Download JNTUA B.Tech 1-1 R13 2019 Dec 13A54102 Mathematics II Question Paper

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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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