List of top Engineering Mathematics Questions asked in GATE Engineering Sciences

The surface area of the portion of the paraboloid \[ z=x^2+y^2 \] that lies between the planes $z=0$ and $z=\tfrac{1}{4}$ is
Let \[ P=\begin{bmatrix}4 & -2 & 2\\ 6 & -3 & 4\\ 3 & -2 & 3\end{bmatrix}, \qquad Q=\begin{bmatrix}3 & -2 & 2\\ 4 & -4 & 6\\ 2 & -3 & 5\end{bmatrix}. \] The eigenvalues of both $P$ and $Q$ are $1,1,2$. Which one of the following statements is TRUE?
Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$?
If the polynomial \[ P(x)=a_0+a_1x+a_2\,x(x-1)+a_3\,x(x-1)(x-2) \] interpolates the points $(0,2)$, $(1,3)$, $(2,2)$, and $(3,5)$, then the value of $P\!\left(\tfrac{5}{2}\right)$ is ____________ (round off to 2 decimal places).
If the quadrature formula \[ \int_{-1}^{1} f(x)\,dx \;\approx\; \frac{1}{9}\Big(c_1 f(-1) + c_2 f\!\left(\tfrac{1}{2}\right) + c_3 f(1)\Big) \] is exact for all polynomials of degree less than or equal to $2$, then
The second smallest eigenvalue of the eigenvalue problem \[ \frac{d^2 y}{dx^2} + (\lambda - 3)\,y = 0,\qquad y(0)=y(\pi)=0, \] is
Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,3$. If \[ B = A^2 + 2A + I, \] where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ are:
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by \[ f(x,y)= \begin{cases} \dfrac{xy}{|x|+y}, & y \neq -|x|, \\[6pt] 0, & \text{otherwise}. \end{cases} \] Then which one of the following statements is TRUE?
In a cosmopolitan city, the population comprises of 30% female and 70% male. Suppose that 5% of females and 30% of males in the population are foreigners. A person is selected at random from this population. Given that the selected person is a foreigner, the probability that the person is a female is ________ (round off to three decimal places).
If the quadrature rule \[ \int_{-1}^1 f(x) \, dx \approx f(\alpha) + \gamma f(\beta), \] where \( \alpha, \beta \) and \( \gamma \) are real constants, is exact for all polynomials of degree \( \leq 3 \), then \( \gamma + 3 (\alpha^2 + \beta^2) + (\alpha^3 + \beta^3) \) is equal to ________.

Let \( f : (0, \infty) \to \mathbb{R} \) be the continuous function such that:

\[ f(x) = 2 + \frac{g(x)}{x} \quad \text{for all} \ x > 0, \quad g(x) = \int_1^x f(t) \, dt \quad \text{for all} \ x > 0. \]

Then \( f(2) \) is equal to:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x, y) = 4xy - 2x^2 - y^4 + 1. \] The number of critical points where \( f \) has local maximum is equal to ________.
Let \[ A = \begin{pmatrix} 2 & 0 & 1 \\ 1 & 2 & 5 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix} \] . Then the sum of the geometric multiplicities of the distinct eigenvalues of \( A \) is equal to ________.

Let \( A \) and \( B \) be \( n \times n \) matrices with real entries. Consider the following statements:

  • P: If \( A \) is symmetric, then rank(\( A \)) = Number of nonzero eigenvalues (counting multiplicity) of \( A \)
  • Q: If \( AB = 0 \), then rank(\( A \)) + rank(\( B \)) \( \leq n \)

Then:

The value of \[ \lim_{x \to 0} \frac{1}{x} \int_{2}^{2+x} \left(t + \sqrt{t^2 + 5}\right) dt \]
Let \( \mathbb{C} = \{ z = x + iy : x \text{ and } y \text{ are real numbers}, i = \sqrt{-1} \} \) be the set of complex numbers. Let the function \( f(z) = u(x, y) + i v(x, y) \) for \( z = x + iy \in \mathbb{C} \) be analytic in \( \mathbb{C} \), where \[ u(x, y) = x^3 y - y x^3 \quad \text{and} \quad v(x, y) = \frac{x^4}{4} + \frac{y^4}{4} - \frac{3}{2} x^2 y^2. \] If \( f'(z) \) denotes the derivative of \( f(z) \), then:
If the partial differential equation \[ (x + 2) \frac{\partial^2 u}{\partial x^2} + 2(x + y) \frac{\partial^2 u}{\partial x \partial y} + 2(y - 1) \frac{\partial^2 u}{\partial y^2} - 3y^2 \frac{\partial u}{\partial y} = 0 \] is parabolic on the circle \( (x - a)^2 + (y - b)^2 = r^2 \), then the values of \( a \), \( b \), and \( r \) are given by:
Let \( y_1(x) \) and \( y_2(x) \) be two linearly independent solutions of the differential equation: \[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0, \quad x>0. \] Let \( W(y_1, y_2)(x) \) denote the Wronskian of \( y_1(x) \) and \( y_2(x) \) at \( x \). If \( W(y_1, y_2)(1) = 1 \), then \( W(y_1, y_2)(2) \) is equal to ________.
Let \( \Gamma \) be the positively oriented circle \( x^2 + y^2 = 9 \) in the xy-plane. If \[ \oint_{\Gamma} (3y + e^x \sin x) \, dx + \left( 7x + \sqrt{e^y + 2} \right) \, dy = \alpha \pi, \] where \( \alpha \) is a real constant, then \( \alpha \) is equal to ________.