DEPARTMENT OF MATHEMATICS â&#x20AC;&#x153;T3Examination, May2017-18â&#x20AC;? Semester:Second Subject:Engg. Mathematics-II Branch: ME Course Type:Core Time: 3 Hours Max.Marks: 80

Date of Exam:21/05/2018 Subject Code:MAH105-T Session: I Course Nature:Hard Program: B.Tech Signature: HOD/Associate HOD:

Note: All questions are compulsory from part A (2*10 = 20 Marks). Attempt any two questions from Part B (15 Marks each). Attempt any two Questions from Part -C (15 Marks each). PART -A Q.1 (a) If đ??š(đ?&#x2018;Ľ ) = đ?&#x2018;Ľ 2 is an even function in (-Ď&#x20AC;, Ď&#x20AC;), then find the values of a0, an& bn. (b) The expansion of đ?&#x2018;&#x201C; (đ?&#x2018;Ľ ) = |cos đ?&#x2018;Ľ | as a Fourier series in (-Ď&#x20AC;, Ď&#x20AC;) does not containsâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śterms. (c) If f(x) is an odd function in (-l, l), then the Fourier Co-efficient an =â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. 1

(d) Find the Fourier sine transform ofđ?&#x2018;Ľ. (e) Write the Fourier transform of dirac-delta function. (f) Is the matrix

cos đ?&#x153;&#x192; 0 â&#x2C6;&#x2019; sin đ?&#x153;&#x192;

0 1 0

sin đ?&#x153;&#x192; 0 orthogonal? cos đ?&#x153;&#x192;

2 â&#x2C6;&#x2019;1 1 (g) The characteristic equation of â&#x2C6;&#x2019;1 2 â&#x2C6;&#x2019;1 isâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś 1 â&#x2C6;&#x2019;1 2 (h) Define Rank of matrix with suitable example. 2 3 â&#x2C6;&#x2019;2 (i) The product of the eigen values of â&#x2C6;&#x2019;2 1 1 isâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.. 1 0 2 (j) The rank of identity matrix of order 3 Ă&#x2014; 3 isâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.. PART - B Q.2 (a) Obtain the Fourier series to represent đ??š(đ?&#x2018;Ľ ) = (

đ?&#x153;&#x2039;â&#x2C6;&#x2019;đ?&#x2018;Ľ 2 2

) , 0 â&#x2030;¤ x â&#x2030;¤ 2Ď&#x20AC;.-

Hence obtain the following relations:

(12) *******

(i) (ii) (iii)

1 2

1

1 2

1

1 12

+ â&#x2C6;&#x2019; +

1 2

2 1

2

2

1 32

+ + +

1 2

3 1

2

3

1 52

+ â&#x2C6;&#x2019;

1 2

4 1

2

4

+ â&#x2039;Ż â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś = â&#x2C6;&#x2018;â&#x2C6;&#x17E; đ?&#x2018;&#x203A;=1 + â&#x2039;Ż â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś = â&#x2C6;&#x2018;â&#x2C6;&#x17E; đ?&#x2018;&#x203A;=1

1

(â&#x2C6;&#x2019;1)

6

đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1

=

đ?&#x2018;&#x203A;2

1

+ â&#x2039;Ż â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś = â&#x2C6;&#x2018;â&#x2C6;&#x17E; đ?&#x2018;&#x203A;=1

đ?&#x153;&#x2039;2

=

đ?&#x2018;&#x203A;2

(2đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1)2

=

đ?&#x153;&#x2039;2 12

đ?&#x153;&#x2039;2

8

(b) Obtain the half range sine series for ex in 0 < x < 1.

(7) đ?&#x2018;&#x2122;

Q.3 (a) Obtain a half range cosine series for đ?&#x2018;&#x201C;(đ?&#x2018;Ľ ) = đ?&#x2018;&#x2DC;đ?&#x2018;Ľ đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; 0 â&#x2030;¤ đ?&#x2018;Ľ â&#x2030;¤ 2 đ?&#x2018;&#x2DC;(đ?&#x2018;&#x2122; â&#x2C6;&#x2019; đ?&#x2018;Ľ )đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; 1

1

đ?&#x2018;&#x2122; 2

(10)

â&#x2030;¤ đ?&#x2018;Ľ â&#x2030;¤ đ?&#x2018;&#x2122;.

1

Deduce the sum of the series 12 + 32 + 52 + â&#x2039;Ż â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś (b) Express the function đ?&#x2018;&#x201C;(đ?&#x2018;Ľ ) = 1, đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; |đ?&#x2018;Ľ | â&#x2030;¤ 1

(5)

0, đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; |đ?&#x2018;Ľ | â&#x2030;Ľ 1 â&#x2C6;&#x17E; sin đ?&#x153;&#x2020;đ??śđ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x2020;đ?&#x2018;Ľ

As a Fourier Integral. Hence evaluate â&#x2C6;Ť0

đ?&#x153;&#x2020;

đ?&#x2018;&#x2018;đ?&#x153;&#x2020;.

1

Q.4 (a) Find Fourier sine transform of đ?&#x2018;Ľ(đ?&#x2018;Ľ 2+đ?&#x2018;&#x17D;2 ) .

(10)

(b) Find the Fourier transform of for đ?&#x2018;&#x201C; (đ?&#x2018;Ľ ) = 1 â&#x2C6;&#x2019; đ?&#x2018;Ľ 2 , đ?&#x2018;&#x2013;đ?&#x2018;&#x201C;|đ?&#x2018;Ľ | < 1 0,

(5)

đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; |đ?&#x2018;Ľ | > 1.

PART -C Q.5 (a) Find rank of the matrix

3 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;1 2 1

2 4 2

by reducing it in its normal form.

(10)

(b) Investigate the values of Îť and Îź so that the equations 2đ?&#x2018;Ľ + 3đ?&#x2018;Ś + 5đ?&#x2018;§ = 9, 7đ?&#x2018;Ľ + 3đ?&#x2018;Ś â&#x2C6;&#x2019; 2đ?&#x2018;§ = 8, 2đ?&#x2018;Ľ + 3đ?&#x2018;Ś + đ?&#x153;&#x2020;đ?&#x2018;§ = đ?&#x153;&#x2021;have- (i) no solution (ii) a unique solution and (iii) an infinite number of solution. cos đ?&#x153;&#x192; Q.6 (a) Verify that the matrix 0 â&#x2C6;&#x2019; sin đ?&#x153;&#x192;

0 sin đ?&#x153;&#x192; 1 0 is orthogonal 0 cos đ?&#x153;&#x192;

(b) Using Cayley-Hamilton Theorem, find the inverse of the matrix â&#x2C6;&#x2019;1 1 â&#x2C6;&#x2019;1

2 2 â&#x2C6;&#x2019;1

(5) 1 â&#x2C6;&#x2019;1 3

2 4 0 3 1 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;2 1 and obtain the model matrix. 0 6 â&#x2C6;&#x2019;2 2 (b)Find eigen values and the corresponding eigen vectors of the matrix â&#x2C6;&#x2019;2 3 â&#x2C6;&#x2019;1 2 â&#x2C6;&#x2019;1 3 Q.7(a) Diagonalise the matrix

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(5)

(10)

(5)

(10)