Engineering Mat...

[1, 1, 2] is an Eigen vector of the matrix, \(A = \left[ {\begin{array}{*{20}{c}} 3&1&{ - 1}\\ 2&2&{ - 1}\\ 2&2&0 \end{array}} \right]\) corresponding to the Eigen value x. Then value of x 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 ________.

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

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?

Evaluate \(\mathop \smallint \limits_c \frac{{{z^2}\;-\;z\;+\;1}}{{(2z\;-\;1)}{(z\;-\;2)}}dz\) where c is the circle |z| = 1.

Laplace transform of t cos (at) 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

The probability that a person  stopping at a gas station will ask to have his tyres checked is 0.12, the probability that he will ask to have his oil checked is 0.29 and the probability that he will ask to have them both checked is 0.07. The probability that a person who has his tyres checked will also have oil checked is

The differential equation satisfying y = \(A{e^{3x}} + B{e^{2x}}\) is 

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 _______.

\(\left[ {\begin{array}{*{20}{c}} 4&5&x\\ 5&6&y\\ 6&k&z \end{array}} \right]\)

For the given matrix, if x, y, z are in AP with a common difference d and the rank of the matrix is 2, then which of the following results is/are always correct?

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

For the differential equation \(\frac{dy}{dx}={{x}^{2}}y-1,~y\left( 0 \right)=1,\) the value of y at x = 0.1, using the Taylors series method, is given by:

\(\smallint \frac{1}{{\left( {x + 1} \right)\sqrt {1 - 2x - {x^2}} }}dx\) is equal to

Consider the hemisphere x² + y² + (z - 2)² = 9, 2 ≤ z ≤ 5 and the vector field F = xi + yj + (z - 2)k The surface integral ∬ (F ⋅ n) dS, evaluated over the hemisphere with n denoting the unit outward normal vector, is

In the Laurent series expansion of \(f\left( z \right)=\frac{1}{z-1}-\frac{1}{z-2}\) valid in the region |z| > 2, then the coefficient of 1/z² is:

A die is tossed 180 times. Using normal distribution find the probability that the face 4 will turn up atleast 35 times (Area under the normal curve between Z = 0 and Z = 1 is 0.3413)

 Evaluate \(\mathop \smallint \nolimits_C \vec F \cdot \overrightarrow {dr} \) where \(\vec F\left( {x,\;y,\;z} \right) = x\hat i + y\hat j + 3\left( {{x^2} + {y^2}} \right)\hat k\) and C is the boundary of the part of the paraboid where z² = 64 – x² – y² which lies above the xy-plane and C is oriented counter clockwise when viewed from above.

If function \(f = {x^2} - {y^2} + 2{z^2}\), point P is (1, 2, 3) and point Q is (5, 0, 4)

Evaluate \(\mathop {\lim }\limits_{x \to 0} {\left[ {\tan \left( {\frac{\pi }{4} + x} \right)} \right]^{1/x}}\)

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,

Let X be a Discrete random variable following binominal probability distribution \({\left( {q + p} \right)^n} = {\left( {\frac{1}{3} + \frac{2}{3}} \right)^{10}}\).

The sum of the standard deviation and the variance of the above distribution is ________.