Engineering Mat...

The lowest eigen value of the 2 × 2 matrix \(\left[ {\begin{array}{*{20}{c}}4&2\\1&3\end{array}} \right]\) is _________

If the Laplace transform of y(t) is given by \(Y\left( s \right) = L\left( {y\left( t \right)} \right) = \frac{5}{{2\left( {s - 1} \right)}} - \frac{2}{{s - 2}} + \frac{1}{{2\left( {s - 3} \right)}}\), then y(0) + y'(0) = _________.

Consider the following differential equation:

\({\left( {\frac{{{d^4}y}}{{d{x^4}}}} \right)^{\frac{1}{2}}} = {\left[ {1 + {{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)}^2}} \right]^{\frac{1}{3}}}\)

Which of the following is/are true regarding the above differential equation?

The value of \(\mathop {\lim }\limits_{{\rm{x}} \to 0} \left( {\frac{{ - \sin {\rm{x}}}}{{2\sin {\rm{x}} + {\rm{x}}\cos {\rm{x}}}}} \right)\) is ________

If \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {-4 +x^2}&{x \le 3}\\ {-2x + a}&{x > 3} \end{array}} \right.\) is a continuous function for all real values of x, then a is equal to ________.

Divergence value of a function \({x^3}y\vec i - \left( {{z^2} - 2y} \right)\vec j + 5{y^2}z\vec k\) at x = 2, y = 3 and z = 4 is

Which is/are true on the interval [4, 5].of the below given function?

f(x) = 3x3 – 40.5x2 + 180x + 7

If a discrete random variable X has the following probability distribution

Evaluate the Standard deviation

A third-degree polynomial f(x) has values 2, 5, 16, 44 at x = 0, 1, 2 and 3 respectively. Estimate the value of \(\int\limits_0^3 {f(x)dx} \) by applying Simpson rule

The probability density of a continuous random variable is given by

P(x) = k e^-|x|, -∞ < x < ∞

Find the value of k.

The value of the following definite integral is \(\displaystyle\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \rm \dfrac{sin^3 \ x \ e^{-x^2}}{x^4}dx\)

A mouse is running on a string whose equation is given by \(y = \frac{2}{3}{x^{3/2}}\). The mouse moves along the string from point A whose x-coordinate is 0 to point B where x = 3. Find the distance ran by the mouse.

If \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1 + x}&{if\;x < 0}\\ {\left( {1 - x} \right)\left( {px + q} \right)}&{if\;x \ge 0} \end{array}} \right.\) satisfies the assumptions of Rolle’s Theorem in the interval [-1, 1], the ordered pair (p, q) is

If r̅ = x i̅ + y j̅ + z k̅ then the possible values of \({\nabla ^2}\left[ {\nabla \cdot \left( {\frac{{\bar r}}{{{r^2}}}} \right)} \right] = \)

What is the value of y(1) if y(0) = 4 and y’(0) = 9 for the differential equation y’’ + 4y’ + 4y = e^2x

The function f(x) = e^x – 1 is to be solved using Newton-Raphson method. If the initial value of x₀ is taken as 1.0, then the absolute error observed at 2^nd iteration is _______.

For the integral \(\mathop \smallint \nolimits_0^{\pi /2} \left( {8 + 4\cos x} \right)dx,\) the absolute percentage error in numerical evaluation with the Trapezoidal rule, using only the endpoints, is ______

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)?

A system comprising of n identical components works if at least one of the components works. Each of the components works with probability 0.8, independent of all other components. The minimum value of n for which the system works with probability at least 0.97 is ________.

Consider the differential equation and choose the correct statements

\({x^2}y'' + 6xy' + 6y = x\)

If C is the path along the curve y = x² – 4x + 4 from (0, 4) to (2, 0), then \(\mathop \oint \nolimits_C \left( {y\hat i - 3x\hat j} \right) \cdot \overrightarrow {dr} \) is

Let \(P = \left[ {\begin{array}{*{20}{c}}1&1&{ - 1}\\2&{ - 3}&4\\3&{ - 2}&3\end{array}} \right]\;and\;Q = \left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 2}&{ - 1}\\6&{12}&6\\5&{10}&5\end{array}} \right]\) be two matrices.

Then the rank of P + Q is ______.

Using the Fourier expansion of \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{0\; - \pi \le x \le 0}\\{\sin x\;0 \le x \le \pi \;}\end{array}} \right.\) 

Find the sum of series,