Explain the following terms:
[3 marks]Inherent Error (ii) Truncation Error (iii) Round-off Error
[ marks]Write an Algorithm for Bisection method.
[4 marks]Derive Newton-Raphson method analytically. Use it to find a root between 0 and 1 of the equation exsinx−1 = 0, correct up to four decimal places.
[7 marks]Evaluate ∫ 1 exdx using trapezoidal rule for n = 4. −1
[3 marks]Fit a straight line to the following data. Also find y at x = 2.5. x 0 1 2 3 y 1 1.8 3.3 4.5 6.3
[4 marks]Apply Budan’s theorem to find the number of roots of the equation x3 −3x2 −4x+13 = 0 in the intervals [−3,−2], [−2,−1], [−1,0], [0,1], [1,2] and [2.3].
[7 marks]Find all the roots of x3 −x−1 = 0 using Bairstow’s method. Start with the initial factor x2 +x +1.
[7 marks]Find a real root of the equation cos x +1 = 3x correct up to three decimal places by the iteration method.
[3 marks]Use Lagrange formula to find f(3) from the following table: x 0 1 2 f(x) 1 3 9 81
[4 marks]Find the cubic spline in the interval [0,2] for the following data: x 0 2 4 y 1 9 41 41 Given M = 0 and M = −12.03
[6 marks]Find the positive root of x−2sin x = 0 using secant method. Correct up to three decimal places starting from x = 2 and x = 1.9.01
[3 marks]Use Newton’s divided difference formula to determine f(5) for the following data: x 1 2 4 f(x) 14 15 5 9
[6 marks]Fit a second degree parabola using least squares method to the following data: x 0 1 2 3 y 1 1.8 1.3 2.5 6.3
[4 marks]18 Evaluate ∫ ydx from the following table using Simpson’s 3/8 rule:0 x 0 3 6 9 12 15 18 y 0 22 29 31 20 4 0
[3 marks]Write an algorithm for Simpson’s 1/3 rule.
[4 marks]Apply Runge-Kutta method of order four to find an approximate value of y when x = 0.2 in steps of 0.1 if dy = x +y2 given that y = 1 when x = 0. dx
[7 marks]If the two lines of regression are 4x −5y+30 = 0 and 20x −9y−107 = 0, which of these lines are lines of regression of x on y and y on x? Find r and 𝜎 xy y when 𝜎 = 3. x
[3 marks]Discuss about the pitfalls in Gauss elimination method.
[4 marks]Solve the following system by Gauss-Seidel method: 5x +y−z = 10; 2x +4y+z = 14; x+y+8z =
[20 marks]Calculate the 3-yearly moving averages of the following data: Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 Sales (millions 3 4 8 6 7 11 9 10 14 of rupees)
[12 marks]The runs scored by two batsmen Aand Bin 9 consecutive matches are given below: A 85 20 62 28 74 5 69 4 13 B 72 4 15 30 59 15 49 27 26 Which of the batsmen is more consistent?
[4 marks]Define central moments. Calculate the first four moments from the following data: x 0 1 2 3 4 5 6 7 f 5 10 15 20 25 20 15 10
[5 marks]Calculate the correlation coefficient between x and y using the following data: x 2 4 5 6 8 11 y 18 12 10 8 7
[5 marks]Find the standard deviation for the following distribution: Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of Students
[4 marks]Find the two regression lines from the following table: x 0 1 2 3 4 5 6 7 8 9 y 43 46 82 98 123 167 199 213 245 272
[7 marks]