List of top Mathematics Questions

Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
If \( f(x) = \begin{cases 2x - 3, & -3 \leq x \leq -2
x + 1, & -2<x \leq 0 \end{cases} \), check the differentiability of \( f(x) \) at \( x = -2 \).}
If \( x = e^{\frac{x}{y}} \), then prove that \( \frac{dy}{dx} = \frac{x - y}{x \log x} \).
If \( |\mathbf{a}| = 2 \), \( |\mathbf{b}| = 3 \) and \( \mathbf{a} \cdot \mathbf{b} = 4 \), then evaluate \( |\mathbf{a} + 2\mathbf{b}| \).
A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function \( Z = 5x + 7y \), where \( x \) and \( y \) are the number of units of X and Y respectively sold. Which of the following statements is correct?
If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?
If \( A \) denotes the set of continuous functions and \( B \) denotes the set of differentiable functions, then which of the following depicts the correct relation between set \( A \) and \( B \)?
If a line makes angles of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) with the positive directions of \( x \), \( y \), and \( z \)-axes respectively, then \( \theta \) is:
The line \( x = 1 + 5\mu, y = -5 + \mu, z = -6 - 3\mu \) passes through which of the following points?
The function \( f(x) = x^2 - 4x + 6 \) is increasing in the interval:
The probability distribution of a random variable \( X \) is given by:
probability distribution of a random variable X
Then \( E(X) \) of distribution is:
If \( E \) and \( F \) are two events such that \( P(E)>0 \) and \( P(F) \neq 1 \), then \( P(E \,|\, F) \) is:
The equation of a line parallel to the vector \( 3 \hat{i} + \hat{j} + 2 \hat{k} \) and passing through the point \( (4, -3, 7) \) is:
The area of the shaded region (figure) represented by the curves \( y = x^2 \), \( 0 \leq x \leq 2 \), and the y-axis is given by:
The area of the shaded region
If the projection of \( \mathbf{a} = \alpha \hat{i} + \hat{j} + 4 \hat{k} \) on \( \mathbf{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \) is 4 units, then \( \alpha \) is:
If \( A \) and \( B \) are square matrices of the same order, then \( (A B^T - B A^T) \) is a:
If \( \int_0^a x \, dx \leq \frac{a}{2} + 6 \), then which of the following holds for \( a \)?
Which of the following can be both a symmetric and skew-symmetric matrix?
The value of \( \cos \left( \frac{\pi}{6} + \cot^{-1}(-\sqrt{3}) \right) \) is:
If \( p \) and \( q \) are respectively the order and degree of the differential equation \( \frac{d}{dx} \left( \frac{dy}{dx} \right)^3 = 0 \), then \( (p - q) \) is:
If \( \mathbf{p} \) and \( \mathbf{q} \) are unit vectors, then which of the following values of \( \mathbf{p} \cdot \mathbf{q} \) is not possible?
The sum of all local minimum values of the function \( f(x) \) as defined below is:

\[ f(x) = \left\{ \begin{array}{ll} 1 - 2x & \text{if } x < -1 \\ \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\ \frac{11}{18} (x-4)(x-5) & \text{if } x > 2 \end{array} \right. \]

Let \(\vec{a} = i + j + k\), \(\vec{b} = 2i + 2j + k\) and \(\vec{d} = \vec{a} \times \vec{b}\). If \(\vec{c}\) is a vector such that \(\vec{a} \cdot \vec{c} = |\vec{c}|\), \(\|\vec{c} - 2\vec{d}\| = 8\) and the angle between \(\vec{d}\) and \(\vec{c}\) is \(\frac{\pi}{4}\), then \(\left|10 - 3\vec{b} \cdot \vec{c} + |\vec{d}|\right|^2\) is equal to:
If \( 7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \cdots \infty \text{ terms}, \) then \( \alpha \) is equal to
Let ABCD be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides AD and BC of the trapezium be parallel to the y-axis. If the diagonal AC is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of ABCD is: