Engineering Mathematics Test 7

Question 1
1 / -0

The inverse Laplace transform of \(\frac{2}{{s + 1}}\) is

A
2(t + 1)

B
2e^-t

C
2e^t

D
e^-2t

Question 2
1 / -0

Which of the following equations represents a one-dimensional wave equation?

A
\(\frac{{\partial u}}{{\partial t}} = {C^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\)

B
\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {C^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\)

C
\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {C^2}\frac{{\partial u}}{{\partial x}}\)

D
\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} + \frac{{{\partial ^2}u}}{{\partial {x^2}}} = 0\)

Question 3
1 / -0

Consider the following partial differential equation:
\(7\frac{{{\partial ^2}{\rm{\omega }}}}{{\partial {{\rm{x}}^2}}} - {\rm{a}}\frac{{{\partial ^2}{\rm{\omega }}}}{{\partial {\rm{x}}\partial {\rm{z}}}} + 7\frac{{{\partial ^2}{\rm{\omega }}}}{{\partial {{\rm{z}}^2}}} + 2\frac{{\partial {\rm{\omega }}}}{{\partial {\rm{x}}}} + {\rm{c}}\frac{{\partial {\rm{\omega }}}}{{\partial {\rm{z}}}} + {\rm{\omega }} = 0\) , where a and c are unknown constants.
For what values of a and c, this equation will be elliptic in nature?

A
For any values of a and c.

B
For any value of a where -14 < a < 14 and any value of c.

C
For any value of a where a > 14 & a < -14 and any value of c.

D
For a = -7 and c = 2.

Question 4
1 / -0

The Fourier series for an even function f(x) is given by

A
\(\frac{{{a_o}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos nx + \mathop \sum \limits_{n = 1}^\infty {b_n}\sin nx\)

B
\(\frac{a_o}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos nx\)

C
\(\mathop \sum \limits_{n = 1}^\infty {b_n}\sin nx\)

D
\(\frac{{{a_o}}}{2}\)

Question 5
1 / -0

The Laplace transform of the function \({\rm{f}}\left( {\rm{t}} \right) = {\cos ^2}4{\rm{t\;}}\) will be where, \({\rm{s}} > 0\)

A
\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} - \frac{{\rm{s}}}{{{{\rm{s}}^2} + 64}}} \right]\)

B
\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} - \frac{8}{{{{\rm{s}}^2} + 64}}} \right]\)

C
\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} + \frac{8}{{{{\rm{s}}^2} + 64}}} \right]\)

D
\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} + \frac{{\rm{s}}}{{{{\rm{s}}^2} + 64}}} \right]\)

Question 6
1 / -0

For the Fourier series of the following function of period 2π \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{ - \pi < x < 0}\\ {1,}&{0 < x < \pi } \end{array}} \right.\) the ratio (to the nearest integer) of the Fourier coefficients of the first and the third harmonic is:

A
1

B
2

C
3

D
6

Question 7
1 / -0

Evaluate \(\mathop \smallint \limits_0^\infty t{e^{ - 2t}}\cos tdt\)

A
3/5

B
4/5

C
3/25

D
4/25

Question 8
1 / -0

The following partial differential equation is associated with heat transfer through a bar of 2 m length. The differential equation is \(\frac{{{\partial ^2}{\rm{h}}}}{{\partial {{\rm{x}}^2}}} = 2.56\frac{{\partial {\rm{h}}}}{{\partial {\rm{t}}}}{\rm{\;\;where\;\;}}0 \le {\rm{x}} \le 2{\rm{\;m\;and\;t}} \ge 0{\rm{\;in\;second}}.\) The boundary conditions associated is \({\rm{h}}\left( {0,{\rm{t}}} \right){\rm{\;}} = {\rm{\;}}0{\rm{\;and\;h}}\left( {2,{\rm{t}}} \right){\rm{\;}} = {\rm{\;}}0\) and the initial condition of the problem is \({\rm{h}}\left( {{\rm{x}},0} \right) = {\rm{sin}}3.2{\rm{x}}.\) Then the solution (ignoring the general form of solution) of the partial differential equation is

A
\({{\rm{e}}^{4{\rm{t}}}}{\rm{sin}}3.2{\rm{x}}\)

B
\({{\rm{e}}^{ - 1.6{\rm{t}}}}{\rm{sin}}3.2{\rm{x}}\)

C
\({{\rm{e}}^{ - 2.56{\rm{t}}}}{\rm{sin}}3.2{\rm{x}}\)

D
\({{\rm{e}}^{ - 4{\rm{t}}}}{\rm{sin}}3.2{\rm{x\;}}\)

Question 9
1 / -0

What will the value of \({\rm{y}}\left( {\frac{1}{3}} \right)\) (corrected up to 3 decimal points) of the differential equation \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}} - 3\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}} + 4{\rm{y}} = {\rm{}}0\) with the given condition \({\rm{y}} = 0{\rm{\;at\;x}} = 0{\rm{\;and\;y}} = 1{\rm{\;at\;x}} = 1{\rm{\;and\;}}\frac{{{\rm{dx}}}}{{{\rm{dt}}}} = 1{\rm{\;at\;x}} = 0\)?

A
0.045-0.049

B
0.0454-0.0491

C
0.0457-0.0496

D
0.0459-0.0492

Question 10
1 / -0

A differential equation for dependent variable y and independent variable t is given by, \(\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{t}}^2}}} = 7{{\rm{y}}^{\frac{2}{5}}}.{\rm{\;}}\)The initial conditions are given as \({\rm{y\;}} = {\rm{\;}}1{\rm{\;and\;}}\frac{{{\rm{dy}}}}{{{\rm{dt}}}} = \sqrt {10} {\rm{\;\;at\;t}} = 0\). What will be the value (corrected up to two decimal places) of \({\rm{y}}\left( {\sqrt {10} } \right)?\)

A
100.50-102.50

B
100.504-102.501

C
100.507-102.506

D
100.509-102.502

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Engineering Mat...

Question 1

2(t + 1)

2e-t

2et

e-2t

Question 2

\(\frac{{\partial u}}{{\partial t}} = {C^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\)

\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {C^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}}\)

\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = {C^2}\frac{{\partial u}}{{\partial x}}\)

\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} + \frac{{{\partial ^2}u}}{{\partial {x^2}}} = 0\)

Question 3

For any values of a and c.

For any value of a where -14 < a < 14 and any value of c.

For any value of a where a > 14 & a < -14 and any value of c.

For a = -7 and c = 2.

Question 4

\(\frac{{{a_o}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos nx + \mathop \sum \limits_{n = 1}^\infty {b_n}\sin nx\)

\(\frac{a_o}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos nx\)

\(\mathop \sum \limits_{n = 1}^\infty {b_n}\sin nx\)

\(\frac{{{a_o}}}{2}\)

Question 5

\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} - \frac{{\rm{s}}}{{{{\rm{s}}^2} + 64}}} \right]\)

\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} - \frac{8}{{{{\rm{s}}^2} + 64}}} \right]\)

\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} + \frac{8}{{{{\rm{s}}^2} + 64}}} \right]\)

\(\frac{1}{2}\left[ {\frac{1}{{\rm{s}}} + \frac{{\rm{s}}}{{{{\rm{s}}^2} + 64}}} \right]\)

Question 6

1

2

3

6

Question 7

3/5

4/5

3/25

4/25

Question 8

\({{\rm{e}}^{4{\rm{t}}}}{\rm{sin}}3.2{\rm{x}}\)

\({{\rm{e}}^{ - 1.6{\rm{t}}}}{\rm{sin}}3.2{\rm{x}}\)

\({{\rm{e}}^{ - 2.56{\rm{t}}}}{\rm{sin}}3.2{\rm{x}}\)

\({{\rm{e}}^{ - 4{\rm{t}}}}{\rm{sin}}3.2{\rm{x\;}}\)

Question 9

0.045-0.049

0.0454-0.0491

0.0457-0.0496

0.0459-0.0492

Question 10

100.50-102.50

100.504-102.501

100.507-102.506

100.509-102.502

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2e^-t

2e^t

e^-2t