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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