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?