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Code: 13A54102

B.Tech I Year (R13) Supplementary Examinations December 2019

MATHEMATICS ? II

(Common to EEE, ECE, EIE, CSE & IT)

Time: 3 hours Max. Marks: 70

PART ? A

(Compulsory Question)

*****

1 Answer the following: (10 X 02 = 20 Marks)

(a) Find the Eigen values and the corresponding of ?

5 4

1 2

?.

(b) Show that ?? = ?

2 3 + 4 ?? 3 ? 4 ?? 2

? is Hermitian.

(c) Define algebraic and transcendental equations with example each.

(d) The value of ?

1

?? 2

1

?? ?? by Simpson?s 1/3 rule (taking n = 4) is________.

(e) If

????

????

= ? ?? , ?? (0) = 1, ? = 0.01 then by Euler?s method the value of ?? 1

is _________.

(f) Write the Fourier series of f(x) in [C, C+2L].

(g)

Find the Fourier cosine transform f(x) = e

ax ?

.

(h) Define convolution theorem.

(i) Write the two dimensional Laplace equation.

(j) Form a partial differential equation by eliminating the arbitrary constants a and b from the equation:

z = ax + by.

PART ? B

(Answer all five units, 5 X 10 = 50 Marks)

UNIT ? I

2

Reduce the matrix ?? = ?

2

1

3

6

3

?1

1

3

?1

?2

3

0

?1

?4

?2

?7

? into its normal form and hence find its rank.

OR

3 Reduce the quadratic form 3x

2

+ 3y

2

+ 3z

2

+ 2xy + 2xz ? 2yz into canonical form using orthogonal

transformation and find its rank, index and signature.

UNIT ? II

4 (a) Using Newton-Raphson method compute ?41 correct to four decimal places.

(b) Find the root of an equation 2 ?? ? log ?? = 6 by Regula-falsi method.

OR

5 (a) Evaluate ? ?? 3

?? ?? 1

0

with five sub-intervals by Trapezoidal rule.

(b) Evaluate ?

?? ?? ?? 2

1

?? ?? using Simpson?s

1

3

rule for n = 4.

UNIT ? III

6

Using Euler?s method, solve for ?? at ?? = 0.1 from

????

????

= ?? + ?? + ?? ?? , ?? (0) = 1 taking step size

? = 0.025.

OR

7 Find the Half range cosine series of ?? ( ?? ) = ?? (1 ? ?? ) ???? [0, 2].

Contd. in page 2

Page 1 of 2

R13

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Code: 13A54102

B.Tech I Year (R13) Supplementary Examinations December 2019

MATHEMATICS ? II

(Common to EEE, ECE, EIE, CSE & IT)

Time: 3 hours Max. Marks: 70

PART ? A

(Compulsory Question)

*****

1 Answer the following: (10 X 02 = 20 Marks)

(a) Find the Eigen values and the corresponding of ?

5 4

1 2

?.

(b) Show that ?? = ?

2 3 + 4 ?? 3 ? 4 ?? 2

? is Hermitian.

(c) Define algebraic and transcendental equations with example each.

(d) The value of ?

1

?? 2

1

?? ?? by Simpson?s 1/3 rule (taking n = 4) is________.

(e) If

????

????

= ? ?? , ?? (0) = 1, ? = 0.01 then by Euler?s method the value of ?? 1

is _________.

(f) Write the Fourier series of f(x) in [C, C+2L].

(g)

Find the Fourier cosine transform f(x) = e

ax ?

.

(h) Define convolution theorem.

(i) Write the two dimensional Laplace equation.

(j) Form a partial differential equation by eliminating the arbitrary constants a and b from the equation:

z = ax + by.

PART ? B

(Answer all five units, 5 X 10 = 50 Marks)

UNIT ? I

2

Reduce the matrix ?? = ?

2

1

3

6

3

?1

1

3

?1

?2

3

0

?1

?4

?2

?7

? into its normal form and hence find its rank.

OR

3 Reduce the quadratic form 3x

2

+ 3y

2

+ 3z

2

+ 2xy + 2xz ? 2yz into canonical form using orthogonal

transformation and find its rank, index and signature.

UNIT ? II

4 (a) Using Newton-Raphson method compute ?41 correct to four decimal places.

(b) Find the root of an equation 2 ?? ? log ?? = 6 by Regula-falsi method.

OR

5 (a) Evaluate ? ?? 3

?? ?? 1

0

with five sub-intervals by Trapezoidal rule.

(b) Evaluate ?

?? ?? ?? 2

1

?? ?? using Simpson?s

1

3

rule for n = 4.

UNIT ? III

6

Using Euler?s method, solve for ?? at ?? = 0.1 from

????

????

= ?? + ?? + ?? ?? , ?? (0) = 1 taking step size

? = 0.025.

OR

7 Find the Half range cosine series of ?? ( ?? ) = ?? (1 ? ?? ) ???? [0, 2].

Contd. in page 2

Page 1 of 2

R13

Code: 13A54102

UNIT ? IV

8

Find the Fourier series for

,0

() , 0

, 0

2

x

fx x x

x

??

?

?

?

?

? ? <<

?

= <<

?

?

?

? =

?

Hence deduce that

1

1

2

+

1

3

2

+

1

5

2

+. . . =

?? 2

8

.

OR

9 (a) Find ?? ( ?? sin ???? ).

(b) Find

( ) ( )

3

1

2

,3

32

z

Zz

zz

?

??

> ??

??

? ?

??

.

UNIT ? V

10

Form the PDE by eliminating arbitrary function

( )

2 22

,0 f x y z xyz ++ = .

OR

11 A bar of length ?? with insulated sides is initially 0 ? temperature throughout the end ?? = 0 is kept at

0 ? for all time and heat is suddenly applied such that

????

????

= 10 at ?? = ?? for all time. Find the

temperature function ?? ( ?? , ?? ).

*****

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This post was last modified on 11 September 2020