropriate answer from the options given below:
Let f(x) be a differentiable function for all values of x with f′(x) ≤ 32 and f(3) = 21, then the maximum value of f(8) is:
The general solution of the differential equation \[ \frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = e^x \cos 2x \] is:
LIST I LIST II
A.d²y/dx² + 13y = 0 I. ex(c1 + c2x)
B.d²y/dx² + 4dy/dx + 5y = cosh 5x II. e2x(c1 cos 3x + c2 sin 3x)
C.d²y/dx² + dy/dx + y = cos²x III. c1ex + c2e3x
D.d²y/dx² - 4dy/dx + 3y = sin 3x cos 2x IV. e-2x(c1 cos x + c2 sin x)
Choose the correct answer from the options given below:
The value of tan(i log 2 − i√3 / 2 + i√3) is
The surface integral \[ \iint_S \mathbf{F} \cdot d\mathbf{S} \] where \( \mathbf{F} = x\hat{i} + y\hat{j} - z\hat{k} \) and \( S \) is the surface of the cylinder \( x^2 + y^2 = 4 \) bounded by the planes \( z = 0 \) and \( z = 4 \), equals:
The value of curl (\(\text{grad } f\)), where \( f = x^2 - 4y^2 + 5z^2 \), is:
Given below are two statements:
Statement (I): If F is an irrotational vector field, then the angular velocity of the vector field is always greater than zero.
Statement (II): For a solenoidal vector function, the divergence is always zero.
Choose the most appropriate answer from the options given below:
Let \( a \) be the magnitude of the directional derivative of the function: \[\phi(x, y) = \frac{x}{x^2 + y^2}\] along a line making an angle of \( 45^\circ \) with the positive x-axis at the point \( (0, 2) \). Then, the value of \( 1/a^2 \) is:
The volume of the solid standing on the area common to the curves \( x^2 = y, y = x \) and cut off by the surface \( z = y - x^2 \) is:
In a submarine telegraph cable, the speed of signaling varies as \( x^2 \log(1/x) \), where \( x \) is the ratio of the radius of the core to that of the covering. To attain the greatest speed, the value of this ratio is:
The asymptote of the spiral \( r = \frac{\theta}{\sin \theta} \) is:
If the radius of curvature of the Folium \( x^3 + y^3 - 3xy = 0 \) at the point \( (3/2, 3/2) \) is 5, then the value of \( b^2 + 2a + 1 \) is:
Given below are two statements:
Statement (I): The nth derivative of the function \( e^x \cos x \cos 2x \) is:
\[\frac{e^x}{2} \big[ (10^n) \cos(3x + n \tan^{-1} 3) + (2^n) \cos(x + \frac{n \pi}{4}) \big]\]
Statement (II): The nth derivative of the function \( x \cos 2x \cos 3x \) is:
\[\frac{1}{2} \big[ (2^n) \cos(2x + \frac{n \pi}{2}) + (4^n) \cos(4x + \frac{n \pi}{2}) + (6^n) \cos(6x + \frac{n \pi}{2}) \big]\]
In light of the above statements, choose the most appropriate answer from the options given below:
If \[ \theta = r e^{-x^2}, \] then for what value of \( n \), the following holds: \[ \frac{1}{2} \left( \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial r^2} \right) = \frac{\partial \theta}{\partial x}. \]
The principal value of \( i^i \) is:
The general value of \( \log(1 + i) + \log(1 - i) \) is:
Statement (I): The determinant of a matrix A and its transpose AT are equal.
Statement (II): The determinant of the product of two matrices A and B is the product of their determinants.
In light of the above statements, choose the most appropriate answer from the options given below:
If \( (\sqrt{3} + 1)^n + (\sqrt{3} - 1)^n = 4 \), then the value of \( n \) is:
In the system of linear equations AX = B, if A is a singular matrix and B is a null matrix, then which of the following is correct?
In the matrix equation: \[ \begin{bmatrix} 3 & -1 \\ 2 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}, \] the values of \( x \) and \( y \) are:
If A is a skew-symmetric matrix of odd order, then the determinant of A is:
Let P and Q be two matrices such that P Q = 0 and P is non-singular, then
(a) Q is also non-singular
(b) Q = 0
(c) Q is singular
(d) P = Q
Choose the correct answer from the options given below: