List of top Mathematics Questions asked in CUET (PG)

If \(1, \alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_{n-1}\) are n roots of the equation, \( x^n = 1 \) then the value of \( (1-\alpha_1)(1-\alpha_2)(1-\alpha_3)\ldots(1-\alpha_{n-1}) \) is

The value of \( \int_2^3 \vec{A} \cdot \frac{d\vec{A}}{dt} dt \) if \( \vec{A}(2) = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{A}(3) = 4\hat{i} - 2\hat{j} + 3\hat{k} \) 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

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; 9x^2 - 12xy + 4y^2 - 74x - 98y + 324 = 0 & (I) \; \text{Hyperbola} \\ (B) \; 12x^2 + 7xy - 12y^2 + 10x + 55y - 125 = 0 & (II) \; \text{A pair of straight lines} \\ (C) \; x^2 + 3xy + 2y^2 + x + y = 0 & (III) \; \text{Ellipse} \\ (D) \; 5x^2 + y^2 - 30x + 1 = 0 & (IV) \; \text{Parabola} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

The general solution of differential equation \( \frac{d^2y}{dx^2} + 9y = \cos(3x) \) is:

If the rank of matrix \[ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & \lambda \end{pmatrix} \] is 2, then the value of \(\lambda\) is:

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \dfrac{y^2}{36} - \dfrac{x^2}{16} = 1 & (I) \; \text{Eccentricity is } 2\sqrt{2} \\ (B) \; 7x^2 + 12xy - 2y^2 - 2x + 4y - 7 = 0 & (II) \; \text{Eccentricity is } \tfrac{3}{2} \\ (C) \; 7x^2 - y^2 = 224 & (III) \; \text{Eccentricity is } \tfrac{\sqrt{13}}{3} \\ (D) \; \dfrac{x^2}{16} - \dfrac{y^2}{20} = \dfrac{1}{9} & (IV) \; \text{Asymptotes are } y = \pm \tfrac{3}{2}x \\ \hline \end{array} \]

Which of the following statements are true?
(A) The curve r = a(1 + cos\(\theta\)) is symmetrical about the initial line
(B) The curve r = 2(1 - 2 sin\(\theta\)) is symmetrical about the initial line
(C) The curve r = a(1 + sin\(\theta\)) is symmetrical about the line \(\theta\) = \(\pi\)/2
(D) The curve r = a sin(3\(\theta\)) is symmetrical about the initial line
(E) The curve r² = a²cos(2\(\theta\)) is symmetrical about pole
Choose the correct answer from the options given below:

If the radius of curvature (\(\rho\)) at (0, 1) of \(y = e^x\) is \(\alpha\sqrt{\beta}\), then \(\alpha^2+\beta\) is:

For what values of n, \( \tan^{-1}3 + \tan^{-1}n = \tan^{-1}\left(\frac{3+n}{1-3n}\right) \) is valid

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; P \text{ and } Q \text{ are two perpendicular forces, acting at a point} & (I) \; R = |P - Q| \\ (B) \; P \text{ and } Q \text{ are equal, forces acting at a point at an angle } \alpha & (II) \; R = P + Q \\ (C) \; P \text{ and } Q \text{ are acting at a point in same direction} & (III) \; R = 2P \cos(\tfrac{\alpha}{2}) \\ (D) \; P \text{ and } Q \text{ are acting at a point in opposite direction} & (IV) \; R = \sqrt{P^2 + Q^2} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

If two stones are thrown vertically upwards with their velocities in the ratio 2:5, then the ratio of the maximum heights attained by the stones is

A necessary and sufficient condition that the general equation of second degree \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) may represent a pair of straight lines is

The asymptotes of the curve \( (x^2 - a^2)(y^2 - b^2) = a^2b^2 \) are

The plane \(x + y + z = \sqrt{3}\lambda\) touches the sphere \(x^2 + y^2 + z^2 - 2x - 2y - 2z - 6 = 0\) if:

The system of equations
x + 2y + z = 6
x + 4y + 3z = 10
x + 4y + \(\lambda\)z = \(\mu\)
is inconsistent if

If R is a closed region in the xy-plane bounded by a simple closed curve C and if M(x, y) and N(x, y) are continuous functions of x and y having continuous derivative in R, then

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

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

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:

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

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

Let \(f\) be a continuous real valued function, defined by, \(f(x) = \sin x\), for all \(x \in [-\frac{\pi}{2}, \frac{\pi}{2}]\). Then which of the following does not hold.