List of top Engineering Mathematics Questions asked in GATE Engineering Sciences

For \( a, b \in \mathbb{R} \), consider the system of linear equations \[ x + y + az = 2 \] \[ 2y + 2z = 1 \] \[ ax + 2z = b \] If the system has infinitely many solutions, then which of the following statements is/are correct?

Consider the function \[ f(x, y) = x^2y + 2xy^2 - 2x^2y^2. \] Then which one of the following statements is correct?

Let \( C \) be the positively oriented boundary of the domain bounded by the curves \( y = 2x^2 \) and \( y^2 = 4x \). Then the value of the line integral \[ \oint_C \left( 2y^2 + 2xy + 4y \right) dx + \left( x^2 + 4xy + 8x \right) dy \] is equal to:

Let \( u(x, t) \) be the solution of the initial boundary value problem \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u = 0, 0<x<\pi, t>0, \] \[ u(x, 0) = 2 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right), 0<x<\pi, \] \[ u(0, t) = u(\pi, t) = 0, t>0. \] Then the value of \( \lim_{t \to \infty} u \left( \frac{3\pi}{4}, t \right) \) is equal to (rounded off to two decimal places):

Suppose the polynomial \( a + bx + cx^2 + dx^3 \) interpolates the data,
\[ (-1,1), (0,3), (1,2), (2,4). \] Then which one of the following statements is correct?

Let \( f(z) \) be an analytic function such that \[ {Re}(f'(z)) = 3x^2 - 4y - 3y^2, f(i) = 0, f'(0) = 0, \] where \( i = \sqrt{-1} \). Then the value of \( f(1) \) is equal to:

Let \( y(x) \) be the solution of the initial value problem, \[ x^2 y'' + xy' - y = 0, x>0, y(1) = 0, y'(1) = 2. \] Then the value of \( y'\left(\frac{1}{2}\right) \) is equal to (Answer in integer) ................

Consider the second order Partial Differential Equation (PDE)
\[ 4x^2 \frac{\partial^2 u}{\partial x^2} + 4(x + y) \frac{\partial^2 u}{\partial x \partial y} + (x^2 + y^2) \frac{\partial^2 u}{\partial y^2} - u = 0. \] Then which one of the following statements is correct?

Consider the infinite series
\[ (P): \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1/n}} {and} (Q): \sum_{n=1}^{\infty} \frac{n^n}{(2n)!}. \] Then which one of the following statements is correct?