List of top Engineering Mathematics Questions asked in GATE IN

The signal \( x(t) = (t - 1)^2 u(t - 1) \), where \( u(t) \) is the unit-step function, has the Laplace transform \( X(s) \). The value of \( X(1) \) is_________

Let \( f(z) = \dfrac{1}{z^2 + 6z + 9} \) defined in the complex plane. The integral \( \oint_{c} f(z) \, dz \) over the contour of a circle \( c \) with center at the origin and unit radius is \(\underline{\hspace{2cm}}\).

The determinant of the matrix \( M \) shown below is \(\underline{\hspace{2cm}}\). \[ M = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 0 & 0 \\ 0 & 0 & 4 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

Given \( A = \begin{pmatrix} 2 & 5 \\ 0 & 3 \end{pmatrix} \), the value of the determinant \( |A^4 - 5A^3 + 6A^2 + 2I| \) is \(\underline{\hspace{2cm}}\).

Consider that \( X \) and \( Y \) are independent continuous valued random variables with uniform PDF given by \( X \sim U(2, 3) \) and \( Y \sim U(1, 4) \). Then \( P(Y \leq X) \) is equal to \(\underline{\hspace{2cm}}\) (rounded off to two decimal places).

The function \[ f(z) = (z - 1)^{-1} - 1 + (z - 1) - (z - 1)^2 + \cdots \] is the series expansion of

Consider the function \( f(x) = -x^2 + 10x + 100 \). The minimum value of the function in the interval \( [5, 10] \) is \(\underline{\hspace{2cm}}\).

Consider the sequence \[ x_n = 0.5x_{n-1} + 1, n = 1, 2, \ldots \text{ with } x_0 = 0. \text{ Then } \lim_{n \to \infty} x_n \text{ is} \]

Consider the row vectors \( v = (1, 0) \) and \( w = (2, 0) \). The rank of the matrix \[ M = 2v^T v + 3w^T w \] where the superscript \( T \) denotes the transpose, is