Annamalai University december 2014 question paper

B.Sc. DEGREE EXAMINATION December 2014

(MATHEMATICS)

(THIRD YEAR)

(PART – III)

720: NUMERICAL METHODS AND TRIGONOMETRY

Time: Three hours Maximum: 100 marks

Answer any FIVE questions (5× 20=100)

1. a) Using Newton’s forward interpolation formula, find the value of y at x=46 from the

following of values.

X 45 50 55 60 65

Y 114.84 96.16 83.32 74.48 68.48

(10)

b) Given the following table, find y(35) using Stirling’s formula.

X 20 30 40 50

Y 512 439 346 243

(10)

2. a) Find the polynomial which takes the following values.

X 0 1 2 3 4

Y 1 2 5 10 17

(10)

b) Find

dx

dy at x=50 from the following table.

X 50 51 52 53 54 55 56

Y 3.6840 3.7084 3.7325 3.7563 3.7798 3.8030 3.8259

(10)

3. a) Apply Simpson’s rule to evaluate ∫

+

2

0

3

1 x

dx to two decimal places by dividing the

range into 4 equal parts.

(10)

b) Using the method of false position, find a real root of the equation x3+x2–1=0

correct to three decimal places.

(10)

4. a) Using Newton Raphson method, find a real root of the equation x3–3x+1=0 which

is between 1 and 2.

(10)

b) Use Graeffe’s method to solve the equation x3–x2–17x–15=0 (10)

2

5. a) Using Gauss Elimination method, solve the following system of equations.

2x+y+4z=12

8x–3y+2z=20

4x+11y–z=33

(10)

b) Solve the following system of equations by Gauss –Seidal method.

3x–y+2z=12

x+2y+3z=11

2x–2y–z=2

(10)

6. a) Using Crout’s method, solve the system of equations

2x–6y+8z=24

5x+4y–3z=2

3x+y+2z=16

(10)

b) Using Taylor series method, find y(0.1), given x y

dx

dy

= −

2

, y(0)=1.

(10)

7. a) Using Euler’s method, find y(0.1), y(0.2) and y(0.3) if y satisfies y

dx

dy

=1− and

y(0)=0

(10)

b) Apply Runge-Kutta method of fourth order to find an approximate value of y

when x=0.2, given that y′=3x+

2

1

y, y(0)=1.

(10)

8. a) Prove that tan–1x + tan–1y=tan–1

−

+

xy

x y

1

(10)

b) If tan–1x+tan–1y+tan-1z=

2

π

show that xy+yz+zx=1

(10)

9. a) Solve the equation x9+x5+x4+1=0 (10)

b)

Express

θ

θ

sin

sin 6

in terms of cos θ

(10)

10. a) Find Log (1+i) (10)

b) Find the sum of n terms of the series cos α+ cos(α+β)+ cos (α+2β)+........ (10)

--------------------

5278

B.Sc. DEGREE EXAMINATION December 2014

(MATHEMATICS)

(THIRD YEAR)

(PART – III)

720: NUMERICAL METHODS AND TRIGONOMETRY

Time: Three hours Maximum: 100 marks

Answer any FIVE questions (5× 20=100)

1. a) Using Newton’s forward interpolation formula, find the value of y at x=46 from the

following of values.

X 45 50 55 60 65

Y 114.84 96.16 83.32 74.48 68.48

(10)

b) Given the following table, find y(35) using Stirling’s formula.

X 20 30 40 50

Y 512 439 346 243

(10)

2. a) Find the polynomial which takes the following values.

X 0 1 2 3 4

Y 1 2 5 10 17

(10)

b) Find

dx

dy at x=50 from the following table.

X 50 51 52 53 54 55 56

Y 3.6840 3.7084 3.7325 3.7563 3.7798 3.8030 3.8259

(10)

3. a) Apply Simpson’s rule to evaluate ∫

+

2

0

3

1 x

dx to two decimal places by dividing the

range into 4 equal parts.

(10)

b) Using the method of false position, find a real root of the equation x3+x2–1=0

correct to three decimal places.

(10)

4. a) Using Newton Raphson method, find a real root of the equation x3–3x+1=0 which

is between 1 and 2.

(10)

b) Use Graeffe’s method to solve the equation x3–x2–17x–15=0 (10)

2

5. a) Using Gauss Elimination method, solve the following system of equations.

2x+y+4z=12

8x–3y+2z=20

4x+11y–z=33

(10)

b) Solve the following system of equations by Gauss –Seidal method.

3x–y+2z=12

x+2y+3z=11

2x–2y–z=2

(10)

6. a) Using Crout’s method, solve the system of equations

2x–6y+8z=24

5x+4y–3z=2

3x+y+2z=16

(10)

b) Using Taylor series method, find y(0.1), given x y

dx

dy

= −

2

, y(0)=1.

(10)

7. a) Using Euler’s method, find y(0.1), y(0.2) and y(0.3) if y satisfies y

dx

dy

=1− and

y(0)=0

(10)

b) Apply Runge-Kutta method of fourth order to find an approximate value of y

when x=0.2, given that y′=3x+

2

1

y, y(0)=1.

(10)

8. a) Prove that tan–1x + tan–1y=tan–1

−

+

xy

x y

1

(10)

b) If tan–1x+tan–1y+tan-1z=

2

π

show that xy+yz+zx=1

(10)

9. a) Solve the equation x9+x5+x4+1=0 (10)

b)

Express

θ

θ

sin

sin 6

in terms of cos θ

(10)

10. a) Find Log (1+i) (10)

b) Find the sum of n terms of the series cos α+ cos(α+β)+ cos (α+2β)+........ (10)

--------------------

5278

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