cosm question bank

QUESTION BANK Subject code: 151601

Subject Name: Computer Oriented Statistical Methods

Prepared by Dr. Shailja Sharma

1 If u =2v6 -5 , find the percentage error in u at v =1 if error in v is 0.05. 3 June11 2 Find the solution of the following equation using floating point arithmetic

with 4-digit mantissa x2 -1000x +25 =0 3 June11

3 Discuss the pitfalls in computing using normalize floating – point numbers. 3 Dec10 4 Explain Floating Point Representation of number with example. 3 June12

5 Explain different types of Errors with it’s propagation during computation & how to improve the accuracy of Numeric Computation.

6 June12

6 Discuss briefly the different types of errors encountered in performing numerical calculations

3 Nov11

7 Find the root of the equation x4 – x – 10 = 0 upto 3 decimal points using Bisection Method.

7 June12

8 Find the root of the equation 2x-log10x-7 = 0 correct to three decimal places using iteration method.

3 June11

9 Find the approximate root of the equation x3 ‐ 4x ‐ 9 = 0 by using False Position Method.

7 June12

10 Use three iterations of Newton Raphson Method to solve the non-linear quations, 09,0722 =+−=+− xyxyx .Take )5.4,5.3(),( 00 =yx as the initial approximation.

6 Dec10

11 Find the real root of the equation x3 - 9x +1=0 by method of Newton Raphson 4 Dec 10 12 Explain Newton Raphson Method in detail 5 June12 13 Prove that Newton-Raphson procedure is second order convergent. 3 Nov11 14 If y(1) = 4, y(3) = 12, y(4) = 19 and y(x) = 7 then find x by Newton’s formula 15 Find the root of the equation by Secant method. 4 June11 16 Write an algorithm for the false position method to find root of the

equation f ( x) = 0 . 3 Nov11

17 Write an algorithm for the successive approximation method to find root of nonlinear equation.

2 Dec10

18 Use the secant method to estimate the root of xexf x −= −)( correct to two significant digits with initial estimate of x-1 =0 and x0 =1.0

4 Nov 11

19 Describe BAIRSTOW method in brief 5 Dec10 20 Find all roots of the equation x3 – 2x2 -5x + 6 = 0 using Graeffe`s method

squaring thrice. 5 7

Dec10, Nov11

21 Use Lagrange’s formula to find third degree polynomial which fits into the data below

X 0 1 3 4 Y -12 0 12 24

Evaluate the polynomial for x = 4.

5 June11

22 State Budan’s theorem. Apply it to find the number of roots of the equation in the interval [-1, 0] and [0,1].

June11

23 Find the root of the equation using Lin-Bairstow’s Method

4 June11

24 Compute f '(0.75) , from the following table x 0.50 0.75 1.00 1.25 1.50 f(x) 0.13 0.42 1.00 1.95 2.35

3 June11




25 Evaluate by (i) Trapezoidal rule (ii) Simpson’s 1/3 rule 4 June 11

26 Represent the function f(x) = 3 in factorial notation and hence show that f (x) =18.

5 June11

27 The distance, s(in km) covered by a car in a given time, t (min) is given in the following table

Time(t) 0 1 2 3 4 5 6 Distance(s) 0 2.5 8.5 15.5 24.5 36.5 50

Estimate the speed and acceleration of the car at t = 5 minutes.

4 June11

28 The distance (s) covered by a car in a given time (t) is given below Time(Minutes) : 10 12 16 17 22 Distance(Km.) : 12 15 20 22 32 Find the speed of car at time t =14 minutes

6 Dec10

29 Obtain cubic spline for every subinterval from the following data x : 0 1 2 3 f(x) : 1 2 33 244 Hence an estimate f(2.5)

6 Dec10

30 Fit cubic splines for first two subintervals from the following data. Utilize the result to estimate the value at x=5. x: 3 4.5 7 9 f(x) : 2.5 1 2.5 0.5

7 Nov11

31 Estimate the function value f (7) using cubic splines from the following data given p0 =p2 =0

i 0 1 2 zi 4 9 16 fi 2 3 4

5 June11

32 Prove the following (i) ∇−Δ=Δ∇=∇Δ (ii) 2/1E∇=δ 4 June11 33 Write an algorithm for Lagrange’s interpolation method to interpolate a value

of dependent variable for given value of independent variable. 2 Dec10

34 Differentiate Interpolation & Extrapolation. 3 June12 35 Estimate the value of f(22) and f(42) from the following data

x: 20 25 30 35 40 45 f(x): 354 332 291 260 231 204

5 June11

36 Explain Cubic Spline Interpolation with it’s conditions. 3 June12 37 Write Langrage Interpolation Algorithm & Solve the following using it:

Find f(x) at x=4. X : 1.5 3 6 f(x) : ‐0.25 2 20

8 June12

38 Consider the following table: x : 20 25 30 f(x) : 0.342 0.423 0.500 Find the value of x where f(x) = 0.399 using Inverse Interpolation. Would you use the difference method or Lagrangian Method?

8 June12

39 Write an algorithm for Trapezoidal Rule to integrate a tabulated function. 3 Nov11




40 Evaluate ∫x2 dx using Trapezoidal Rule by taking h=1/8. 4 June12 41

Evaluate : ∫5

110log xdx , taking 8 subintervals, correct to four decimal places by

Trapezoidal rule.

6 Dec10

42 The table gives the distance in nautical miles of the visible horizon for the given heights in feet above the earth’s surface. Find the values of y when x=390ft. height(x): 100 150 200 250 300 350 400 Distance(y): 10.63 13.03 15.04 16.81 18.42 19.90 21.27

4 Nov11

43 A train is moving at the speed of 30 m/sec. suddenly brakes are applied. The speed of the train per second after t seconds is given by the following table. Time(t): 0 5 10 15 20 25 30 Speed(v): 30 24 19 16 13 11 10 Apply Simpson’s three-eighth rule to determine the distance moved by the train in 30 seconds.

4 Nov11

44 Explain Simpson 1/3 Rule in detail. 4 June12 45 Using Simpson’s rule, find the volume of the solid of revolution formed by

rotating about x-axis. The area between the x-axis, the lines x = 0 and x = 1 and a curve through the points (0,1), (0.25,0.9896), (0.50,0.9589), (0.75,0.9089) and (1,0.8415).

46 A slider in a machine moves along a fixed straight rod. Its distance x cm. along the rod is given below for various values of the time t seconds. Find the velocity of the slider when t = 0.1 second. t: 0 0.1 0.2 0.3 0.4 0.5 0.6 x: 30.13 31.62 32.87 33.64 33.95 33.81 33.24

7 Nov11

47 Write an algorithm for simpson`s three-eight rule to integrate a tabulated function.

2 Dec10

48 Compute f’(0.75),from the following table X: 0.50 0.75 1.00 1.25 1.50 F(x):0.13 0.42 1.00 1.95 2.35

3 June11

49 Evaluate dx

x∫ +

6

021

1 by (i)Trapezoidal rule (ii) Simpson’s 1/3 rule

4 June11

50 The following data gives pressure and volume of superheated steam V: 2 4 6 8 10 P: 105 42.7 25.3 16.7 13 Find the rate of change of pressure w.r.t. volume when V=8

6 Dec10

51 Following table shows speed in m/s and time in second of a car t : 0 12 24 36 48 60 72 84 96 108 120 v : 0 3.60 10.08 18.90 21.60 18.54 10.26 5.40 4.50 5.40 9.00 Using simpson`s one-third rule find the distance travelled by the car in 120 second

6 Dec10

52 Given where y = 0 when x = 0 find y(0.2) and y(0.4) using

Runga Kutta method

5 June11




53 Solve the dy/dx = x2– y, y(0) = 1. Find y(0.1) and y(0.2), h=0.1 using Runge Kutta’s 2nd Order Method.

7 June 12

54 Given that 2yx

dxdy

+= , y(0) = 1. Using Runge-kutta method find

approximate value of y 0.2,take step size 0.1

6 Dec10

55 Given that yx

dxdy

+= with initial condition y(0)=1.Use Runge-Kutta fourth

order method to find y(0.1).

4 Nov11

56 Write an algorithm for Euler`s method to solve ODE ),( yxf

dxdy

= 2 Dec10

57 Solve dy/dx = 2x – y, y(0) = 2 in the range 0 ≤ x ≤ 0.3 by taking h=0.1 using Euler’s Method

7 June12

58 Using Euler`s method, compute y(0.5) for differential equation 22 xy

dxdy

−= with y = 1 when x = 0(taking h = 0.1)

4 June11

59 Solve the differential equation yx

dxdy

+= [0,1] x 1, = y(0)with ∈ by Taylor’s

series expansion to obtain y for x = 0.

4 June11

60 Use Taylor series to find approximate value of cos(-8 ) to 5 significant digits. 7 Nov11 61

Use Heun’s predictor-corrector method to integrate yedxdy x 5.04 8.0 −= from

x= 0 to x = 3 with a step size of 1. The initial condition at x=0 is y=2.(Perform only one iteration in corrector step)

7 Nov11

62 Using Gauss’s quadrature formula, evaluate ∫ +

4

2

2 )2( dxxx 3 June11

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