List of top Engineering Mathematics Questions asked in GATE Instrumentation Engineering

Let the difference equation \( y[n] = \alpha y[n-1] + x[n] \), where \( \alpha>1 \) and \( \alpha \) is real, represent a causal discrete-time linear time invariant system. The system is initially at rest. If \( x[n] = -\delta[n-p] \), where \( p>10 \), the value of \( y[p+1] \) is:
Choose the correct statement(s) from the following options, regarding Cauchy's theorem on complex integration \( \oint_C f(z)\,dz \), where \( C \) is a simple closed path in a simply connected domain \( D \).
The value of the integral \( \int_{-\pi}^{\pi} (\cos^6 x + \cos^4 x) \, dx \) is:
Newton-Raphson method is used to compute the inverse of the number 1.6. Among the following options, the initial guess of the solution that results in non-convergence of the iterative process is:
Consider the function \( f(z) = \frac{2z + 1}{z^2 - 2z} \), where \( z \) is a complex variable. The sum of the residues at singular points of \( f(z) \) is:
If one of the eigenvectors of the matrix
\[ A = \begin{bmatrix} 1 & 1 \\ -4 & x \end{bmatrix} \] is along the direction of
\[ \begin{bmatrix} 2\alpha \\ \alpha \end{bmatrix} \] where \( \alpha \) is any non-zero real number, then the value of \( x \) is ________ (in integer).
The solution of the differential equation \(\dfrac{dy}{dx} = \dfrac{y}{x}\) represents:
A \( 2n \times 2n \) matrix \( A = [a_{ij}] \) has its elements defined as: \[ a_{ij} = \begin{cases} \beta(i + j), & \text{if } i + j \text{ is odd} \\ -\beta(i + j), & \text{if } i + j \text{ is even} \end{cases} \] where \( n \) is an integer greater than 2, and \( \beta \) is any non-zero real number. What is the rank of matrix \( A \)?
Choose the eigenfunction(s) of stable linear time-invariant continuous-time systems from the following options.
The probability of a student missing a class is \( 0.1 \). In a total number of 10 classes, the probability that the student will not miss more than one class is _______ (rounded off to two decimal places).
The value of the surface integral $\iint_S (2x + z) dy dz + (2x + z) dz dx + (2z + y) dx dy$ over the sphere $S: x^2 + y^2 + z^2 = 9$ is
What is $\lim_{x \to 0} f(x)$, where $f(x) = x \sin \frac{1}{x}$?
The rank of the matrix $A$ given below is one. The ratio $\dfrac{\alpha}{\beta}$ is ______ (rounded off to the nearest integer). \[ A = \begin{bmatrix} 1 & 4 \\ -3 & \alpha \\ \beta & 6 \end{bmatrix} \]
Consider the real-valued function \( g(x)=\max\{(x-2)^2,\,-2x+7\}\), \(x\in(-\infty,\infty)\). The minimum value attained by \(g(x)\) is _____(rounded off to one decimal place).
Let $f(z)=j\,\dfrac{1-z}{1+z}$, where $z$ is complex and $j=\sqrt{-1}$. The inverse function $f^{-1}(z)$ maps the {real axis} to the ______.
$X$ is a discrete random variable taking values $0,1,2$ with $P(X=0)=0.25$, $P(X=1)=0.5$, and $P(X=2)=0.25$. With $E[ . ]$ denoting expectation, compute $E[X]-E[X^2]$ (rounded off to one decimal place).
The solution $x(t)$, $t\ge 0$, to $\ddot{x}=-k\dot{x}$ ($k>0$) with $x(0)=1$ and $\dot{x}(0)=0$ is:
$F(z)=\dfrac{1}{1-z}$, when expanded as a power series around $z=2$, would result in $F(z)=\sum_{k=0}^{\infty} a_k (z-2)^k$ with ROC $|z-2|<1$. The coefficients $a_k,\;k\ge 0$, are given by the expression ________.
Choose solution set $S$ corresponding to the system of two equations \[ x-2y+z=0,\qquad x-z=0 \] (Note: $\mathbb{R}$ denotes the set of real numbers).