Previous Year Questions & Detailed Solutions

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1 Previous Year Questions & Detailed Solutions. The rate of convergence in the Gauss-Seidal method is as fast as in Gauss Jacobi smethod ) thrice ) half-times ) twice 4) three by two times. In application of Simpsons rd rule, theinterval h for closer approximation should be ) small ) odd and large ) large 4) even and small. Which of the following is a step by step method? ) Taylor s method ) Picards method ) Adams Bashforth method 4) Eulers method 4. The interpolation polynomial from the data x f(x) ) x x + 5 ) x + x +5 ) x x + 5 4) x 5x +5 a+h a 5. The value of the integration f x is approximately equal to ) h [f(a)+f(a+h)+f(a+h)] ) h [f(a)+4f(a+h)+f(a+h)] ) h [f(a)+f(a+h)+f(a+h)] 4) h [f(a)+4f(a+h)+f(a+h)] 6. The finite difference approximation for d y at x = x is ) h (y(x -h)-y(x )+y(x +h)] ) h [y(x -h)+y(x )+y(x +h)] ) h [y(x -h)-y(x )+y(x +h)] 4) h [y(x -h)+y(x )+y(x +h) 7. For the following data X 4 6 Y - 7 the straight line y=mx+c by the method of least square is ) y=-x-l ) y=x-l ) y=l-x 4) y=x-l 8. The velocity v (km/min) of a train which starts from rest, is given at fixed intervals of time t(min) as follows: t v The approximate distance covered by Simpson's rule is ) 6. ) 9. ). 4) Find the cubic polynomial by Newton s forward difference which takes the following: X f(x) then f(4) is ) 4 ) 4 ) 9 4) 4. The first derivative dy at x= for the given data X f(x) 5

2 ) ) - ) - 4). Error in Simpson s rule is of the order ) h ) h ) h 4 4) h. If α, β and γ are the roots of x + px + qx+ =, then the equation whose roots are,, is α β γ ) x + qx px l= ) x +qx -px+= ) x -qx +px-l = 4) x - qx +px+l=. In Jacobi s iteration method to get the (r+ ) th iterates, we use the ) values of the r th iterates ) values of the (r -) th iterates ) values of the (r+ ) th iterates 4) latest available values 4. If h is the length of the intervals, then the error in the trapezoidal rule is of order ) h ) h ) h 4) h 4 5. If E is the translation operator, then the central difference operator δ is ) E / + E -/ ) E / - E -/ ) (E/ + E -/ ) 4) - (E/ + E -/ ) 6. If Δf(x) = f(x + h)-f(x), E f(x) =f(x + h) and δf(x)= x + h f x h then Δ is ) δ ) δ ) δ / 4 δ / + δ- / 7. Expression for backward interpolation d y x= using Newton s ) h f n + f n + 4 f n + ) h Δ Δ 4 f n ) h Δ + Δ + Δ 4 + f n 4) h Δ Δ Δ4 f n 8. By Taylor s series method solution of y'=x + y ; y() = is ) y=l+x x ) y = -x+x ) y=x+ x +x 4) y=l+x+x + 4 x 9. Error in the Trapezoidal rule is of the order ) -h ) h ) h 4 4) h. A river is 8 ft wide, the depth d in feet at a distance x ft. from one bank is given by the following table: x y Approximately the area of cross section by Simpson s / rule is ) 7 sq.ft. ) 68 sq.ft. ) 7 sq.ft. 4) 6 sq.ft.. The first derivative dy at x= 4 for the data. X 4 f(x) ) -4 ) 4 ) 6 4) 6. If x and x are two points, f(x ) and f(x ) are of opposite signs, then the root x by regular falsi method is equal to ) x f x x f x f x f x ) x f x x f x f x f x ) x f x x f x f x f x 4) x f x x f x f x f x. The Newton-Raphson iteration scheme to find an approximate real root of the equationf(x) = is ) + = + f ) + = - f

3 ) + = + f 4) + = - f 4. Power method is used to find ) all eigen values only ) all eigen vectors only ) all eigen values and eigen vectors 4) numerically largest eigen value and the corresponding eigen vector 5. Lagrangean interpolation polynomial which passes through the points (, ), (, ), (, ) is ) x + l ) x - ) x +l 4) x - 6. d y is approximately equal to ) y i+ y i h ) y i++ y i h ) y i y i +y i+ 4) y i +y i +y i+ h h 7. The scheme a+h a f(x) = h [f(a) + 4f(a + h)+f(a + h) is called ) Trapezoidal rule ) Simpson s / rule ) Simpson s /8 rule 4) Weddle s rule 8. The Lagrange s interpolation polynomial curve passing through the points (, ) (, ) and (, ) is ) x -x+l ) -4x + x + ) x x + 4) x x + 9. Which of the following formulas is a particular case of Runge-Kutta formula of second order? ) Taylar series formula ) Picard s formula ) Euler s modified formula 4) Milne s predictor formula. What is the highest degree of a polynomial whose definite integration using Simpson s rule gives the exact value? ) ) ) 4) 4. The global error arising in using Runge - Kutta method of fourth order to solve an ordinary differential equation is of ) (h ) ) (h ) ) (h 4 ) 4) (h 5 ). Simpson s one-thid rule integration is exact for all polynomial of degree not exceeding ) ) ) 4 4) 5. With usual notations, the value of (+Δ) (- ) is ) ) ) 4) 4 4. The modified Newton-Raphson formula of finding the root of the equation f(x)= is ) + = f ) + = ) + = f 4) + = + 5. The value of f fn f f f f +x Trapezoidal rule with h =. is ).687 ).877 ).77 4) Simpson s one-third rule of integration is exact for all polynomial of degree not exceeding ) ) ) 4 4) 5

4 7. Which of the following formula is a particular case of Runge - Kutta formula of the second order? ) Taylor series formula ) Picard s formula ) Euler s modified formula 4) Milne s predictor formula 8. The Newton Raphson method of finding the root of the equation f(x)= is by using the formula ) + = f ) + = f ) + = + f 4) + = + f 9. The value of +x by Trapezoidal Ruletaking n= is ).645 ).64 ).58 4).6 4. Trapezoidal rule for the evaluation of the integral b a f x requires the Interval to be divided into ) only an even number of intervals ) only an odd number of intervals ) any number of intervals 4) a minimum of ten intervals.6 4. If f(x) - k[f() + 4f(.)+f(.6)] then thevalue of k when Simpson s rule is applied is ). ). ). 4).4 4. The Newton - Raphson formula to find an approximate value of a is ) + = + a ) + = a 4 ) + = + a 4) + = a 4. The iteration formula given by Newton- Raphson method to find to root of the equation x sin x+cos x= is )+ = sin +cos cos )+ = cos sin +cos )+ = sin +cos cos +sin 4)+ = cos + sin 44. The interpolating polynomial for the data x - f -5 ) + x x ) + x + x ) + x x 4) x + x + x 45. Given that f(l) = l, f(.5) =.875, f() = 7, f(.5) = 4.5 and f() = 5, the value of f(x) obtained by Simpson s rule is ) 7.5 ) 8.5 ) 8. 4) A root of x - 4x -9 = using bisection method in two stages is ).75 ).56 ) ) The lagrange s interpolation polynomial curve passing through the points (, ) (, ) and (, ) is ) x - x + ) -4x + x + ) x -x + 4) x x+ 48. What is the highest degree polynomial whose definite integration using Simpson s rule gives the exact value? ) ) ) 4) 4

5 49. Milne s predictor value of y(, 4) given thaty'= y- x y, y() = y'(.)=.9. y'(.) =.8, y'(.) =.7 is ) 5 ) 6 5 ) 5 4) The recurrence formula for finding the reciprocal of the n th root of M, by Newton's method is ) x k+ = n x k n +M nx k n ) x k+ = k n+ Mx k n ) x k+ = k n + Mx k 4) x k+ = n n x k + Mx k 5. The value of ( + x ) by simpson s one-third rule with two strips is ) 6 ) 4 ) 47 4) Which of the following formulas is a particular case of Runge - Kutta formula of second order? ) Taylor series formula ) Picard's formula ) Euler s modified formula 4) Milne's predictor formula DETAILED SOLUTIONS. () The rate of convergence of Gauss - Seidel method is roughly twice that of Gauss - Jacobi.. (4) In application of Simpsons rd rule, the intervalh for closer approximation should be even and small.. (4) 5 Euler s method is a step by step method. 4. () Consider f(x) = x -x+5 then f() = -()+5 = 5 f(l) = l -(l)+5 = 4 f() = -()+5 = 8 f(x) = x -x+5 is the requiredpolynomial. 5. () 6. () a+h a f(x) = h [f(a) + 4f(a + h) + f(a + h)] d y (x = x ) = f f +f h h [y(x h) -y (x ) + y (x + h)] 7. (4) x 4 6 y -l 7 Consider y = x-l x = y = ()- = - x = y=()-l = x = 4 y=(4)-l = 7 x = 6 y= (6)- = y=x-l is the required straight line. 8. (*) y = h [(y +y n ) + 4 (y +y +...+y n )+ (y +y 4 + y n- )] = [(+)+4(8+9++5)+(5+ + +)] = 9 9. () f(x) = y + pδf x + p p Δ f(x ) + p p (p )! where p = x x h Δf x = Δ f(x ) = - Δ f(x )!

6 Δ f(x ) = p = x f(x) = + x + x(x ) (-)! x x x () +! = + x x (x - )+x(x-) (x-) f(4) = = = 4. () x f(x) Δy Δ y Δ y - Δy Δ y Δ y α, β, γ are root ofx + px + qx + = x + px +qx+,, are roots of α β γ + p + q x x x + = x + qx + px + =,, are roots of α β γ x - qx +px-l =. () For the system of equations a x + b y + c z = d a x + b y+c z = d a x +b y+c z = d Then the (r+ ) th iteration in Jacobi s method is x r+ = a d b y r c z r y r+ = b z r+ = c d a x r c z r d a x r b y r 5 First derivative by Newton s forward difference formula y = h Δy Δ y + Δ y = + () = --=-. () Error in Simpson s rule is of order h4. () Results: Let a, a,..., a n be the roots of the equation +p - + p p n = than the roots of -p - +p (-) n P n = are a, -a,... -a n Given 6 To find (r+) th iteration we use r iterations. 4. () Error in the trapezoidal rule is of the order h. 5. () δ f(x) = (E / E -/ ) f(x) 6. () δ = E ½ - E -/ e Δ = E ; Δ = E - where E = f(x+h) E - = f(x-h) E / = f(x+ h/); E -/ = f(x - h/) Δ = (E - ) (- E - ) = E + E - = E + E () dδ = (E / E -/ ) = E-E / E -/ + E - = E - + E -...()

7 () = () Δ = δ 7. (4) Standard formula d y x= h Δ Δ + Δ4 f n 8. (4) Taylor s series y = y + h! y + h! y + h! y +... Given y() = y = ; x = y' = x + y (Given) y '= + = y'' = x+ yy' y " = ()+.. = y'" = + (yy'' + (y')) = + (. + ) = +( + l) = 8 putting h - x then y = + x! + x! + x! 8 = l + x + x + 4 x 9. (4)Error in thetrapezoidal rule is of the order h.. () Simpson s rule x = +nh f x x = h y + 4 y + y + + y + y 4 + +yn = h [y +y n +4(sum of odd ordinates) +sum of even ordinates)] Cross Section Area 7 = [(++4( )+(7++4)] = [+44+66] = 7 sq.ft.. () x y=f(x) First difference Second difference - Third difference - 4 Fourth difference y = ; y = y = 4 y = = Newtons formula for Backward difference f'(x) = h [ y + y + y...] = + = 4. () By regular - falsi method the root is given by x f x x f(x ) f(x ) f(x ). () The Newton - Rapson Iteration scheme to find an approximate real root of the equation f(x)= is

8 + = f( ) 4. (4) Power method is used to determine numerically largest eigen value and the corresponding eigen vector of a matrix A. 5. () Let x = ; y = x = ; y = x = ; y = x x y = x x y x x (x x ) + x x (x x ) y x x (x x ) + = + 6. () x x (x x ) y x x (x x ) x (x )_ x (x ) + x 4x + x (x ) = x 4x+ x +9x+5x 5x = x + = x + + x x + 5x 5x d y = h y i + r + y i + = h y approximately = y h i+ y i = y h i+ y i+ (y i+ y i ) = y h i+ y i+ + y i which is also equal to d y = y h i+ y i + yi 7. () 8. () a+h a is called f(x) = h [f(a) + 4f(a + h) + f(a + h)] Simpson s rule. 8 Since only values of x are given. Assume y = f(x) = a + a x+a x(x-l) put x=, y= = a +a ()+a () a = put x -, y= = a + a +a () = a + a = l+ a a=- put x=, y= = a + a +6a f(x) = l x+x(x-l) = l x+ x -x = x -x+ 9. () Paricular case of R.K. formula of second order is Euler s modified formula.. () Simpson s one-third rule is x +nh x y = h y + y n + 4 y + y + y n + y + y 4 + y n In this case we reglect all differences above the second. So y will be a polynomial of second degree only, i.e., y = ax +bx+c. () The error in R.K. method of fourth order to solve an ordinary differential equation is of (h 4 ).. () Required degree =. () Formula ( + Δ) (- ) =

9 4. () Newton Rapson formula 5. (4) X + = f y = + x y..965 y.4.86 y.6.75 y y 4.5 y 5 By Trapezoidal Rule, + x = h [(y + y 5 ) + (y + y + y + y 4 )] =. [.5+(.687)] = () Simpsons rule of Integration is exact for allpolynomial of degree not exceeding. 7. () Particular case of Runge Kutta formula of second order is Euler s modified formula. 8. () Newton Raphson formula of finding the root of f(x)= is + = - f 9. () 9 h = = x = y = +x + x y - =.5 + x y -. + By Trapezoidal rule = +x = h y + y + y [ + (.5)+.] = () Any number of intervals. 4. () h = interval length =. By Simpsons rule k = h =. =. 4. () Let x = a x -a = Let f(x) = x -a thenf'(x) = x + = - f = - a = +a = + a 4. () Newton Raphson Method + = - f f(x) = x sin x+ cos x

10 f'(x) = x cosx+sin x-sinx = x cos x By Newton Raphson method + = - sin +cos cos 44. () Since only 4 values of x are given assume f(x) to be a polynomial of degree Let f(x) = a + a x + a x (x-) + a x (x-) (x-) put x = -, f(x) = = a a + a 6a... () x = then f(x) = = a + a +a +a a =... () put x = then f(x) = l = a + a a i =... () put x = then -5 = a + a + a l++ a = -5 a = -6 a = -... (4) put the values of a, a, a in () l 6-6a = 6a = -6 a = - f(x) = + x + (-) x (x-) + x(x-) (x-) = - x + x x (x - x +) = - x + x - x + x - x = +x-x f(x) = +x-x 45. () x = ; y = x =.5 ; y =.875 x = ; y = 7 x =.4 ; y = 4.5 x 4 = ; y 4 = 5 Simpson s formula = h [y +y n +4(y +y +...)+(y +y )] =.5 [+5+4( )+(7)] = () If a function f(x) is continuous between a and b and f(a) and f(b) are of opposite sign then there exists atleast one root between a and b. f(x) = x - 4x -9 f() = = -9 = -ve f() = 7-9 = 6 = +ve Hence, the root lies between and By first approximation x = + =.5 Again f(.5) = = -ve Hence, the root lies between.5 and Second approximation.5 + = () Since only values of x are given assume f(x) be a polynomial of degree Let f(x) = a + a x+ a x (x-l) when x= f(x) = = a + a +a a = when x=l ; f(x) = = a + a a = - when x=, f(x)= = a + a +6a a = Nowf(x) = l x+x(x-l) = l x+ x -x = x -x+ Required polynomial

11 = x -x+ 48. () Required degree = 49. () y n+,p = y n- + 4h put n = y 4,p = y + 4h [y - y +y ] y n y n + y n = + 4 (.) ( ) = +.4 ( ) = +.4 (.4) = +.. = () n x = M = M Let f(x) = M f (x) = n- By Newtons formula = x k x k n M nx k n x k+ = x k f x k f x k = nx k x k n xk n +M nx k n = x k n n +M nx k n 5. (4) h=/ X y = +x 4 5 +x = = 6 = h y + y + 4y +5+ = 6 47 = () Euler s modified formula is the particular case of Runge - Kutta formula. English Examsdaily Join Us on FB : Tamil Examsdaily Tamil Whatsapp Group English - Click Here Tamil - Click Here