Numerical Analysis & Computer Programming

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1 Numerical Analysis & Computer Programming Previous year Questions from 07 to 99 Ramanasri Institute W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : / 6 6

2 07. Eplain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matri [0 Marks] 6 8. Write the Boolean epression z( y z)( y z) in the simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given epression and for its simplest form. [0 Marks]. For given equidistant values u, u0, u and u a values is interpolated by Lagrange s formula. Show that it may y( y ) ( ) be written in the form u yu0 u u0, where y. [5 Marks]!! b h 4. Derive the formula yd [( y0 yn ) ( y y y a 4 y5... yn ) ( y y6 yn )]. Is there any 8 restriction on n? State that condition. What is the error bounded in the case of Simpson s 8 rule? [0 Marks] 5. Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method. [5 Marks] (i) 4096 (ii) (iii) Convert the following decimal numbers to univalent binary and headecimal numbers: (0 marks) 7. Let f( ) e cos for [0,]. Estimate the value of f(0.5) Using Lagrange interpolating polynomial of degree over the nodes 0, 0., 0.6 and.also compute the error bound over the interval[0,] and the actual error E (0.5) (0 marks) 8. For an integral f( ) d show that the two point Gauss quadrature rule is given by 4 f( ) d f f using this rule estimate e d (5 marks) 9. Let A, B, C be Boolean variable denote complement A, A B of is an epression for A OR B and BAis. an epression for AANDB.Then simplify the following epression and draw a block diagram of the simplified epression using AND andor gates. A.( A B, C).( A B C).( A B C).( A B C). (5 marks) Find the principal (or canonical) disjunctive normal form in three variables p, q, r for the Boolean epression( p q) r ( p q) r. Is the given Boolean epression a contradiction or a tautology?. Find the Lagrange interpolating polynomial that fits the following data: (0 Marks) Reputed Institute for IAS, IFoS Eams Page

3 4 f ( ) 69 Find f (.5) dy. Solve the initial value problem ( y ), d y () in the interval[,.4]using the Runge- Kutta fourth-order method with step size h 0. (5 Marks). Find the solution of the system using Gauss-Seidel method (make four iterations) (5 Marks) Apply Newton-Raphson method to determine a root of the equation cos e 0 correct up to four decimal places. (0 Marks) 5. Use five subintervals to integrate d using trapezoidal rule. 0 (0 Marks) 6. Use only AND and OR logic gates to construct a logic circuit for the Boolean epression z y uv (0 Marks) 7. Solve the system of equations 7 using Gauss-Seidel iteration method (perform three iterations) (5 Marks) 8. dy Use Runge-Kutta formula of fourth order to find the value of y at 0.8, where d y, y(0.4) 0.4. Take the step length h Draw a flowchart for Simpson s one-third rule. (5 Marks) 0. For any Boolean variables and y, show that y. (5 Marks) 0. In an eamination, the number of students who obtained marks between certain limits were given in the following table: Marks No. of students Reputed Institute for IAS, IFoS Eams Page

4 Using Newton forward interpolation formula, find the number of students whose marks lie between 45 and 50. (0 Marks). Develop an algorithm for Newton-Raphson method to solve f ( ) 0 starting with initial iterate, 0 n be the number of iterations allowed, epsilon be the prescribed relative error and delta be the prescribed lower bound for f'( ). Use Euler s method with step size h 0.5 to compute the approimate value of y (0.6), correct up to five decimal places from the initial value problem. y' ( y ), y(0) (5 Marks) 4. The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in km/hour. t v Estimate approimately the total distance run in 0 minutes by using composite Simpson s rule. 0 (5 Marks) 5. Use Newton-Raphson method to find the real root of the equation cos correct to four decimal places ( Marks) dy 6. Provide a computer algorithm to solve an ordinary differential equation f(, y) d in the interval [ ab, ] for n number of discrete points, where the initial value is ya ( ), using Euler s method. (5 Marks) 7. Solve the following system of simultaneous equations, using Gauss-Seidel iterative method : 0y z 8 0 y z 7 y 0z 5 8. Find dy at 0. d from the following data: : y : In a certain eamination, a candidate has to appear for one major & two major subjects.the rules for declaration of results are marks for major are denoted by M and for minors by M and M.If the candidate obtains 75% and above marks in each of the three subjects, the candidate is declared to have passed the eamination in first class with distinction. If the candidate obtains 60% and above marks in each of the three subjects, the candidate is declared to have passed the eamination in first class. If the candidate obtains 50% or above in major, 40% or above in each of the two minors and an average of 50% or above in all the three subjects put together, the candidate is declared to have passed the eamination in second class. All those candidates, who have obtained 50% and above in major and 40% or above in minor, are declared to have passed the eamination. If the candidate obtains less than 50% in major or less than 40% in anyone of the two minors, the candidate is declared to have failed in the eaminations. Draw a flow chart to declare the results for the above. Reputed Institute for IAS, IFoS Eams Page 4

5 d Calculate (up to places of decimal) by dividing the range into 8 equal parts by Simpson s rule. ( Marks). (i) Compute (05) 0 to the base 8. (ii) Let A be an arbitrary but fied Boolean algebra with operations and, ' and the zero and the unit element denoted by 0 and respectively. Let, y, z... be elements of A. If, y A be such that y 0 and y then prove that y '... ( Marks). A solid of revolution is formed by rotating about the - ais, the area between the - ais, the line 0 and and a curve through the points with the following co-ordinates: y Find the volume of the solid.. Find the logic circuit that represents the following Boolean function. Find also an equivalent simpler circuit: f(, y, z) y z Draw a flow chart for Lagrange s interpolation formula Find the positive root of the equation 0e 0 correct up to 6 decimal places by using Newton-Raphson method. Carry out computations only for three iterations. ( Marks) 6. (i) Suppose a computer spends 60 per cent of its time handling a particular type of computation when running a given program and its manufacturers make a change that improves its performance on that type of computation by a factor of 0. If the program takes 00 sec to eecute, what will its eecution time be after the change? (ii) If A B AB' A' B, find the value of y z. (6+6= Marks) 7. Given the system of equations y 4y z y 6z Aw 4 4z Bw C Reputed Institute for IAS, IFoS Eams Page 5

6 State the solvability and uniqueness conditions for the system. Give the solution when it eists. 8. Find the value of the integral 5 log 0 d by using Simpson s rd rule correct up to 4 decimal places. Take 8 subintervals in your computation. 9. (i) Find the headecimal equivalent of the decimal number (5876) 0 (ii) For the given set of data points (, f( ),(, f( ),...( n, f( n) write an algorithm to find the value of f ( ) by using Lagrange s interpolation formula (iii) Using Boolean algebra, simplify the following epressions (a) a a' b a' b' c a' b' c' d... (b) ' y' z yz z where ' represents the complement of (5+0+5=5 Marks) (i) The equation a b 0 has two real roots and. Show that the iterative method ( ak b) given by : k, k 0,,... is convergent near, if k (ii) Find the values of two valued Boolean variables A, B, C, D by solving the following simultaneous equations: A AB 0 AB AC AB AC CD CD where represents the complement of (6+6= Marks) 4. (i) Realize the following epressions by using NAND gates only : g ( a b c) d( a e) f where represents the complement of (ii) Find the decimal equivalent of(57.) 8 (6+6= Marks) 4. Develop an algorithm for Regula-Falsi method to find a root of f ( ) 0 starting with two initial iterates 0 and sign f( 0) sign f( ). Take n as the maimum number of iterations allowed and epsilon be the prescribed error. (0 Marks) 4. Using Lagrange interpolation formula, calculate the value of f() from the following table of values of and f ( ) : to the root such that (5 Marks) f ( ) Find the value of y(.) using Runge-Kutta fourth order method with step sizeh 0. from the initial value problem: y' y, y() (5 Marks) Find the smallest positive root of equation e cos 0 using Regula-Falsi method. Do three iterations. ( Marks) 46. State the principle of duality Reputed Institute for IAS, IFoS Eams Page 6

7 (i) in Boolean algebra and give the dual of the Boolean epressionsx Y. X. Z. Y Z and XX 0 (ii) Represent A B CA B CA B C in NOR to NOR logic network. 47. (i) The following values of the function f( ) sin cos are given: f ( ) (6+6= Marks) Construct the quadratic interpolating polynomial that fits the data. Hence calculate f. Compare with eact value. (ii) Apply Gauss-Seidel method to calculate, y, z from the system: y 6z 4 6 y z. 6y z with initial values4.67, 7.6, Carry out computations for two iterations (5+5=0 Marks) 48. Draw a flow chart for solving equation F ( ) 0 correct to five decimal places by Newton-Raphson method (0 Marks) Use the method of false position to find a real root of lying between and and correct to places of decimals. ( Marks) 50. Convert: (i) given to be in the decimal system into one in base 6. (ii) (0.0) into a number in the decimal system. (6+6= Marks) 5. (i) Find from the following table, the area bounded by the - ais and the curve y f( ) between 5.4 and 5.40 using the trapezoidal rule: f ( ) (5 Marks) (ii) Apply the second order Runge-Kutta method to find an approimate value of y at Evaluate takingh 0., given that y satisfies the differential equation and the initial condition y' y, y(0) (5 Marks) I e d by the Simpson s rule b f( ) d f( 0) 4 f( ) f( ) 4 f( )... f( n) 4 f( n ) f( n) a with n 0, 0., 0, 0.,...,.0 ( Marks) (i) Given the number in decimal system. Write its binary equivalent. Reputed Institute for IAS, IFoS Eams Page 7

8 (ii) Given the number 898 in decimal system. Write its equivalent in system base 8. (6+6= Marks) 54. If Q is a polynomial with simple roots,,... n and if P is a polynomial of degree n, show that n P ( ) P( k ). Hence prove that there eists a unique polynomial of degree with given Q( ) Q'( )( ) k k k values c k at the point k, k,,... n.. (0 Marks) 55. Draw a flowchart and algorithm for solving the following system of linear equations in unknowns, & : C X D with,, C c X D d (0 Marks) ij i, j j j i i Use appropriate quadrature formulae out of the Trapezoidal and Simpson s rules to numerically d integrate 0 with h 0.. Hence obtain an approimate value of. Justify the use of particular quadrature formula. ( Marks) 57. Find the headecimal equivalent of (489) 0 and decimal equivalent of(0.0) ( Marks) 58. Find the unique polynomial P ( ) of degree or less such that P(), P() 7, P(4) 64. Using the Lagrange s interpolation formula and the Newton s divided difference formula, evaluate P (.5) (0 Marks) 59. Draw a flow chart and also write algorithm to find one real root of the non linear equation ( ) by the fied point iteration method. Illustrate it to find one real root, correct up to four places of decimals, of 5 0. (0 Marks) The velocity of a particle at distance from a pint on it s path is given by the following table: S(meters) V(m/sec) Estimate the time taken to travel the first 60 meters using Simpson s rd rule. Compare the result with Simpson s th rule. 8 ( Marks) 6. (i) If( AB, CD) 6 ( ) ( y) 8 ( z) 0 then find, y & z (ii) In a 4-bit representation, what is the value of in signed integer form, unsigned integer form, signed s complement form and signed s complement form? (6+6= Marks) 6. How many positive and negative roots of the equation e 5sin 0 eist? Find the smallest positive root correct to decimals, using Newton-Raphson method. (0 Marks) 6. Using Gauss-Siedel iterative method, find the solution of the following system: 4 y 8z 6 5 y z 6 0y z up to three iterations. (5 Marks) Reputed Institute for IAS, IFoS Eams Page 8

9 64. Evaluate 00 e d by employing three points Gaussian quadrature formula, finding the required 0 weights and residues. Use five decimal places for computation. ( Marks) 65. (i) Convert the following binary number into octal and hea decimal system: (ii) Find the multiplication of the following binary numbers:00. and 0.(6+6= Marks) 66. Find the positive root of the equation e using Newton-Raphson method correct to four decimal places. Also show that the following scheme has error of second order: a n n n (0 Marks) 67. Draw a flow chart and algorithm for Simpson s b rd rule for integration d correct to 0 6 a (0 Marks) Find a real root of the equation f( ) 5 0 by the method of false position. ( Marks) into its binary equivalent. 69. (i) Convert 0 (ii) Multiply the binary numbers.0 and 0. and check with its decimal equivalent 70. (i) Find the cubic polynomial which takes the following values: y(0), y() 0, y() & y() 0. Hence, or otherwise, obtain y (4) (4+8= Marks) (ii) Given: dy ywhere y(0), using the Runge-Kutta fourth order method, find (0.) d y and y (0.). Compare the approimate solution with its eact solution. e.057, e.4. (0+0=0 Marks) 7. Draw a flow chart to eamine whether a given number is a prime. (0 Marks) Show that the truncation error associated with linear interpolation of f ( ), using ordinates at 0 and with 0 is not larger in magnitude than ( 0) 8 M where M ma f ''( ) in 0. Hence show that if interpolation of f ( ) in 0 cannot eceed t f( ) e dt, the truncation error corresponding to linear 0 ( 0). e ( Marks) Reputed Institute for IAS, IFoS Eams Page 9

10 7. (i) Given A. B' A'. B C show that A. C' A'. C B (ii) Epress the area of the triangle having sides of lengths 6,, 6 units in binary number system. (6+6= Marks) 74. Using Gauss Seidel iterative method and the starting solution 0, determine the solution of the following system of equations in two iterations Compare the approimate solution with the eact solution (0 Marks) 75. Find the values of the two-valued variables A, B, C & D by solving the set of simultaneous equations A' A. B 0 A. B A. C A. B A. C ' C. D C '. D 000 (5 Marks) 76. (i) Using Newton-Raphson method, show that the iteration formula for finding the reciprocal of the p th i p Ni root of N is i p (ii) Prove De Morgan s Theorem( p q)' p'. q' (6+6= Marks) 77. (i) d Evaluate, by subdividing the interval (0, ) into 6 equal parts and using Simpson s onethird rule. Hence find the value of and actual error, correct to five places of decimals 0 (ii) Solve the following system of linear equations, using Gauss-elimination method: (5+5=0 Marks) 999 b 78. Obtain the Simpson s rule for the integral I f( ) d and show that this rule is eact for polynomials of a degree n. In general show that the error of approimation for Simpson s rule is given by 5 ( b a) iv R f ( ), 0,. 880 d Apply this rule to the integral and show that R du 79. Using fourth order classical Runge-Kutta method for the initial value problem tu, u(0), dt with h 0. on the interval [0, ], calculate u (0.4) correct to si places of decimal. Reputed Institute for IAS, IFoS Eams Page 0

11 d Evaluate by Simpson s rule with 4 strips. Determine the error by direct integration. 8. By the fourth order Runge-Kutta method. tabulate the solution of the differential equation dy y, y (0) 0 in [0, 0.4] with step length 0. correct to five places of decimals d 0y 4 8. Use Regula-Falsi method to show that the real root of log0. 0 lies between and Apply that fourth order Runge-Kutta method to find a value of y correct to four places of decimals dy at 0., when y' y, y(0) d 84. Show that the iteration formula for finding the reciprocal of N is n n Nn, n 0, Obtain the cubic spline approimation for the function given in the tabular form below: 0 and M0 0, M 0 f ( ) Describe Newton-Raphson method for finding the solutions of the equation f ( ) 0 and show that the method has a quadratic convergence. 87. The following are the measurements t made on a curve recorded by the oscillograph representing a change of current i due to a change in the conditions of an electric current: t i Applying an appropriate formula interpolate for the value of i when t.6 dy dz 88. Solve the system of differential equations z, y for 0. given that y 0 d d and z when 0, using Runge-Kutta method of order four Find the positive root of log cos nearest to five places of decimal by Newton-Raphson method. 90. Find the value of.4.6 e f( ) d from the following data using Simpson s rd rule for the interval 8 (.6,.) and th rule for (.,.4) : 8 Reputed Institute for IAS, IFoS Eams Page

12 f ( ) f ( ) Find the positive root of the equation e e correct to five decimal places Fit the following four points by the cubic splines. i y i i Use the end conditions Use the end conditions y" 0 y" 0 Hence compute (i) y (.5) (ii) y '() 9. Find the derivative of f ( ) at 0.4 from the following table: y f( ) Find correct to decimal places the two positive roots of e Evaluate approimately 4 d Simpson s rule by taking seven equidistant ordinates. Compare it with the value obtained by using the trapezoidal rule and with eact value. 96. Solve dy y for.4 by Runge-Kutta method, initially, d y (Take h 0. ) Compute to 4 decimal placed by using Newton-Raphson method, the real root of 4sin Solve by Runge-Kutta method dy y d with the initial conditions 0 0, y0 correct up to 4 decimal places, by evaluating up to second increment of y (Take h 0. ) 99. Fit the natural cubic spline for the data. Reputed Institute for IAS, IFoS Eams Page

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