GATE Instrumentation

Consider a function f(z) = z² + z + 1 where z ∈ C is a complex variable. A simple closed contour γ in z-plane encloses the point z = 1 + 0j. The value of integral ∮_γ (f(z))/(z-1)...

Consider a matrix $A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & 0 \\ 0 & 1 & 2 \end{bmatrix}$ Let $b = \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}$ and $x = \begin{bmatrix} x_1 \\ x_2 \...

Let $$z=x+iy$$ where $$i = \sqrt { - 1} .$$ Then $$\overline {\cos \,z} = $$

The angle between two vectors $${X_1} = {\left[ {\matrix{ 2 & 6 & {14} \cr } } \right]^T}$$ and $${X_2} = {\left[ {\matrix{ { - 12} & 8 & {16} \cr } } \right]^T}$$ in radian is ___...

The probability that a communication system will have high fidelity is $$0.81.$$ The probability that the system will have both high fidelity and high selectivity is $$0.18$$. The...

If $$V$$ is a non-zero vector of dimension $$3 \times 1,$$ then the matrix $$\,A = V{V^T}$$ has a rank $$=$$ ________

The eigen values of the matrix $$A = \left[ {\matrix{ 1 & { - 1} & 5 \cr 0 & 5 & 6 \cr 0 & { - 6} & 5 \cr } } \right]$$ are

In the neighborhood of $$z=1,$$ the function $$f(z)$$ has a power series expansion of the form $$f\left( z \right) = 1 + \left( {1 - z} \right) + {\left( {1 - z} \right)^2} + ........

Let $$\,\,f:\left[ { - 1, - } \right] \to R,\,\,$$ where $$\,f\left( x \right) = 2{x^3} - {x^4} - 10.$$ The minimum value of $$f(x)$$ is _______.

$$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} + n} - \sqrt {{n^2} + 1} } \right)\,\,$$ is ________.

The vector that is NOT perpendicular to the vectors $$\,\,\left( {i + j + k} \right)\,\,$$ and $$\,\left( {i + 2j + 3k} \right)\,\,$$ is _________.

An urn contains $$5$$ red and $$7$$ green balls. A ball is drawn at random and its colour is noted. The ball is placed back into the urn along with another ball of the same colour....

Consider the matrix $$A = \left( {\matrix{ 2 & 1 & 1 \cr 2 & 3 & 4 \cr { - 1} & { - 1} & { - 2} \cr } } \right)$$ whose eigen values are $$1, -1$$ and $$3$$. Then trace of $$\left(...

The value of the integral $${1 \over {2\pi j}}\int\limits_c {{{{z^2} + 1} \over {{z^2} - 1}}} dz$$ where $$z$$ is a complex number and $$C$$ is a unit circle with center at $$1+0j$...

A straight line of the form $$y=mx+c$$ passes through the origin and the point $$(x, y)=(2,6).$$ The value of $$m$$ is

The probability density function of a random variable $$X$$ is $$\,{P_x}\left( x \right) = {e^{ - x}}\,\,$$ for $$\,\,x \ge 0\,\,$$ and $$0$$ otherwise. The expected value of the f...

The double integral $$\,\,\int_0^a {\int_0^y {f\left( {x,y} \right)\,dx\,dy\,\,\,} } $$ is equivalent to

The value of $$\oint\limits_c {{1 \over {{z^2}}}dz} $$ where the contour is the unit circle traversed clock - wise, is

Let $$A$$ be an $$n \times n$$ matrix with rank $$r\left( {0 < r < n} \right).$$ Then $$AX=0$$ has $$p$$ independent solutions, where $$p$$ is

A coin is tossed thrice. Let $$X$$ be the event that head occurs in each of the first two tosses. Let $$Y$$ be the event that a tail occurs on the third toss. Let $$Z$$ be the even...

The probability that a thermistor randomly picked up from a production unit is defective is $$0.1.$$ The probability that out of $$10$$ thermistors randomly picked up, $$3$$ are de...

The magnitude of the directional derivative of the function $$f\left( {x,y} \right) = {x^2} + 3{y^2}$$ in a direction normal to the circle $$\,{x^2} + {y^2} = 2,$$ at the point $$(...

For the matrix $$A$$ satisfying the equation given below, the eigen values are $$$\left[ A \right]\left[ {\matrix{ 1 & 2 & 3 \cr 7 & 8 & 9 \cr 4 & 5 & 6 \cr } } \right] = \left[ {\...

Given $$x\left( t \right) = 3\,\sin \,\left( {1000\pi t} \right)\,\,$$ and $$\,\,y\left( t \right) = 5\cos \,\left( {1000\pi t{\pi \over t}} \right)$$ The $$x$$-yplot will be

A scalar valued function is defined as $$f\left( x \right){x^T}Ax + {b^T}x + c,$$ where $$A$$ is a symmetric positive definite matrix with dimension $$n \times n;$$ $$b$$ and $$x$$...

Given that $$x$$ is a random variable in the range $$\left[ {0,\infty } \right]$$ with a probability density function $${{{e^{ - {x \over 2}}}} \over K},$$ the value of the constan...

The iteration step in order to solve for the cube roots of a given number $$'N'$$ using the Newton-Raphson's method is

The type of the partial differential equation $${{\partial f} \over {\partial t}} = {{{\partial ^2}f} \over {\partial {x^2}}}\,\,is$$

While numerically solving the differential equation $$\,{{dy} \over {dx}} + 2x{y^2} = 0,y\left( 0 \right) = 1\,\,$$ using Euler's predictor corrector (improved Euler- Cauchy) metho...

The maximum value of the solution $$y$$ $$(t)$$ of the differential equation $$\,\,y\left( t \right) + \mathop y\limits^{ \bullet \,\, \bullet } \left( t \right) = 0\,\,\,$$ with i...

For a vector $$E,$$ which one of the following statements is NOT TRUE?

A continuous random variable $$X$$ has a probability density function $$f\left( x \right) = {e^{ - x}},0 < x < \infty .$$ Then $$P\left\{ {X > 1} \right\}$$ is

One pair of eigenvectors corresponding to the two eigen values of the matrix $$\left[ {\matrix{ 0 & { - 1} \cr 1 & {0 - } \cr } } \right]$$

The dimension of the null space of the matrix $$\left[ {\matrix{ 0 & 1 & 1 \cr 1 & { - 1} & 0 \cr { - 1} & 0 & { - 1} \cr } } \right]$$ is

The unilateral Laplace transform of $$f(t)$$ is $$\,{1 \over {{s^2} + s + 1}}.$$ The unilateral Laplace transform of $$t$$ $$f(t)$$ is

If $$x\left[ N \right] = {\left( {1/3} \right)^{\left| n \right|}} - {\left( {1/2} \right)^n}\,u\left[ n \right],$$ then the region of convergence $$(ROC)$$ of its $$Z$$-transform...

The maximum value of $$f\left( x \right) = {x^3} - 9{x^2} + 24x + 5$$ in the interval $$\left[ {1,6} \right]$$ is

Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the value of $${A^3}$$ is

Two independent random variables $$X$$ and $$Y$$ are uniformly distributed in the interval $$\left[ { - 1,1} \right].$$ The probability that max $$\left[ {X,Y} \right]$$ is less th...

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

Consider the differential equation $${{{d^2}y\left( t \right)} \over {d{t^2}}} + 2{{dy\left( t \right)} \over {dt}} + y\left( t \right) = \delta \left( t \right)$$ with $$y\left( t...

The direction of vector $$A$$ is radially outward from the origin, with $$\left| A \right| = K\,{r^n}$$ where $${r^2} = {x^2} + {y^2} + {z^2}$$ and $$K$$ is constant. The value of...

With initial condition $$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$ the solution of the differential equation, $$\,\,\,t{{dx} \over {dt}} + x = t\,\,\,$$ is

The box $$1$$ contains chips numbered $$3, 6, 9,$$ $$12$$ and $$15$$. The box $$2$$ contains chips numbered $$6, 11, 16, 21$$ and $$26$$. Two chips, one from each box are drawn at...

The matrix $$M = \left[ {\matrix{ { - 2} & 2 & { - 3} \cr 2 & 1 & 6 \cr { - 1} & { - 2} & 0 \cr } } \right]$$ has eigen values $$-3, -3, 5.$$ An eigen vector corresponding to the e...

The series $$\,\,\sum\limits_{m = 0}^\alpha {{1 \over {{4^m}}}{{\left( {x - 1} \right)}^{2m}}\,\,\,} $$ converges for

Consider the differential equation $$\mathop y\limits^{ \bullet \bullet } + 2\,\mathop y\limits^ \bullet + y = 0\,\,$$ with boundary conditions $$y(0)=1$$ & $$y(1)=0.$$ The value o...

The integral $$\int\limits_{ - \alpha }^\alpha \delta \left( {t - {\pi \over 6}} \right)6\,\sin \,\left( t \right)dt$$ evaluates to

Consider the differential equation $${{dy} \over {dx}} + y = {e^x}$$ with $$y(0)=1.$$ Then the value of $$y(1)$$ is

A real $$n \times n$$ matrix $$A$$ $$ = \left[ {{a_{ij}}} \right]$$ is defined as follows $$\left\{ {\matrix{ {{a_{ij}} = i,} & {\forall i = j} \cr { = 0,} & {otherwise} \cr } .} \...

$$X$$ and $$Y$$ are non-zero square matrices of size $$n \times n$$. If $$XY = {O_{n \times n}}$$ then

$$u(t)$$ represents the unit step function. The Laplace transform of $$u\left( {t - \tau } \right)$$ is

A screening test is carried out to detect a certain disease. It is found that $$12$$% of the positive reports and $$15$$% of the negative reports are incorrect. Assuming that the p...

The eigen values of a $$2 \times 2$$ matrix $$X$$ are $$-2$$ and $$-3$$. The eigen values of matrix $${\left( {X + 1} \right)^{ - 1}}\left( {X + 5{\rm I}} \right)$$ are

A sphere of unit radius is centered at the origin. The unit normal at a point $$(x, y, z)$$ on the surface of the sphere is the vector.

Given $$y = {x^2} + 2x + 10\,\,\,$$ the value of $$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 1}}\,\,$$ is equal to

$$\,\mathop {Lim}\limits_{x \to 0} {{\sin x} \over x}\,\,\,$$ is

The expression $${e^{ - ln\,x}}$$ for $$x > 0$$ is equal to

It is known that two roots of the non-linear equation $$\,{x^3} - 6{x^2} + 11x - 6 = 0\,\,$$ are $$1$$ and $$3.$$ The third root will be

$${P_x}\left( X \right) = M{e^{\left( { - 2\left| x \right|} \right)}} + N{e^{\left( { - 3\left| x \right|} \right)}}\,\,$$ is the probability density function for the real random...

A random variable is uniformly distributed over the interval $$2$$ to $$10.$$ Its variance will be

Consider a Gaussian distributed random variable with zero mean and standard deviation $$\sigma .\,\,\,$$ The value of its cumulative distribution function at the origin will be

Consider the differential equation $${{dy} \over {dx}} = 1 + {y^2}.$$ Which one of the following can be particular solution of this differential equation ?

Consider the function $$\,\,y = {x^2} - 6x + 9.\,\,\,$$ The maximum value of $$y$$ obtained when $$x$$ varies over the interval $$2$$ to $$5$$ is

Consider the function $$\,\,f\left( x \right) = {\left| x \right|^3},\,\,\,$$ where $$x$$ is real. Then the function $$f(x)$$ at $$x=0$$ is

Let $$j\, = \,\sqrt { - 1} $$. Then one value of $${j^j}$$ is

Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and $${{a_{ij}} = i.j.}$$ Then the rank of $$A$$ is

For the function $${{\sin z} \over {{z^3}}}$$ of a complex variable z, the point z = 0 is

The value of $$\,\int\limits_0^\infty {\int\limits_0^\infty {{e^{ - {x^2}}}{e^{ - {y^2}}}} dx\,dy\,\,\,\,} $$ is

Let $$A$$ be $$n \times n$$ real matrix such that $${A^2} = {\rm I}$$ and $$Y$$ be an $$n$$-diamensional vector. Then the linear system of equations $$Ax=y$$ has

The polynomial $$\,p\left( x \right) = {x^5} + x + 2\,\,$$ has

Identity the Newton $$-$$ Raphson iteration scheme for the finding the square root of $$2$$

For real $$x,$$ the maximum value of $${{{e^{Sin\,x}}} \over {{e^{Cos\,x}}}}\,\,$$ is

Assume that the duration in minutes of a telephone conversation follows the exponential distribution $$\,f\left( x \right) = {1 \over 5}{e^{ - x/5}},\,x \ge 0.\,\,\,$$ The probabil...

For a given $$2x2$$ matrix $$A,$$ it is observved that $$A\left[ {\matrix{ 1 \cr { - 1} \cr } } \right] = - 1\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]$$ and $$A\left[ {\matrix{...

A system of linear simultaneous equations is given as $$AX=b$$ where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\...

A system of linear simultaneous equations is given as $$AX=b$$ where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\...

For initial value problem $$\,\mathop y\limits^{ \bullet \bullet } + 2\,\mathop y\limits^ \bullet + \left( {101} \right)y = \left( {10.4} \right){e^x},y\left( 0 \right) = 1.1\,\,$$...

If a vector $$\overrightarrow R \left( t \right)$$ has a constant magnitude then

Let $${z^3}\, = \,\overline z $$, where z is a complex number not equal to zero. Then z is a solution of