GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- IV EXAMINATION — SUMMER 2020
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Subject Code: 2140706 Date: 29/10/2020
Subject Name: NUMERICAL AND STATISTICAL METHODS FOR COMPUTER ENGINEERING
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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MARKS
Q.1 (a) Find the relative error if the number X =0.004997 is 03
- truncated to three decimal places.
- rounded off to three decimal places.
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(b) Find the negative root of x’ —7x+3 = 0 by the bisection method 04 correct up to three decimal places.
(c) Using Gauss Jacobi method solve the following system of the equations: 07
8x—y+2z=13
x—-10y+3z=17
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3x+2y+12z=25Q.2 (a) Using trapezoidal rule to evaluate ? dx, dividing the interval into four equal parts. 03
02 1/(2+ x2)
(b) By using Lagrange’s interpolation formula, find y(10). 04
X 5 6 9 11
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y 12 13 14 16(c) Using the Runge-Kutta method of fourth order, solve 10 dy/dx =x2+3y, y(0)=1 at x=0.1, x =0.2 taking h =0.1 07
OR
(c) Using Euler’s method find the approximate value of y at x=1.5 taking h =0.1. Given that dy/dx =(y-x)/vx and y(1)=2. 07
Q.3 (a) Using Newton Raphson method find the positive root of x2 —x—10 =0 correct up to three decimal places. 03
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(b) Fit a least square quadratic curve to the following data: 04
X 1 2 3 4
y 1.7 1.8 2.3 3.2
Estimate y(2.4).
(c) Find the regression coefficients bxy and byx hence, find the correlation coefficient between x and y for the following data 07
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X 4 2 3 4 2y 2 3 2 4 4
Q.4 (a) Using Simpson’s 1/3 rule, find ? ex dx, by taking n = 6. 03
00.6
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(b) Using Newton’s divided difference formula, compute f(10.5) from the following data: 04
X 10 11 13 17
f(x) 2.3026 2.3979 2.5649 2.8332
(c) Solve x4 —8x3 +39x2 —62x + 50 by using Lin Bairstow method up to third iteration starting with p0 = q0 =0. 07
Q.4 (a) Find a real root of the equation xlog10 x=1.2 by the regula falsi method. 03
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(b) The first four moments of distribution about x =2 are 1, 2.5, 5.5 and 16. Calculate the four moments about X and about zero. 04
(c) Given that dy/dx = (1/2)x2 + (1/2)y2, y(0) =1, y(0.1)=1.06, y(0.2)=1.12, y(0.3) =1.21 evaluate y(0.4) by Milne’s predictor-corrector method. 07
OR
Q.4 (a) Find the arithmetic mean form the following data: 03
Marks less than 10 20 30 40 50 60
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No. of students 10 30 60 110 150 180(b) (i) Obtain relation between ? and E. 04
(ii) Obtain relation between D and E.
(c) Obtain cubic spline for every subinterval from the following data 07
X 0 1 2 3
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f(x) 1 2 33 244Q.5 (a) Two unbiased coins are tossed. Find expected value of number of heads. 03
(b) By Simpson’s 3/8 rule, evaluate ? dx taking h = 1/6 04
01 1/x
(c) From the following table, estimate the number of students who obtained marks between 40 and 45. 07
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Marks 30-40 40-50 50-60 60-70 70-80No. of students 31 42 51 35 31
OR
Q.5 (a) Using Budan’s theorem find the number of roots of the equation S(x)=x5—4x3 +3x2 ~10x+8 =0 in the interval [-1,0]. 03
(b) Find the positive solution of x—2sinx =0, correct up to three decimal places starting from x0=2 and x1 =1.9. Using secant method. 04
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(c) Using Gauss Siedel method solve the following system of the equations: 07
3x—-0.1y-0.2z=17.85
0.1x+7y-0.3z=-19.3
0.3x-0.2y+10z=71.4
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