GATE EC Engineering Mathematics questions

Let $f\left ( x,y \right )=\dfrac{ax^{2}+by^{2}}{xy},$ where $a$ and $b$ are constants. If $\dfrac{\partial f}{\partial x}=\dfrac{\partial f}{\partial y}$ at $x = 1$ and $y = 2$, then the relation between $a$ and $b$ is

Consider matrix $A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix}$ and vector $x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}.$ The number of distinct real value of $k$ for which the equation $Ax=0$ has infinitely many solutions is _________.

Let $X_{1},\:X_{2},\:X_{3}$ and $X_{4}$ be independent normal random variable with zero mean and unit variance. The probability that $X_{4}$ is the smallest among the four is ________.

Taylor series expansion of $f\left ( x \right )=\int ^{x}_{0}e^{-\left ( \frac{t^{2}}{2} \right )}dt$ around $x=0$ has the form $$f\left ( x \right )={a}_{0}+a_{1}x+a_{2}x^{2}+...$$ The coefficient $a_{2}$ (correct to two decimal places) is equal to ________.

A curve passes through the point $\left ( x=1,y=0 \right )$ and satisfies the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x^{2}+y^{2}}{2y}+\dfrac{y}{x}.$ The equation that describes the curve is

A random variable $X$ takes values $-0.5$ and $0.5$ with probabilities $\dfrac{1}{4}$ and $\dfrac{3}{4}$, respectively. The noisy observation of $X\:\text{is}\:Y=X+Z,$ where $Z$ has uniform probability density over the interval $(-1,1).\: X$ and $Z$ are independent. If the MAP rule based detector outputs $\hat{X}$ as …

The position of a particle $y\left ( t \right )$ is described by the differential equation: $$\frac{d^{2}y}{dt^{2}}=-\frac{dy}{dt}-\frac{5y}{4}.$$ The initial conditions are $y\left ( 0 \right )=1$ and $\frac{dy}{dt}\mid_{t=0}=0$. The position (accurate to two decimal places) of the particle at $t=\pi$ is _________.

The contour $C$ given below is on the complex plane $z=x+j y,$ where $j=\sqrt{-1}.$ The value of the integral $\displaystyle{}\dfrac{1}{\pi j}\oint _{C}\dfrac{dz}{z^{2}-1}$ is _______.

Let $r=x^{2}+y-z$ and $z^{3}-xy+yz+y^{3}=1.$ Assume that $x$ and $y$ are independent variables. At $\left( x,y,z \right)=\left ( 2,-1,1 \right ),$ the value (correct to two decimal places) of $\dfrac{\partial r}{\partial x}$ is _________ .

Consider the 5 $\times$ 5 matrix $$\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$$ It is given that A has only one real eigenvalue. Then the real eigenvalue of A is

The rank of the matrix $$\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$$ is

Consider the following statements about the linear dependence of the real valued function $y_{1}=1,y_{2}=x$ and $y_{3}=x^{2}$ over the field of real numbers. Which one among the following is correct? Both I and II are true Both I and III are true Both II and IV are true Both III and IV are true

Three fair cubical dice are thrown simultaneously . The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place)________.

Let $f(x)=e^{x+x^{2}}$ for real $x$ . From among the following, choose the Taylor series approximation of $f(x)$ around $x=0$, which includes all powers of $x$ less than or equal to $3$.

A three dimensional region $R$ of finite volume is described by $x^2 + y^2 \leq z^3; \: \: 0 \leq z \leq 1,$ where $x,y,z$ are real. The volume of $R$ (up to two decimal places) is _________

Let $I=\int_{c}\left ( 2zdx+2ydy+2xdx \right )$ where $x,y,z$ are real, and let $C$ be the straight line segment from point $A:(0,2,1)$ to point $B:(4,1,-1)$.The value of $I$ is ____________.

Which one of the following is the general solution of the first order differential equation $$\frac{dy}{dx}=(x+y-1)^{2},$$ where $x,y$ are real?

Starting with $x=1$, the solution of the equation $x^{3}+x=1$, after two iterations of Newton-Raphson’s method (up to two decimal places) is__________.

Which one of the following options correctly describes the locations of the roots of the equation $s^{4}+s^{2}+1=0$ on the complex plane?

The rank of the matrix $\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0& 0& 1& -1& \end{bmatrix}$ is ________.