GATE Engineering Sciences

For the complex function f(z) = f(x + iy) = x³ – 4xy² + i(4x²y – 2y³), consider the following two statements P: f is not analytic at (0, 0). Q: f does not satisfy the Cauchy-Rieman...

Let C be the circle (x – 1)² + y² = 25 oriented counterclockwise. Then, the value of the line integral $\oint_C [(x^5 – 3y)dx + (−2x + e^{y^2})dy]$ is

Let W(x) denote the Wronskian of two linearly independent solutions y₁(x) and y₂(x) of the differential equation $(x-1)\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + xe^xy = 0, \quad x > 1$....

Which of the following statements is/are true?

Consider the matrix A = $\begin{bmatrix} 1 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Then, which of the following statements is/are true?

Let L be the lamina of the form $x^2 + 4y^2 \le 64$, $0 \le y \le 4$, with density $\rho(x, y) = |x|y$. Then, the mass of L (in integer) is _________

Let X be a random variable having probability density function (pdf) $f_X(x) = \begin{cases} \frac{e}{2(e-1)}e^{-x}, & 0 < x < 1, \\ \frac{1}{2}, & 1 < x < 2, \\ 0, & \text{elsewhe...

Match each entry of List-1 with a suitable entry in List-2 and choose the correct option. List-1 P The sum of the series $\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}$ is equal to Q $\...

Consider the following partial differential equation (PDE) $(y - 1)\frac{\partial^2 u}{\partial x^2} - (x - 3)^2\frac{\partial^2 u}{\partial y^2} + y^2\frac{\partial u}{\partial x}...

Let $\begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & -10 \end{bmatrix} \begin{bmatrix} 1 & u_{12} & u_{13} \\ 0 & 1 & u_{23} \\ 0 & 0 & 1 \end{bmatrix}$...

The initial value problem $\frac{du}{dt} = u^2 + t^2$, $t \ge 0$, with $u(0) = 1$, is solved by using the explicit Euler method with step size $h = 0.2$. Then, the value of $u(0.4)...