List of top Engineering Mathematics Questions asked in GATE EE

Let \( (x, y) \in \mathbb{R}^2 \). The rate of change of the real-valued function \[ V(x, y) = x^2 + x + y^2 + 1 \] at the origin in the direction of the point \( (1, 2) \) is __________ (round off to the nearest integer).

Consider ordinary differential equations given by \[ \frac{dx_1(t)}{dt} = 2x_2(t), \quad \frac{dx_2(t)}{dt} = r(t) \] with initial conditions \( x_1(0) = 1 \) and \( x_2(0) = 0 \). If \[ r(t) = \begin{cases} 1, & t \geq 0 \\ 0, & t < 0 \end{cases} \] then at \( t = 1 \), \( x_1(t) = \underline{2cm} \) (round off to the nearest integer).
Let \( R \) be a region in the first quadrant of the \( xy \)-plane enclosed by a closed curve \( C \) considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region \( R \)?
Let \(\vec{E}(x, y, z) = 2x^2 \hat{i} + 5y \hat{j} + 3z \hat{k}\). The value of \[ \iiint_V (\vec{\nabla} \cdot \vec{E}) \, dV, \] where \( V \) is the volume enclosed by the unit cube defined by \( 0 \leq x \leq 1, 0 \leq y \leq 1, \) and \( 0 \leq z \leq 1 \), is
Let \( f(x, y, z) = 4x^2 + 7xy + 3xz^2 \). The direction in which the function \( f(x, y, z) \) increases most rapidly at point \( P = (1, 0, 2) \) is
Let the probability density function of a random variable \( x \) be given as \[ f(x) = ae^{-2|x|} \] The value of ‘a’ is ________.
Consider a matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -2 \\ 0 & 1 & 1 \end{bmatrix} \). The matrix \( A \) satisfies the equation \( 6A^{-1} = A^2 + cA + dI \), where \( c \) and \( d \) are scalars and \( I \) is the identity matrix. Then \( (c + d) \) is equal to
Let \( f(x) = \int_0^x e^t (t - 1)(t - 2) \, dt \). Then \( f(x) \) decreases in the interval
e\(^A\) denotes the exponential of a square matrix A. Suppose \( \lambda \) is an eigenvalue and \( \mathbf{v} \) is the corresponding eigen-vector of matrix A. Consider the following two statements: Statement 1: e\(^\lambda\) is an eigenvalue of e\(^A\). Statement 2: \( \mathbf{v} \) is an eigen-vector of e\(^A\). Which one of the following options is correct?
Consider a 3 x 3 matrix A whose \( (i, j) \)-th element, \( a_{i,j} = (i - j)^3 \). Then the matrix A will be