paper - iv - Acharya Nagarjuna University
paper - iv - Acharya Nagarjuna University
paper - iv - Acharya Nagarjuna University
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III B.A./B.Sc. Mathematics<br />
6<br />
Paper IV (Elect<strong>iv</strong>e -1) - Curriculum<br />
55. Use Simpson’s 1 3 rule to prove that log e<br />
7 is approximately 1⋅9587<br />
using<br />
7<br />
dx<br />
∫ .<br />
x<br />
1<br />
56. Find the value of the integral<br />
1<br />
∫<br />
0<br />
dx<br />
1+<br />
x<br />
2<br />
by using Simpson’s 1 3 and 3 8<br />
rule. Hence obtain the approximate<br />
value of π in each case.<br />
57. Evaluate<br />
1<br />
∫<br />
0<br />
1<br />
dx by Boole’s rule.<br />
1+<br />
x<br />
x<br />
58. Evaluate the integral e dxby Boole’s Rule.<br />
4<br />
∫<br />
0<br />
52 ⋅<br />
∫<br />
59. Evaluate the integral log xdx, using Weddle’s rule.<br />
4<br />
π / 2<br />
60. Integrate numerically<br />
∫<br />
0<br />
sin<br />
x dx by Weddle’s rule.<br />
61. Solve the equations 3x+ 2y+ 4z = 7; 2x+ y+ z = 7; x+ 3y+ 5z<br />
= 2 by matrix inversion method.<br />
62. Solve the equations x+ y+ z = 9, 2x+ 5y+ 7z = 52, 2x+ y− z = 0by Cramer’s rule.<br />
63. Solve the equations 2x− y+ 4z = 12; 8x− 3y+ 2z = 23; 4x+ 11y− z = 33 by Gauss elimination<br />
method.<br />
64. Solve the system of equations 2x+ y+ z = 10, 3x+ 2y+ 3z = 18, x+ 4y+ 9z<br />
= 16by Gauss elimination<br />
method.<br />
65. Solve the equations 3x+ 2y+ 4z = 7; 2x+ y+ z = 7; x+ 3y+ 5z<br />
= 2by Factorization method.<br />
66. Solve the equations 2x+ 3y+ z = 9, x+ 2y+ 3z = 6, 3x+ y+ 2z<br />
= 8 by LU decomposition method.<br />
67. Solve the following equations by Gauss-Jacobi method :10x− y+ z = 12; x− 10y+ z = 12;<br />
x+ y− 10z<br />
= 12 correct to 3 decimals.<br />
68. Solve the following equations by Jacobi method : 20x+ y− 2z<br />
= 17; 3x+ 20y− z = − 18;<br />
2x− 3y+ 20z<br />
= 25 .<br />
69. By Gauss-seidel iterat<strong>iv</strong>e method solve the linear equations x + 10x + x = 6; 10x + x + x = 6 and<br />
x + x + 10x<br />
= 6 .<br />
1 2 3<br />
1 2 3 1 2 3<br />
70. Solve the following system by Gauss-Seidel method : 10x+ 2y+ z = 9, 2z+ 20y− 2z<br />
= −44,<br />
− 2x+ 3y+ 10z = 22 .<br />
dy<br />
71. Solve the differential equation = x + y , with y( 0) = 1, x∈ [ 0, 1]<br />
by Taylor series expansion to obtain<br />
dx<br />
y for x = 01. ⋅