Engineering Mat...

Let \(M = \left[ {\begin{array}{*{20}{c}} 9&2&7&1\\ 0&7&2&1\\ 0&0&{11}&6\\ 0&0&{ - 5}&0 \end{array}} \right]\). Then, the value of det((8I – M)³) is

If the directional derivative of the function z = y²e^2x at (2, -1) along the unit vector \(\vec b = \alpha \hat i + \beta \hat j\) is zero, then |α + β| equals

 If a function y(x) is described by the initial value problem, \(\frac{{{d^2}y}}{{d{x^2}}} + 5\frac{{dy}}{{dx}} + 6y = 0\), with initial conditions y(0) = 2, and \({\left( {\frac{{dy}}{{dx}}} \right)_{x = 0}} = 0,\) then the value of y at x = 1 is ________. (Round off to 2 decimal places)

Let X be a random variable having Poisson (2) distribution. Then \(E\left( {\frac{1}{{1 + x}}} \right)\) equals

Let E and F be two events. Then which one of the following statements is NOT always TRUE?

A packet contains 10 distinguishable firecrackers out of which 4 are defective. If three firecrackers are drawn at random (without replacement) from the packet, then the probability that all three firecrackers are defective equals

Let X be a random variable having U(0, 10) distribution and Y = X – [X], where [X] denotes the greatest integer less than or equal to X. Then P(Y > 0.25) equals ______

The line integral of the vector function \(u\left( {x,y} \right) = 2y\hat i + x\hat j\) along the straight line from (0, 0) to (2, 4) is ________

The value(s) of the integral \(\mathop \smallint \limits_{ - \pi }^\pi \left| x \right|\cos nxdx,\;n \ge 1\) is (are)

For real constants a and b, let \(M = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\ a&b \end{array}} \right]\) be an orthogonal matrix. Then which of the following statements is/are always TRUE?

Evaluate \(\underset{c}{\overset{{}}{\mathop \oint }}\,\frac{{{e}^{2z}}}{{{\left( z+1 \right)}^{4}}}dz\) where C is circle |z| = 3

Let f(x, y) = e^x sin y, x = t³ + 1 and y = t⁴ + t. Then \(\frac{{df}}{{dt}}\) at t = 0 is _______. (rounded off to two decimal places)

Consider the differential equation \(\frac{{dy}}{{dx}} + 10y = f\left( x \right),x > 0\), where f(x) is continuous function such that \(\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 1\). Then the value of \(\mathop {\lim }\limits_{x \to \infty } y\left( x \right)\) is ________.

Consider the following system of linear equations

x + y + 5z = 3, x + 2y + mz = 5 and x + 2y + 4z = k

The system is consistent if

Consider a sequence of independent Bernoulli trials with probability of success in each trial being \(\frac{1}{5}\). Then which of the following statements is/are TRUE?

The first four terms of the Taylor series expansion of \(f\left( z \right) = \frac{{z + 1\;}}{{\left( {z - 3} \right)\left( {z - 4} \right)}},\) when z = 2 is 

Let u(x,y) = x³ + a x² y + b x y² + 2y³ be a harmonic function and v(x, y) its harmonic conjugate. If v(0, 0) = 1, then |a + b + v(1, 1)| is equal to _____

Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB) = rank(A) rank(B)

II. det(AB) = det(A) det(B)

III. rank(A + B) ≤ rank(A) + rank(B)

IV. det(A + B) ≤ det(A) + det(B)

Let A be (n x n) real valued square symmetric matrix of rank 2 with \(\mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n A_{ij}^2 = 50\). Consider the following statements.

(I) One eigenvalue must be in [–5, 5]

(II) The eigenvalue with the largest magnitude must be strictly greater than 5

The solution of the partial differential equation \({{x}^{2}}\frac{\partial z}{\partial x}+{{y}^{2}}\frac{\partial z}{\partial y}=\left( x+y \right)z\) is

The particular integral of the differential equation (2D³ – 7D² + 7D – 2) y = e^-8x is

The equations of the two lines of regression are: 4x + 3y + 7 = 0 and 3x + 4y + 8 = 0. The correlation coefficient between x and y is