Self Studies

Engineering Mat...

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  • Question 1
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    [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 _______

  • Question 2
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    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 ________.

  • Question 3
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    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

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

  • Question 5
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    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?

  • Question 6
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    Evaluate \(\mathop \smallint \limits_c \frac{{{z^2}\;-\;z\;+\;1}}{{(2z\;-\;1)}{(z\;-\;2)}}dz\) where c is the circle |z| = 1.

  • Question 7
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    Laplace transform of t cos (at) is

  • Question 8
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    Which is/are true on the interval [4, 5].of the below given function?

    f(x) = 3x3 – 40.5x2 + 180x + 7

  • Question 9
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    If a discrete random variable X has the following probability distribution

    X

    2

    -1

    p(x)

    \(\frac{1}{3}\)

    \(\frac{2}{3}\)


    Evaluate the Standard deviation

  • Question 10
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    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

  • Question 11
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    The differential equation satisfying y = \(A{e^{3x}} + B{e^{2x}}\) is 

  • Question 12
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    The function f(x) = ex – 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 2nd iteration is _______.

  • Question 13
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    \(\left[ {\begin{array}{*{20}{c}} 4&5&x\\ 5&6&y\\ 6&k&z \end{array}} \right]\)

    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?

  • Question 14
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    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

  • Question 15
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    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:

  • Question 16
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    \(\smallint \frac{1}{{\left( {x + 1} \right)\sqrt {1 - 2x - {x^2}} }}dx\) is equal to

  • Question 17
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    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

  • Question 18
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    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:

  • Question 19
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    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)

  • Question 20
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     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.

  • Question 21
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    If function \(f = {x^2} - {y^2} + 2{z^2}\), point P is (1, 2, 3) and point Q is (5, 0, 4)

  • Question 22
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    Evaluate \(\mathop {\lim }\limits_{x \to 0} {\left[ {\tan \left( {\frac{\pi }{4} + x} \right)} \right]^{1/x}}\)

  • Question 23
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    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,

    \(\frac{1}{{1.3}} - \frac{1}{{3.5}} + \frac{1}{{5.7}} - \frac{1}{{7.9}} + \ldots \infty \) (Correct up to 3 decimal).

  • Question 24
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    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 ________.

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