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III B.A./B.Sc. Mathematics ACHARYA NAGARJUNA UNIVERSITY CURRICULUM - B.A / B.Sc MATHEMATICS - PAPER - IV (ELECTIVE - 1) NUMERICAL ANALYSIS QUESTION BANK FOR PRACTICALS UNIT - I Paper IV (Elective -1) - Curriculum 1. i) Which of the following numbers has the greatest precision. a) 4⋅ 3201 b) 432 ⋅ c) 4⋅320106 ii) How many digits are to be taken in computing 20 so that the error does not exceed 0⋅ 01% . 2. i) Sum the numbers 0⋅1532, 15⋅45, 0⋅000354, 305⋅1, 8⋅12, 143⋅3, 0⋅0212, 0⋅643 and 0⋅1734where in each of which all the given digits are correct. ii) If u= 5xy 2 / z 3 then find relative maximum error in u , given that ∆x = ∆y = ∆z = 0001and ⋅ x = y = z =1. 3 3. Find a real root of the equation f ( x) = x −x− 1= 0 by bisection method. 3 4. Find a real root of the equation x −6x− 4= 0 by bisection method. 5. Find a positive root of the equation xe x = 1, which lies between 0 and 1 by bisection method. 6. Find the root of tan x+ x =0 upto two decimal places, which lies between 2 and 21 ⋅ by bisection method. 7. Find a real root of the equation xlog 10 x = 12 ⋅ by bisection method. 3 8. Find a real root of the equation f ( x) = x −2x− 5= 0by the method of false position upto three places of decimals. 3 2 9. Find a real root of the equation x −x − 2= 0 by Regula-Falsi method. 10. Find the root of the equation xe x = cos x using the Regula Falsi method correct to three decimal places. 3 11. Find the root of x + x− 1= 0by iteration method, give that root lies near 1. 12. Find a real root of the equation cos x = 3x−1 correct to three decimal places, using iteration method. 13. Find by the iteration method, the root near 38 ⋅ , of the equation 2x− log 10 x = 7 correct to four decimal places. 2 14. Find the real root of the equation x − 5x+ 2= 0by Newton-Raphson’s method. 15. Find by Newton’s method, the root of the equation e x = 4xwhich is near to 2 correct to three places of decimals. 16. Using Newton-Raphson method, establish the iterative formula x = 1 ⎛ N n+ x + ⎞ 1 ⎜ n ⎟ to calculate the square 2 ⎝ x ⎠ root of N. Hence find the square root of 8. 2 n
III B.A./B.Sc. Mathematics 3 Paper IV (Elective -1) - Curriculum 17. Using Newton-Raphson method, establish the iterative formula x = 1 ⎛ N n+ x + ⎞ 1 ⎜ n 2 ⎟ 3 ⎝ 2 to calculate the x ⎠ cube root of N. Hence find the cube root of 12 applying the Newton-Raphson formula twice. 18. 3 2 Find a double root of the equation f ( x) = x −x − x+ 1= 0by generalized Newton’s method. 19. Find a root of the equation xe x = 1 by Ramanujan’s method. 20. 3 Find the root of the equation yx ( ) = x −2x− 5= 0, which lies between 2 and 3 by Muller’s method. 21. Show that i) ( 1+ ∆ )( 1−∇ ) = 1 ii) E∇=∆ iii) −12 / −12 / δ = E ∆ = ∆ E iv) 2 ∇ − ∆ = ∆∇ = δ v) µ = 1+ 1 δ 4 2 2 n 22. Evaluate i) ∆2 x 3 3 Ex ii) ⎛ ∆ 2 ⎞ 3 ⎜ ⎟ ⎝ E ⎠ x , the interval of differencing being unity. 2 3 2 3 23. Prove that i) u = u + ∆u + ∆ u + ∆ u ii) u = u + ∆u + ∆ u + ∆u 3 2 1 24. Find the missing term in the following data. x : 0 1 2 3 4 y: 1 3 9 81 0 0 4 3 2 3 25. Form a table of differences for the function f ( x) = x + 5x−7for x =−1012345 , , , , , , and continue the table to obtain f ( 6) and f ( 7 ). 3 2 26. Find the function whose first difference is x + 3x + 5x+ 12, if 1 be the interval of differencing. 27. The population of a country in the decennial census were as under. Estimate the population for the year 1895. Year ( x ): 1891 1901 1911 1921 1931 Population ( y )(in thousands) : 46 66 81 93 101 28. From the following find y value at x = 026. ⋅ x 010 ⋅ 015 ⋅ 020 ⋅ 025 ⋅ 030 ⋅ y = Tanx 0⋅1003 0⋅1511 02027 ⋅ 0⋅2553 0⋅3093 29. From the following table, find the number of students who obtain less than 56 marks. Marks : 30-40 40-50 50-60 60-70 70-80 No. of students : 31 42 51 35 31 30. Find the cubic polynomial which takes the following values. x : 0 1 2 3 f ( x): 0 2 1 10 1 1
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