Engineering Mat...

In the Laurent series expression of \(f\left( z \right) = \frac{1}{{\left( {z - 1} \right)\left( {z - 2} \right)}}\) valid for 0 < |z - 1|< 1, the co-efficient of \(\frac{1}{{\left( {z - 1} \right)}}\;is\)

The function f(z) = |z|² is:

Consider the matrix \(A = \left[ {\begin{array}{*{20}{c}} x&0&0\\ 0&y&{ - 1}\\ 0&1&{ - 2} \end{array}} \right]\)

Which of the following conditions must satisfy to get all the Eigenvalues of A as negative.

A 3 by 3 matrix B is known to have eigenvalues 0, 1, 2.

If \(x = \mathop \sum \limits_{k = 1}^\infty {a_k}\sin kx\), for -π ≤ x ≤ π, the value of a₂ is ______

If  \(u = \log \frac{{{x^2}}}{y},\;then\;\;x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}\;\) is equal to

Let the random variable X have the density f(x) = kx if 0 ≤ x ≤ 3. Find P(|X − 1.8| < 0.6) up to two decimal places.

A machine produces items of which 1% at random are defective. How many items can be packed in a box while keeping the chance of one or more defectives in the box to be no more than 0.5?

If the potential function is log (x² + y²) then the function is

The solution of the differential equation

t²y’ – y² – yt = 0, t > 0 is

Consider the following initial value problem

y’’ + 3y’ = e^2t, y(0) = 1, y’(0) = 0

The value of y(1) is __________ (upto two decimal places).

Let \(I = \mathop \smallint \limits_C^\; \frac{{f\left( z \right)}}{{\left( {z - 1} \right)\left( {z - 2} \right)}}dz,\) where \(f\left( z \right) = \sin \frac{{\pi z}}{2} + \cos \frac{{\pi z}}{2}\) and C is the curve |z| = 3 oriented anti-clockwise. Then the value of I is

Let X have the density f(x) = 2x if 0 ≤ x ≤ 1 and f(x) = 0 otherwise. The mean and variance of the random variable Y = −2X + 3 respectively are

Two events A and B are such that P(A) = 0.5, P(B) = 0.3 and P(A ∩ B) = 0.1. Which of the following is/are true?

The first two columns of an orthogonal 3 × 3 matrix are \({V_1} = \left[ {\begin{array}{*{20}{c}}{\cos x\cos y}\\{\sin x}\\{\cos x\sin y}\end{array}} \right],\;{V_2} = \left[ {\begin{array}{*{20}{c}}{ - \sin y}\\0\\{\cos y}\end{array}} \right]\)

Which of the following represents the third column of this matrix.

Consider the matrix \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 1}&0\\{ - 1}&2&{ - 1}\\0&{ - 1}&1\end{array}} \right]\)

Which of the following are the Eigenvectors of the matrix (A³ + 5I)?

Consider the following three systems of equations:

Case a: x₁ + x₂ = 1, 2x₁ – x₂ = 0

Case b: x₁ + x₂ = 1, 2x₁ + 2x₂ = 2

Case c: x₁ + x₂ = 1, x₁ + x₂ = 0

The value of the integral \(\displaystyle\int_0^{2\pi}\left(\dfrac{3}{9+\sin^2 \theta }\right)d\theta \) is

If \(f\left( {x,y} \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{xy\left( {{x^2} - {y^2}} \right)}}{{{x^2} + {y^2}}},\;\left( {x,y} \right) \ne \left( {0,\;0} \right)}\\{0,\;\left( {x,y} \right) = \left( {0,\;0} \right)}\end{array}} \right.\)

Consider the initial value problem,

y⁽⁴⁾ = -sin t + cos t, y’’’(0) = 7, y’’(0) = y’(0) = -1, y(0) = 0. The value of y(π / 2) is ______ (upto two decimal places).

The solution of the initial value problem

Solve \(2\frac{{{\partial ^2}z}}{{\partial {x^2}}} + 5\frac{{{\partial ^2}z}}{{\partial x\partial y}} + 2\frac{{{\partial ^2}z}}{{\partial {y^2}}} = 0\)