IGNOU MCA MCSE-004 5th semester term-end examination (NUMERICAL AND STATISTICAL COMPUTING) books/block,term-end exam notes,upcoming guess paper,important questions,study materials,previous year papers download.
Notes-2
1. (a) Show that a(b-c) ≠ ab-ac, where
Use 4-digit precision floating point and significant digit rounded off.
(b) Solve the following linear system of equations using Gauss Elimination method with partial pivoting :
x1 + x2 + x3 = 3
4x1 + 3x2 + 4x3 = 11
9x1 + 3x2 + 4x3 = 16
(c) Estimate the missing term in the following data using forward differences :
x: 1 2 3 4 5
f(x) : 3 7 ? 21 31
using Simpson's 1/3 rule with h = 0.5.
(e) A filling machine is set to pour 952 ml of oil into bottles. The amount to fill is normally distributed with a mean of 952 ml and a standard deviation of 4 ml.Use the standard normal table to find the probability that the bottle contains oil between 952 and 956 ml.
(f) What is the utility of residual plots ? What is the disadvantage of residual plots ?
(g) If 𝛑= 314159265, then find out to how many decimal places the approximate value of 22/ 7 is accurate.
(h) Three bags of same type have the following balls :
Bag 1 : 2 black 1 white
Bag 2 : 1 black 2 white
Bag 3 : 2 black 2 white
One of the bags is selected and one ball is drawn. It turns out to be white. What is the probability of drawing a white ball again, the first one not having been returned ?
(i) Define Poisson Distribution.
2. (a) Find the smallest positive root of the quadratic equation
x2 - 8x + 15 = 0,
using Newton-Raphson method.
(b) Find the Lagrange interpolating polynomial of degree 2 approximating the function y = ln x. Hence determine the value of ln 2.7. Also find the error.
x 2 2.5 3.0
y = ln x 0-69315 0.91629 1.09861
(c) What are the sources of errors in numerical data and processing ? How does error measure accuracy ?
using Gauss-Legendre three-point formula.
(b) Solve the initial value problem u' = - 2t u2 with u(0) = 1 and h 0.2 on the interval [0, 1]. Use the fourth order classical Runge-Kutta method.
(c) Evaluate
using Composite Simpson's rule with 5 points.
4. (a) Calculate the correlation coefficient for the following heights (in inches) of fathers (X)
and their sons (Y) :
X: 65 66 67 67 68 69 70 72
Y: 67 68 65 68 72 72 69 71
Obtain the equations of lines of regression. Also estimate the value of X for Y = 70.
(b) A manufacturer of cotter pins knows that 5% of his product is defective. If he sells cotter pins in boxes of 100 and guarantees that not more than 10 pins will be defective, what is the approximate
probability that a box will fail to meet the guaranteed quality ?
5. (a) What do you mean by pseudo-random number generation ? What is the practical advantage of the concept of random number generation ?
(b) For the data given in the table, compute R
(c) If a bank receives on an average 𝛌 = 6 bad cheques per day, what is the probability that it will receive 4 bad cheques on any given day, where 𝛌, denotes the average arrival rate per day ?
Notes-3
1. (a) Solve the quadratic equation 4x2 + 8x - 21 = 0 using two decimal digit arithmetic with rounding, using any one of the following methods :
(i) Regula-Falsi
(ii) Secant
(iii) Bisection
(b) Round off the number 4.5126 to 4 significant figures and find the relative percentage error.
(c) Obtain the positive root of the equation x2 - 1 = 0 by Newton-Raphson method,correct to two decimal places.
(d) Explain the two pitfalls in the Gauss Elimination Method.
(e) Solve the following system of linear equations using LU decomposition method :
6x1 - 2x2 = 14
9x1 - x2 + x3 = 21
3x1 + 7x2 + 5x3 = 9.
(f) What is the lowest degree polynomial which satisfies the following set of values,using forward difference polynomial ? Also find the polynomial.
x 0 1 2 3 4 5 6 7
f(x) 0 7 26 63 124 215 342 511
(g) Calculate the value of the integral
Assume h = 0.2.Compare the numerical solution with the exact solution.
2. (a) What do you mean by the terms "Accuracy" and "Precision" ? How are they related to significant digits ?
(b) Show that the equation x3 - 6x - 1 = 0 has a root in the interval ] - 1, 0[. Obtain this root using Successive Iteration or Bisection method.
(c) Find the Lagrange interpolating polynomial of degree 2 approximating the function y = ln x defined by the following values mentioned in the table. Hence determine the value of In 2.7.
x 2 2.5 3.0
y = In x 0.69315 0.91629 1.09861
3. (a) Solve the initial value problem u' = - 2tu2, with u(0) = 1, h = 0.2 on the interval [0, 1]. Use the fourth order classical Runge-Kutta method.
(b) Solve the following system of equations using Gauss elimination with partial pivoting :
2x + y + z = 10
3x + 2y + 3z = 18
x + 4y + 9z = 16
(c) What is the utility of residual plots ? What is the disadvantage of residual plots ?
4. (a) If a bank receives on an average 𝛌 = 6 bad cheques per day, what is the probability that it will receive 4 bad cheques on any given day, where 𝛌 denotes the average arrival rate per day ?
(b) A hosiery mill wants to estimate how its monthly costs are related to its monthly output rate. For that the firm collects data regarding its cost and output for a sample of nine months as given by the following table :
Output (tons) Production Cost (thousands of dollars)
1 2
2 3
4 4
8 7
6 6
5 5
8 8
9 8
7 6
(i) Draw the scatter diagram for the data.
(ii) Find the regression equation when the monthly output is the dependent variable (x) and monthly cost is the independent variable (y).
(iii) Use this regression line to predict the firm's monthly cost if they decide to produce 4 tons per month.
(c) An individual's IQ score has a N(100, 15) distribution. Find the probability that an individual's IQ score is between 91 and 121.
two-point Gauss-Legendre formulae.Compare with the exact solution and the exact value is I = 1.
(b) The following values of the function f(x) for the values of x are f(1) = 4, f(2) = 5, f(7) = 5 and f( 8) = 4.Find the value of f(6) and also the value of x in the interval [1, 8] for which f(x) is maximum or minimum.
(c) Round off the number 4.5126 to four significant figures and find the relative percentage error.
IGNOU MCA MCSE-004 5th semester term-end examination (NUMERICAL AND STATISTICAL COMPUTING) term-end exam notes Download
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Notes-1
1. (a) Determine the root of the equation 2x = cos x + 3 correct to three decimal places.
(b) Solve the following system of equations by using Gauss Elimination method.
2x+ y+ z =10
3x + 2y + 3z = 18
x + 4y + 9z = 16
(c) Using Lagrange interpolation, determine the value of log10 301, from the tabulated data given below :
X 300 304 305 307
log10 X 2.4771 2.4829 2.4843 2.4871
(d) Ten coins are thrown simultaneously, Find the probability of getting at least seven heads.
(e) What do you mean by "Goodness to fit test" ? What for the said test is required ?
(f) Calculate the value of the integral
by using Simpsons 3/8 rule.
(g) Find the probability that an individual's IQ score is between 91 and 121. Provided,the individuals IQ score has a Normal distribution N (100, 152).
(h) Write short note on following :
(i) Non Linear Regression
(ii) Acceptance Rejection Method
2. (a) Determine the value of expression X = -√3+√5+√7;
accurate up to 4 significant digits, also find the absolute and relative errors.
(b) Determine the value of Y using Euler's method, when X = 0.1 Given Y(0) =1 and Y'=X2 +Y.
(c) Find the value of ∆ tan-1 x, where ∆ is the difference operator, with differencing step size 'h'.
(d) Solve the following system of equations by using LU Decomposition method.
x+y=2 ; 2x+3y=5
3. (a) Solve the initial value problem given below,By using Runge - Kutta Method.
dy/dx=y-x with y(0) = 2 and h= 0.1 also find y(0.1) and y(0.2) correct to four decimal places.
(b) Determine the Goodness to fit parameter 'R' for the data given below.
X 100 110 120 130 140 150 160 170 180 190
Y 45 51 54 61 66 70 74 78 85 89
Analyse the results and comment on whether the predicted line fits well into the data or not.
4. (a) Develop the difference table for the data given below and use it to find the first and tenth term for the given data.
X 3 4 5 6 7 8 9
Y 2.7 6.4 12.5 21.6 34.3 51.2 72.9
(b) Find the smallest root of the equation f(x) = x 3-6x2+11x-6 = 0 by using Newton - Raphson method. Give two drawbacks of Newton - Raphson method.
5. (a) In a partially destroyed laboratory record of an analysis of correlation data,the following data are only legible :
(i) Variance of X = 9
(ii) Regression equation :
8X-10Y+66 = 0
40X-18Y = 214
Notes-2
1. (a) Show that a(b-c) ≠ ab-ac, where
Use 4-digit precision floating point and significant digit rounded off.
(b) Solve the following linear system of equations using Gauss Elimination method with partial pivoting :
x1 + x2 + x3 = 3
4x1 + 3x2 + 4x3 = 11
9x1 + 3x2 + 4x3 = 16
(c) Estimate the missing term in the following data using forward differences :
x: 1 2 3 4 5
f(x) : 3 7 ? 21 31
using Simpson's 1/3 rule with h = 0.5.
(e) A filling machine is set to pour 952 ml of oil into bottles. The amount to fill is normally distributed with a mean of 952 ml and a standard deviation of 4 ml.Use the standard normal table to find the probability that the bottle contains oil between 952 and 956 ml.
(f) What is the utility of residual plots ? What is the disadvantage of residual plots ?
(g) If 𝛑= 314159265, then find out to how many decimal places the approximate value of 22/ 7 is accurate.
(h) Three bags of same type have the following balls :
Bag 1 : 2 black 1 white
Bag 2 : 1 black 2 white
Bag 3 : 2 black 2 white
One of the bags is selected and one ball is drawn. It turns out to be white. What is the probability of drawing a white ball again, the first one not having been returned ?
(i) Define Poisson Distribution.
2. (a) Find the smallest positive root of the quadratic equation
x2 - 8x + 15 = 0,
using Newton-Raphson method.
(b) Find the Lagrange interpolating polynomial of degree 2 approximating the function y = ln x. Hence determine the value of ln 2.7. Also find the error.
x 2 2.5 3.0
y = ln x 0-69315 0.91629 1.09861
(c) What are the sources of errors in numerical data and processing ? How does error measure accuracy ?
using Gauss-Legendre three-point formula.
(b) Solve the initial value problem u' = - 2t u2 with u(0) = 1 and h 0.2 on the interval [0, 1]. Use the fourth order classical Runge-Kutta method.
(c) Evaluate
using Composite Simpson's rule with 5 points.
4. (a) Calculate the correlation coefficient for the following heights (in inches) of fathers (X)
and their sons (Y) :
X: 65 66 67 67 68 69 70 72
Y: 67 68 65 68 72 72 69 71
Obtain the equations of lines of regression. Also estimate the value of X for Y = 70.
(b) A manufacturer of cotter pins knows that 5% of his product is defective. If he sells cotter pins in boxes of 100 and guarantees that not more than 10 pins will be defective, what is the approximate
probability that a box will fail to meet the guaranteed quality ?
5. (a) What do you mean by pseudo-random number generation ? What is the practical advantage of the concept of random number generation ?
(b) For the data given in the table, compute R
(c) If a bank receives on an average 𝛌 = 6 bad cheques per day, what is the probability that it will receive 4 bad cheques on any given day, where 𝛌, denotes the average arrival rate per day ?
Notes-3
1. (a) Solve the quadratic equation 4x2 + 8x - 21 = 0 using two decimal digit arithmetic with rounding, using any one of the following methods :
(i) Regula-Falsi
(ii) Secant
(iii) Bisection
(b) Round off the number 4.5126 to 4 significant figures and find the relative percentage error.
(c) Obtain the positive root of the equation x2 - 1 = 0 by Newton-Raphson method,correct to two decimal places.
(d) Explain the two pitfalls in the Gauss Elimination Method.
(e) Solve the following system of linear equations using LU decomposition method :
6x1 - 2x2 = 14
9x1 - x2 + x3 = 21
3x1 + 7x2 + 5x3 = 9.
(f) What is the lowest degree polynomial which satisfies the following set of values,using forward difference polynomial ? Also find the polynomial.
x 0 1 2 3 4 5 6 7
f(x) 0 7 26 63 124 215 342 511
(g) Calculate the value of the integral
Assume h = 0.2.Compare the numerical solution with the exact solution.
2. (a) What do you mean by the terms "Accuracy" and "Precision" ? How are they related to significant digits ?
(b) Show that the equation x3 - 6x - 1 = 0 has a root in the interval ] - 1, 0[. Obtain this root using Successive Iteration or Bisection method.
(c) Find the Lagrange interpolating polynomial of degree 2 approximating the function y = ln x defined by the following values mentioned in the table. Hence determine the value of In 2.7.
x 2 2.5 3.0
y = In x 0.69315 0.91629 1.09861
3. (a) Solve the initial value problem u' = - 2tu2, with u(0) = 1, h = 0.2 on the interval [0, 1]. Use the fourth order classical Runge-Kutta method.
(b) Solve the following system of equations using Gauss elimination with partial pivoting :
2x + y + z = 10
3x + 2y + 3z = 18
x + 4y + 9z = 16
(c) What is the utility of residual plots ? What is the disadvantage of residual plots ?
4. (a) If a bank receives on an average 𝛌 = 6 bad cheques per day, what is the probability that it will receive 4 bad cheques on any given day, where 𝛌 denotes the average arrival rate per day ?
(b) A hosiery mill wants to estimate how its monthly costs are related to its monthly output rate. For that the firm collects data regarding its cost and output for a sample of nine months as given by the following table :
Output (tons) Production Cost (thousands of dollars)
1 2
2 3
4 4
8 7
6 6
5 5
8 8
9 8
7 6
(i) Draw the scatter diagram for the data.
(ii) Find the regression equation when the monthly output is the dependent variable (x) and monthly cost is the independent variable (y).
(iii) Use this regression line to predict the firm's monthly cost if they decide to produce 4 tons per month.
(c) An individual's IQ score has a N(100, 15) distribution. Find the probability that an individual's IQ score is between 91 and 121.
two-point Gauss-Legendre formulae.Compare with the exact solution and the exact value is I = 1.
(b) The following values of the function f(x) for the values of x are f(1) = 4, f(2) = 5, f(7) = 5 and f( 8) = 4.Find the value of f(6) and also the value of x in the interval [1, 8] for which f(x) is maximum or minimum.
(c) Round off the number 4.5126 to four significant figures and find the relative percentage error.
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