List of top Mathematics Questions

The integral \[ \int \frac{dx}{x^2 \left( x^4 + 1 \right)^{3/4}} \] equals:

The value of \( \int_{-1}^1 \sin^5 x \cos^4 x \, dx \) is:

The value of \( \int \frac{dx}{(x+1)(x+2)} \) is:

The value of \( \int_0^{\frac{2\pi}{0}} \left( 1 + \sin \left( \frac{x}{2} \right) \right) \, dx \) is:

If \( y = a \sin^3 t \), \( x = a \cos^3 t \), then \( \frac{dy}{dx} \) at \( t = \frac{3\pi}{4} \) is:

The minimum value of \( 1 - \sin x \) is:

Consider the following statements:
% Statement Statement (I): If either \( |\vec{a}| = 0 \) or \( |\vec{b}| = 0 \), then \( \vec{a} \cdot \vec{b} = 0 \).
% Statement Statement (II): If \( \vec{a} \times \vec{b} = 0 \), then \( \vec{a} \) is perpendicular to \( \vec{b} \).
Which of the following is correct?

The equation of the line through the point \( (0, 1, 2) \) and perpendicular to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{-2} \] is:

If a line makes angles \( 90^\circ, 60^\circ \) and \( \theta \) with \( x, y \) and \( z \) axes respectively, where \( \theta \) is acute, then the value of \( \theta \) is:

If \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - \hat{j} + 4\hat{k} \), and \( \vec{c} = \hat{i} + \hat{j} + \hat{k} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then the value of \( \lambda \) is:

If \( A \) is a square matrix satisfying the equation \( A^2 - 5A + 7I = 0 \), where \( I \) is the identity matrix and \( 0 \) is the null matrix of the same order, then \( A^{-1} \) is:

Let \( X \) and \( Y \) be continuous random variables with probability density functions \( P_X(x) \) and \( P_Y(y) \), respectively. Further, let \( Y = X^2 \) and \[ P_X(x) = \begin{cases} 1, & x \in (0,1] \\ 0, & \text{otherwise} \end{cases} \] Which one of the following options is correct?

Consider discrete random variables \( X \) and \( Y \) with probabilities as follows:

\( P(X=0 \text{ and } Y=0) = \frac{1}{4}, \)
\( P(X=1 \text{ and } Y=0) = \frac{1}{8}, \)
\( P(X=0 \text{ and } Y=1) = \frac{1}{2}, \)
\( P(X=1 \text{ and } Y=1) = \frac{1}{8}. \)

Given \( X = 1 \), the expected value of \( Y \) is

Consider the set \( S \) of points \( (x, y) \in \mathbb{R}^2 \) which minimize the real-valued function \( f(x, y) = (x + y - 1)^2 + (x + y)^2 \). Which of the following statements is true about the set \( S \)?

If the function
\[ f(x) = \begin{cases} \sin(2x), & \text{for } x > 0, \\ a + bx, & \text{for } x \leq 0, \end{cases} \]
where \( a \) and \( b \) are constants, is differentiable at \( x = 0 \), then \( a + b \) is equal to:

If $$ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6, \text{ then } f(1) \text{ is equal to:} $$ 

Which one of the following plots represents \( f(x) = -\frac{|x|}{x} \), where \( x \) is a non-zero real number?
Note: The figures shown are representative.