List of top Mathematics Questions

Let $r = \text{Min}\{\alpha, \beta, \gamma\}$, $R = \text{Max}\{\alpha, \beta, \gamma\}$, $f(z) = \frac{z}{(z-\alpha)(z-\beta)(z-\gamma)}$. $I_1 = \oint_{C_1} f(z)dz$ and $I_2 = \oint_{C_2} f(z)dz$, where $C_1 : |z| < r$ and $C_2 : |z| = R+1$, then $I_1 + I_2 = $}

The iterative formula for finding the approximate root of $f(x) = 0$ using Newton-Raphson method is:

Choose a possible probability density function from the given functions:

If $f(x) = x^3$, $0 \le x \le 4$, $f(x+4) = f(x)$ $\forall x \in \mathbb{R}$ and the Fourier series of $f(x)$ is $f(x) = \sum_{n=0}^{\infty} \left(a_n \cos \frac{n\pi x}{2} + b_n \sin \frac{n\pi x}{2}\right)$, then $a_0 =$}

If $A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 5 & 4 \\ 1 & 0 & 5 \end{bmatrix}$, then the determinant of $\left(A^{2026} - 11A^{2025} - 9A^{2023}\right)$ is equal to:}

If $z = x^2 y + e^{xy^2}$, then $\left(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial x \partial y}\right)$ evaluated at $(1,0)$ is:}

The particular integral of $\left(D^4 - D^3 - 9D^2 - 11D - 4\right)y = e^{-x}$, where $D = \frac{d}{dx}$, is:}

An Eigen value of the matrix $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$ is $1$. An eigen vector corresponding to it is:}

For a Binomial distribution, mean is \(15\) and variance is \(6\). If \[ P(X\ge2) = 1-\left(\frac25\right)^{25}k, \] then \(k=\)

The probability density function of a continuous random variable is \[ f(x)= \begin{cases} \dfrac35 e^{-3x/5}, & x>0 \\ 0, & x\le0 \end{cases} \] The mean of the distribution is

If \(\alpha, \beta\) are two distinct eigenvalues of a matrix \(A\), then the corresponding eigenvectors are

On the Argand plane, eigenvalues of any unitary matrix lie on

Given \[ P(A)=\frac{7}{20}, \quad P(B)=\frac{1}{2}, \quad P(C)=\frac{9}{20}, \] \[ P(A\cap B\cap C)=\frac{1}{20}, \quad P(A\cap B)=\frac{3}{20}, \] \[ P(A\cap C)=\frac{1}{8}, \quad P(B\cap C)=\frac{1}{5}, \] then \(P(B|\overline{A})\) is

\(f:\mathbb{R}\rightarrow\mathbb{R}\) is increasing in \((-\infty,0)\cup(1,\infty)\) and decreasing in \((0,1)\). If \(f(2)=6\) and the continuous curve \(y=f(x)\) passes through \((0,0)\), then

If \(Z\) is a standard normal variable, \[ P(Z>z_1)=\alpha, \] \[ P(Z<z_2)=\beta \] and \(z_2<z_1\), then a possible value of \[ P(z_2<Z<z_1) \] is

The maximum value of \(x^{4}y^{3}\) such that \(x+y=42\) exists at \(x=\alpha,\; y=\beta\). Then \(\dfrac{\alpha}{\beta}\) in its lowest form is

Evaluate \[ \int_{-1}^{1}\frac{x^{2}\sin x}{x^{4}+1}\,dx. \]

Let \(A\) and \(B\) be \(n\times n\) real matrices. Which of the following statements is correct?

The differential equation \[ 2\frac{\partial^2 z}{\partial x^2} + 5\frac{\partial^2 z}{\partial x\partial y} + 2\frac{\partial^2 z}{\partial y^2} + 7\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} =0 \] is classified as

If \(\mu_1,\mu_2\) and \(\mu_3\) are the mean, median and mode respectively of a normal distribution, then \[ 3(\text{Median})-2(\text{Mode}) = ? \]

Approximate positive root of the equation \[ x^2-7x+9=0 \] using Newton-Raphson method with initial guess \(x_0=2\).

The joint probability density function of a two-dimensional random variable \((X,Y)\) is \[ f(x,y)= \begin{cases} 2, & 0\\ 0, & \text{otherwise}. \end{cases} \] Which of the following is correct?

Evaluate \[ \iint_D x^2y\,dx\,dy, \] where \[ D:\;x^2+y^2\le16. \]

Evaluate \[ \oint_C \frac{z+2}{z^2+2z+2}\,dz, \] where \(C\) is the circle \[ |z-\alpha|=1, \qquad \alpha=-1+\frac{3}{2}i. \]

If \[ \overline{V}=x^{2}y\,\overline{i}-yz^{2}\,\overline{j}+xyz\,\overline{k} \] is the linear velocity of a particle in circular motion at the point \((1,1,1)\), then its angular velocity is