List of top Mathematics Questions

The number of accidents occurring in AU region in a month follows Poisson distribution with mean $\lambda = 5$. The probability of less than two accidents in a randomly selected month is ______
Let $X$ be an exponential random variable with mean parameter one. Then the conditional probability $P(X>10 | X>5)$ is equal to
A machine produces 0, 1 or 2 defective items in a day with probabilities of \( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \) respectively. Then, the standard deviation of the number of defective items produced by the machine in a day is ............
If \( X \) is a continuous random variable with the probability density
\[ f(x) = \begin{cases} K(1 - x^3), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Then, the value of \( K \) is ............
If \( A = \begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix} \) is a \( 2 \times 2 \) matrix, then the eigenvalues of the matrix \( 2A^2 - 4A + 5I \) are ........., where \( I \) is the \( 2 \times 2 \) unit matrix.

How many possible words can be created from the letters R, A, N, D (with repetition)?

The particular integral of the differential equation
\[ \frac{d^2y}{dx^2} - 6 \frac{dy}{dx} + 9y = e^{3x} \text{ is ............} \]
If $(a, b, c)$ is the unique solution of the system of linear equations
$x + y + z = 2, 2x + y - z = 3, 3x + 2y + z = 4$,
then $b^2 + c^2 = ........$
If \( i = \sqrt{-1} \), then \[ \sum_{n=2}^{30} i^n + \sum_{n=30}^{65} i^{n+3} = \]
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 ________
A machine produces 0, 1 or 2 defective items in a day with probabilities of \( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \) respectively. Then, the standard deviation of the number of defective items produced by the machine in a day is ...........
Let \( A, B \) be two events and \( \overline{A} \) be the complement of \( A \). If \( P(\overline{A}) = 0.7 \), \( P(B) = 0.7 \), and \( P(B|A) = 0.5 \), then \( P(A \cup B) = \) ...........
If the Laplace transform of a function \( f(t) \) is given by \( \frac{2s + 1}{(s + 1)(s + 2)} \), then \( f(0) \) is equal to ...........
If \( y(x) \) satisfies the differential equation \( x \frac{dy}{dx} + (x - y) = 0 \) subject to the condition \( y(1) = 0 \), then \( y(e) \) is ...........
If the matrix \( A = \begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix} \) has three distinct eigenvalues and one of its eigenvectors is \( \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \), then which of the following can be another eigenvector of \( A \)?
Determine the value of $\lambda$ and $\mu$ for which such that the system of equations $x + 2y + z = 6$, $x + 4y + 3z = 10$, and $2x + 4y + \lambda z = \mu$ has infinite number of solutions.
A project requires 12 workers to complete in 20 days. If 4 workers leave after 8 days, how many additional days will the remaining workers take to finish the project?
If $ 0 \le x \le 3,\ 0 \le y \le 3 $, then the number of solutions $(x, y)$ for the equation: $$ \left( \sqrt{\sin^2 x - \sin x + \frac{1}{2}} \right)^{\sec^2 y} = 1 $$
If \( A \) and \( B \) are two mutually exclusive events with \( P(B) \ne 1 \), then the conditional probability \[ P(A \mid \overline{B}) = \,? \] where \( \overline{B} \) is the complement of \( B \)
If \( X \) is a continuous random variable with the probability density function
\[ f(x) = \begin{cases} cx^3, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases} \] then \( P\left( \frac{1}{2} < X < \frac{3}{2} \right) \) is ..........