List of top Mathematics Questions

\[ \sec^{-1} \left( -\sqrt{2} \right) - \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \] is equal to:

The following graph is a combination of:

If \[ A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \] then $A$ is a/an:
The probability of a component being defective is 0.01. There are 100 such components in a machine. Then the probability of two or more defective components in the machine is _______
If the function \( f(z) = -x^2 + xy + y^2 + i(ax^2 + bxy + cy^2) \) of complex variable \(z = x + iy\) is analytic in the complex plane, then the values of \(a, b, c\) are _______
Let \( A \) be a \(3 \times 3\) matrix and \( B = 2A^2 + A^{-1} - I \), where \( I \) is a \(3 \times 3\) identity matrix. If the eigenvalues of \( A \) are 1, –1 and 2, then the trace of \( B \) is ________
Let \( A = \begin{bmatrix} a+1 & b & c \\ a & b+1 & c \\ a & b & c+1 \end{bmatrix} \). If determinant of the matrix \( A \) is zero, then \( (a + b + c)^3 = \_\_\_\_\_\_ \)
The general solution of the differential equation \( x^2 y'' - xy' + 5y = 0 \) is _______
The value of the real variable \( x>0 \) that minimizes the function \( f(x) = x^x e^{-x} \) is ________

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \text{ and } \mathbf{a} \times \mathbf{b} \neq 0 \text{ Show that } \mathbf{b} = \mathbf{c}$.

The angle between vectors $ \mathbf{a} = \hat{i} + \hat{j} - 2\hat{k} $ and $ \mathbf{b} = 3\hat{i} - \hat{j} + 2\hat{k} $ is:
If \[ f(x) = \begin{cases} \frac{\sin^2(ax)}{x^2}, & x \neq 0 \\ 1, & x = 0 \end{cases} \] is continuous at \( x = 0 \), then the value of \( a \) is:

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\) and \(\mathbf{a} \times \mathbf{b} \neq 0. Show \;that \;\mathbf{b} = \mathbf{c}\).

If \( A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 3 & 0 & 5 \\ 0 & 0 & 5 \end{array} \right] \), then \( A^3 \) is:

If $y = 5 \cos x - 3 \sin x$, prove that  $\frac{d^2y}{dx^2} + y = 0$.

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ such that $|\vec{a}| = 3$, $|\vec{b}| = 5$, $|\vec{c}| = 7$, then find the angle between $\vec{a}$ and $\vec{b}$.

Find the foot of the perpendicular drawn from the point \( (1, 1, 4) \) on the line \( \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} \).
Find \( \frac{dy}{dx} \) if \( x^3 + y^3 + x^2 = a^b \), where \( a \) and \( b \) are constants.
Find the interval/intervals in which the function \( f(x) = \sin 3x - \cos 3x \), \( 0<x<\frac{\pi}{2} \) is strictly increasing.
Solve the differential equation: \[ x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x \]
Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]
Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]
\( \int \frac{\cos 2x - \cos 2\theta}{\cos x - \cos \theta} dx \) is equal to:
Evaluate: \( \int_0^1 \frac{2x}{5x^2 + 1} dx \)
Let \( A \) and \( B \) be two matrices of suitable orders. Then, which of the following is not correct?