Download GTU BE/B.Tech 2019 Winter 4th Sem Old 140001 Mathematics Iv Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 4th Sem Old 140001 Mathematics Iv Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (Old) EXAMINATION ? WINTER 2019
Subject Code: 140001 Date: 07/12/2019

Subject Name: Mathematics-IV

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a) Find the fifth root of unity. 07
(b) Define interpolation. Using Lagrange?s interpolation, find ?? (2).
?? ?1 0 1 3
?? 2 1 0 ?1

07

Q.2 (a) Define bilinear transformation. Find bilinear transformation which maps points
0,1, ? into points ?1, ??? , 1 respectively.
07
(b)
Expand
1
?? 2
?3?? +2
in Laurent series in region (i) 1 < |?? | < 2 (ii) |?? | > 2.
07
OR
(b)
Prove that tan ?
?1
?? =
1
2
?????? (
1+?? 1??? )
07

Q.3 (a)
State Cauchy?s integral formula. Evaluate ?
3?? 2
+?? ?? 2
?1
???? ;
?? where |?? ? 1| = 1.
07
(b) Define (i) Analytic function (ii) Harmonic function.
If ?? (?? ) = ?? (?? , ?? ) + ???? (?? , ?? ) is analytic and ?? (?? , ?? ) = ?? 3
? 3?? 2
?? , then find
?? (?? , ?? ).
07
OR
Q.3 (a) Derive Cauchy ? Riemann equations in polar form. 07
(b)
Evaluate ? ?? exp ( ?? ?? ? )???? ;
?? where ?? is the boundary of square with vertices
0, 1, 1 + ?? , ?? with counterclockwise direction.
07

Q.4 (a)
Prove that (i) (1 + ?)(1 ? ?) = 1 (ii) ?? = ?? ???
07
(b) Using Runge-Kutta 4
th
order method find ?? (0.1).
???? ???? = ?? 2
+ ?? 2
; ?? (0) = 1
07
OR
Q.4 (a) Find Newton?s forward interpolating polynomial and hence find ?? (5).
?? 4 6 8 10
?? 1 3 8 16

07
(b)
Evaluate ?
????
1+?? 2
6
0
(take h =1) using
(i) Trapezoidal rule (ii) Simpson?s 1/3 rule and (iii) Simpson?s 3/8 rule.
07

Q.5 (a) Set up Newton iteration for computing square root of given positive number
?? and find ?2.
07
(b) Evaluate using 2 point and 3 point Gaussian integration.
?
???? 1 + ?? 1
0

07




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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (Old) EXAMINATION ? WINTER 2019
Subject Code: 140001 Date: 07/12/2019

Subject Name: Mathematics-IV

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a) Find the fifth root of unity. 07
(b) Define interpolation. Using Lagrange?s interpolation, find ?? (2).
?? ?1 0 1 3
?? 2 1 0 ?1

07

Q.2 (a) Define bilinear transformation. Find bilinear transformation which maps points
0,1, ? into points ?1, ??? , 1 respectively.
07
(b)
Expand
1
?? 2
?3?? +2
in Laurent series in region (i) 1 < |?? | < 2 (ii) |?? | > 2.
07
OR
(b)
Prove that tan ?
?1
?? =
1
2
?????? (
1+?? 1??? )
07

Q.3 (a)
State Cauchy?s integral formula. Evaluate ?
3?? 2
+?? ?? 2
?1
???? ;
?? where |?? ? 1| = 1.
07
(b) Define (i) Analytic function (ii) Harmonic function.
If ?? (?? ) = ?? (?? , ?? ) + ???? (?? , ?? ) is analytic and ?? (?? , ?? ) = ?? 3
? 3?? 2
?? , then find
?? (?? , ?? ).
07
OR
Q.3 (a) Derive Cauchy ? Riemann equations in polar form. 07
(b)
Evaluate ? ?? exp ( ?? ?? ? )???? ;
?? where ?? is the boundary of square with vertices
0, 1, 1 + ?? , ?? with counterclockwise direction.
07

Q.4 (a)
Prove that (i) (1 + ?)(1 ? ?) = 1 (ii) ?? = ?? ???
07
(b) Using Runge-Kutta 4
th
order method find ?? (0.1).
???? ???? = ?? 2
+ ?? 2
; ?? (0) = 1
07
OR
Q.4 (a) Find Newton?s forward interpolating polynomial and hence find ?? (5).
?? 4 6 8 10
?? 1 3 8 16

07
(b)
Evaluate ?
????
1+?? 2
6
0
(take h =1) using
(i) Trapezoidal rule (ii) Simpson?s 1/3 rule and (iii) Simpson?s 3/8 rule.
07

Q.5 (a) Set up Newton iteration for computing square root of given positive number
?? and find ?2.
07
(b) Evaluate using 2 point and 3 point Gaussian integration.
?
???? 1 + ?? 1
0

07




2
OR
Q.5 (a) Find a root of ?? 3
? 4?? ? 9 = 0 correct upto three decimal places using
Bisection method.
07
(b) Solve by Gauss-Siedel method correct upto 3 decimal places.
27?? + 6?? ? ?? = 85
6?? + 15?? + 2?? = 72
?? + ?? + 54?? = 110

07

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