GATE Electrical Engineering

Consider the set $S$ of points $(x, y) \in R^2$ which minimize the real valued function $$ f(x, y)=(x+y-1)^2+(x+y)^2 $$ Which of the following statements is true about the set $S$...

Let $v_1$ and $v_2$ be the two eigen vectors corresponding to distinct eigen values of a $3 \times 3$ real symmetric matrix. Which one of the following statements is true?

Let $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1\end{array}\right]$ and $b=\left[\begin{array}{c}1 / 3 \\ -1 / 3 \\ 0\end{array}\right]$, then the system of l...

Let $P=\left[\begin{array}{ccc}2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^2$ is equal to

Consider discrete random variable $X$ and $Y$ with probabilities as follows: $$ \begin{aligned} & P(X=0 \text { and } Y=0)=\frac{1}{4} \\ & P(X=1 \text { and } Y=1)=\frac{1}{8} \\...

Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y=X^2$ and $P_X(x)=\left\{\begin{array}{cc}1, &...

Let $a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $f(R)=a_R \frac{1}{R^...

Consider ordinary differential equations given by $\dot{x}_1(t)=2 x_2(t), \dot{x}_2(t)=r(t)$ with initial conditions $x_1(0)=1$ and $x_2(0)=0$. If $r(t)=\left\{\begin{array}{ll}1,...

Let $C$ be a clockwise oriented closed curve in the complex plane defined by $|\lambda|=1$. Further, let $f(x)=j z$ be a complex function, where $j=\sqrt{-1}$. Then, $\oint_C f(z)...

Let $(x, y) \in \Re^2$. 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 nea...

Which one of the following matrices has an inverse?

Let $X$ be a discrete random variable that is uniformly distributed over the set {$-10, -9, \cdots, 0, \cdots, 9, 10$}. Which of the following random variables is/are uniformly dis...

Which of the following complex functions is/are analytic on the complex plane?

Consider the complex function $f(z) = \cos z + e^{z^2}$. The coefficient of $z^5$ in the Taylor series expansion of $f(z)$ about the origin is ______ (rounded off to 1 decimal plac...

The sum of the eigenvalues of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2$ is ______ (rounded off to the nearest integer).

Consider a vector $\vec{u} = 2\hat{x} + \hat{y} + 2\hat{z}$, where $\hat{x}$, $\hat{y}$, $\hat{z}$ represent unit vectors along the coordinate axes $x$, $y$, $z$ respectively. The...

Let $f(t)$ be a real-valued function whose second derivative is positive for $- \infty

Consider the function $f(t) = (\text{max}(0,t))^2$ for $- \infty

Which of the following differential equations is/are nonlinear?

For a given vector $${[\matrix{ 1 & 2 & 3 \cr } ]^T}$$, the vector normal to the plane defined by $${w^T}x = 1$$ is

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation $$\sigma_0$$. The $$\sigma_0$$ i...

In the following differential equation, the numerically obtained value of $$y(t)$$, at $$t=1$$ is ___________ (Round off to 2 decimal places). $${{dy} \over {dt}} = {{{e^{ - \alpha...

Three points in the x-y plane are ($$-$$1, 0.8), (0, 2.2) and (1, 2.8). The value of the slope of the best fit straight line in the least square sense is _________ (Round off to 2...

Consider the following equation in a 2-D real-space. $$|{x_1}{|^p} + |{x_2}{|^p} = 1$$ for $$p > 0$$ Which of the following statement(s) is/are true.

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________...

Consider a 3 $$\times$$ 3 matrix A whose (i, j)-th element, a i,j = (i $$-$$ j) 3 . Then the matrix A will be

e 4 denotes the exponential of a square matrix A. Suppose $$\lambda$$ is an eigen value and v is the corresponding eigen-vector of matrix A. Consider the following two statements:...

Let $$f(x) = \int\limits_0^x {{e^t}(t - 1)(t - 2)dt} $$. Then f(x) decreases in the interval.

Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equation 6A $$-$$1 = A 2 + cA + dI, where c and d...

Let $$f(x,y,z) = 4{x^2} + 7xy + 3x{z^2}$$. The direction in which the function f(x, y, z) increases most rapidly at point P = (1, 0, 2) is

Let $$\overrightarrow E (x,y,z) = 2{x^2}\widehat i + 5y\widehat j + 3z\widehat k$$. The value of $$\mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt V} {(\overrightarrow \nabla...

Let the probability density function of a random variable x be given as f(x) = ae $$-$$2|x| The value of a is _________.

Let $p$ and $q$ be real numbers such that $p^2+q^2=1$. The eigen values of the matrix $\left[\begin{array}{cc}p & q \\ q & -p\end{array}\right]$ are

Let $A$ be a $10 \times 10$ matrix such that $A^5$ is null matrix and let $I$ be the $10 \times 10$ identity matrix. The determinant of $A+I$ is $\_\_\_\_$ .

Let $P(z)=z^3+(1+j) z^2+(2+j) z+3$, where $z$ is complex number. Which one of the following is true?

Suppose the probability that a coin toss shows "head" is $p$, where $0

Suppose the circles $x^2+y^2=1$ and $(x-1)^2+(y-1)^2=r^2$ intersect each other orthogonally at the point $(u, v)$. Then $u+v=$ $\_\_\_\_$ .

Let $(-1-j),(3-j),(3+j)$ and $(-1+j)$ be the vertices of rectangle $C$ in the complex plane. Assuming that $C$ is traversed in counter-clockwise direction, the value of contour int...

Let $f(x)$ be a real-valued function such that $f^{\prime}\left(x_0\right)=0$ for some $x_0 \in(0,1)$ and $f^{\prime \prime}\left(x_0\right)>0$ for all $x \in(0,1)$. Then $f(x)$ ha...

In the open interval $(0,1)$, the polynomial $p(x)=x^4-4 x^3+2$ has

A function $$f(x)$$ is defined as $$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$ Which one of...

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most $$3$$ tosses. Let the random variable $$Y$$ denotes the number of heads. The value of var...

The value of the contour integral in the complex - plane $$\oint {{{{z^3} - 2z + 3} \over {z - 2}}} dz$$ along the contour $$\left| z \right| = 3,$$ taken counter-clockwise is

The matrix $$A = \left[ {\matrix{ {{3 \over 2}} & 0 & {{1 \over 2}} \cr 0 & { - 1} & 0 \cr {{1 \over 2}} & 0 & {{3 \over 2}} \cr } } \right]$$ has three distinct eigen values and o...

For a complex number $$z,$$ $$\mathop {Lim}\limits_{z \to i} {{{z^2} + 1} \over {{z^3} + 2z - i\left( {{z^2} + 2} \right)}}$$ is

The eigen values of the matrix given below are $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & { - 3} & { - 4} \cr } } \right]$$

Let $$x$$ and $$y$$ be integers satisfying the following equations $$$2{x^2} + {y^2} = 34$$$ $$$x + 2y = 11$$$ The value of $$(x+y)$$ is _________.

Consider the differential equation $$\left( {{t^2} - 81} \right){{dy} \over {dt}} + 5ty = \sin \left( t \right)\,\,$$ with $$y\left( 1 \right) = 2\pi .$$ There exists a unique solu...

Consider a function $$f\left( {x,y,z} \right)$$ given by $$f\left( {x,y,z} \right) = \left( {{x^2} + {y^2} - 2{z^2}} \right)\left( {{y^2} + {z^2}} \right).$$ The partial derivative...

An urn contains $$5$$ red balls and $$5$$ black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball...

Let $$g\left( x \right) = \left\{ {\matrix{ { - x} & {x \le 1} \cr {x + 1} & {x \ge 1} \cr } } \right.$$ and $$f\left( x \right) = \left\{ {\matrix{ {1 - x,} & {x \le 0} \cr {{x^{2...

Assume that in a traffic junction, the cycle of the traffic signal lights is $$2$$ minutes of green (vehicle does not stop) and $$3$$ minutes of red (vehicle stops). Consider that...

Let $$\,{y^2} - 2y + 1 = x$$ and $$\,\sqrt x + y = 5.\,\,$$ The value of $$\,x + \sqrt y \,\,$$ equals ________. (Given the answer up to three decimal places)

The value of line integral $$\,\,\int {\left( {2x{y^2}dx + 2{x^2}ydy + dz} \right)\,\,} $$ along a path joining the origin $$(0, 0, 0)$$ and the point $$(1, 1, 1)$$ is

The value of the integral $$\oint\limits_c {{{2z + 5} \over {\left( {z - {1 \over 2}} \right)\left( {{z^2} - 4z + 5} \right)}}} dz$$ over the contour $$\left| z \right| = 1,$$ take...

Consider the function $$f\left( z \right) = z + {z^ * }$$ where $$z$$ is a complex variable and $${z^ * }$$ denotes its complex conjugate. Which one of the following is TRUE?

Let $$\,\,S = \sum\limits_{n = 0}^\infty {n{\alpha ^n}} \,\,$$ where $$\,\,\left| \alpha \right| < 1.\,\,$$ The value of $$\alpha $$ in the range $$\,\,0 < \alpha < 1,\,\,$$ such t...

The maximum value attained by the function $$f(x)=x(x-1) (x-2)$$ in the interval $$\left[ {1,2} \right]$$ is _________.

The line integral of the vector field $$\,\,F = 5xz\widehat i + \left( {3{x^2} + 2y} \right)\widehat j + {x^2}z\widehat k\,\,$$ along a path from $$(0, 0, 0)$$ to $$(1,1,1)$$ param...

Consider $$3 \times 3$$ matrix with every element being equal to $$1.$$ Its only non-zero eigenvalue is __________.

Candidates were asked to come to an interview with $$3$$ pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a...

The solution of the differential equation, for $$t > 0,\,\,y''\left( t \right) + 2y'\left( t \right) + y\left( t \right) = 0$$ with initial conditions $$y\left( 0 \right) = 0$$ and...

A function $$y(t),$$ such that $$y(0)=1$$ and $$\,y\left( 1 \right) = 3{e^{ - 1}},\,\,$$ is a solution of the differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {...

Let the eigenvalues of a $$2 \times 2$$ matrix $$A$$ be $$1,-2$$ with eigenvectors $${x_1}$$ and $${x_2}$$ respectively. Then the eigenvalues and eigenvectors of the matrix $${A^2}...

The value of the integral $$\,\,2\int_{ - \infty }^\infty {\left( {{{\sin \,2\pi t} \over {\pi t}}} \right)} dt\,\,$$ is equal to

Let the probability density function of a random variable $$X,$$ be given as: $$${f_x}\left( x \right) = {3 \over 2}{e^{ - 3x}}u\left( x \right) + a{e^{4x}}u\left( { - x} \right)$$...

$$A$$ $$3 \times 3$$ matrix $$P$$ is such that , $${p^3} = P.$$ Then the eigen values of $$P$$ are

Let $$y(x)$$ be the solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 4y = 0\,\,$$ with initial conditions $$y(0)=0$$ and $$\,\,{\left. {...

Let $$P = \left[ {\matrix{ 3 & 1 \cr 1 & 3 \cr } } \right].$$ Consider the set $$S$$ of all vectors $$\left( {\matrix{ x \cr y \cr } } \right)$$ such that $${a^2} + {b^2} = 1$$ whe...

Let $$A$$ be a $$4 \times 3$$ real matrix which rank$$2.$$ Which one of the following statement is TRUE ?

We have a set of $$3$$ linear equations in $$3$$ unknown. $$'X \equiv Y'$$ means $$X$$ and $$Y$$ are equivalent statements and $$'X \ne Y'$$ means $$X$$ and $$y$$ are not equivalen...

Two coins $$R$$ and $$S$$ are tossed. The $$4$$ joint events $$\,\,\,\,\,\,{H_R}{H_S},\,\,\,\,{T_R}{T_S},\,\,\,\,{H_R}{T_S},\,\,\,\,{T_R}{H_S}\,\,\,\,\,\,\,$$ have probabilities $$...

The volume enclosed by the surface $$f\left( {x,y} \right) = {e^x}$$ over the triangle bounded by the lines $$x=y;$$ $$x=0;$$ $$y=1$$ in the $$xy$$ plane is ________.

The Laplace transform of $$f\left( t \right) = 2\sqrt {t/\pi } $$$$\,\,\,\,\,$$ is$$\,\,\,\,\,$$ $${s^{ - 3/2}}.$$ The Laplace transform of $$g\left( t \right) = \sqrt {1/\pi t} $$...

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selected, one from each set. What i...

A differential equation $$\,\,{{di} \over {dt}} - 0.21 = 0\,\,$$ is applicable over $$\,\, - 10 < t < 10.\,\,$$ If $$i(4)=10,$$ then $$i(-5)$$ is

The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a & { - 2} \cr } } \right]$$ has three linearly independent real...

If a continuous function $$f(x)$$ does not have a root in the interval $$\left[ {a,b} \right],\,\,$$ then which one of the following statements is TRUE?

A random variable $$X$$ has probability density function $$f(x)$$ as given below: $$$\,\,f\left( x \right) = \left\{ {\matrix{ {a + bx} & {for\,\,0 < x < 1} \cr 0 & {otherwise} \cr...

If the sum of the diagonal elements of a $$2 \times 2$$ matrix is $$-6$$, then the maximum possible value of determinant of the matrix is ____________.