NORTH MAHARASHTRA UNIVERSITY, JALGAON Question Bank Bank/S.Y.B.Sc...2016-09-22NORTH MAHARASHTRA...

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  • NORTH MAHARASHTRA UNIVERSITY,

    JALGAON

    Question Bank

    New syllabus w.e.f. June 2008

    Class : S.Y. B. Sc. Subject : Mathematics

    Paper : MTH 222 (B) (Numerical Analysis)

    Prepared By :

    1) Prof. R.P Chopara Haed, Department of Mathematics,

    Arts, Science and Commerce College Chalisgaon.

    2) Prof J . G. Chavan Department of Mathematics, Rashtriya College Chalisgaon.

  • 1

    Question Bank

    Paper : MTH 222 (B)

    Numerical Analysis

    Unit I

    1 : Questions of 2 marks 1) What is meant by Inherent error?

    2) Define Rounding error.

    3) Define Truncation error.

    4) Explain : Absolute error and relative error.

    5) What is meant by Percentage error?

    6) State with usual notation the Newton Raphson formula.

    7) In the method of false position, state the formula for the first

    approximation of the root of given equation, where symbol have

    their usual meaning.

    8) Find the root of the equation x3 x 1 = 0 lying between 1 and 2

    by Bisection method up to first iteration.

    9) Show that a real root of the equation x3 4x 9 = 0 lies between 2

    and 3 by Bisection method.

    10) Using Bisection method, show that a real root of the equation 3x -

    xsin1+ = 0 lies between 0 and 1.

    11) Find the first approximation of x for the equation x =

    0.21sin(0.5+x) by iteration method starting with x = 0.12.

  • 2

    12) Find an iterative formula to find N where N is a positive number

    by Newton Raphson method.

    13) Using Newton Raphson method find first approximation x1 for

    finding 10 , taking x0 = 3.1.

    14) Using Newton Raphson method find first approximation x1 for

    finding 3 13 , taking x0 = 2.5.

    15) Obtain Newton Raphson formula for finding a rth root of a given

    number c.

    16) Show that a real root of a equation xlog10x 1.2 = 0 lies between 2

    and 3.

    17) What is meant by significant figure? Find the significant figures in

    0.00397.

    18) If true value of a number is 36.25and its absolute error is 0.002.

    find the relative error and percentage error.

    19) If the absolute error is 0.005 and relative error is 3.26410-6, then

    find the true value and percentage error.

    2 : Fill in the blanks/Multiple choice Questions of 1

    marks 1) If X is the true value of the quantity and X1 is the approximate

    value then the relative error is ER = - - - - and percentage error is

    EP = - - - -

    2) If X is the true value and X1 is the approximate value of the given

    quantity then its absolute error is EA = - - - - and relative is error

    ER = - - - -

    3) Every algebraic equation of the nth degree has exactly - - - -

    roots.

  • 3

    4) After rounding of the number 2.3762 to the two decimal places, we

    get the number - - - -.

    5) Rounding off the number 32.68673 to 4 significant digits, we get a

    number - - - -

    a) 32.68 b) 32.69 c) 32.67 d) 32.686

    6) In bisection method if roots lies between a and b then f(a) f(b) is - - - - a) < 0 b) = 0 c) > 0 d) none of these

    7) If percentage error of a number is 3.26410-4 then its relative

    error is - - - -

    a) 3.26410-5 b) 3.26410-6

    c) 3.26410-7 d) none of these

    8) The root of the equation x3 2x 5 = 0 lies between - - - -

    a) 0 and 1 b) 1 and 2 c) 2 and 3 d) 3 and 4

    9) In Newton Raphson method for finding the real root of equation

    f(x) = 0, the value of x is given by - - - -

    a) x0 - )0(xf'

    )0f(x b) x0 c) )0(xf'

    )0f(x d) none of these

    3 : Questions of 4 marks

    1) Explain the Bisection method for finding the real root of an equation

    f(x) = 0.

    2) Explain the method of false position for finding the real root of an

    equation f(x) = 0.

    3) Explain the iteration method for finding the real root of an equation

    f(x) = 0. Also state the required conditions.

  • 4

    4) State and prove Newton-Raphson formula for finding the real root of

    an equation f(x) = 0.

    5) Explain in brief Inherent error and Truncation error. What is meant by

    absolute, relative and percentage errors? Explain.

    6) Using the Bisection method find the real root of each of the equation :

    (i) x3 x 1 = 0. (ii) x3 + x2 + x + 7 = 0.

    (iii) x3 4x 9 = 0. (iv) x3 x 4 = 0.

    (v) x3 18 = 0. (vi) x3 x2 1 = 0.

    (vii) x3 2x 5 = 0. (viii) x3 9x + 1 = 0.

    (ix) x3 10 = 0. (x) 8x3 2x 1 = 0.

    (xi) 3x xsin1+ = 0. (xii) xlog10x = 1.2.

    (xiii) x3 5x + 1 = 0. (xiv) x3 16x2 + 3 = 0.

    (xv) x3 20x2 3x + 18 = 0. (up to three iterations).

    7) Using Newton-Raphson method, find the real root of each of the

    equations given bellow (up to three iterations) :

    (i) x2 5x + 3 = 0 (ii) x4 x 10 = 0

    (iii) x3 x 4 = 0 (iv) x3 2x 5 = 0

    (v) x5 + 5x + 1 = 0 (vi) sinx = 1 x

    (vii) tanx = 4x (viii) x4 + x2 80 = 0

    (ix) x3 3x 5 = 0 (x) xsinx + cosx = 0

    (xi) x3 + x2 + 3x + 4 = 0 (xii) x2 5x + 2 = 0

    (xiii) 3x = cosx + 1 (xiv) xlog10x 1.2 = 0

    (xv) x5 5x + 2 = 0 (xvi) x3 + 2x2 + 10x 20 = 0

    8) Using Newton-Raphson method, find the value of each of :

    (i) 10 (ii) 3 13 (iii) 17 (iv) 29 (v) 3 10

    9) Using Newton-Raphson method, find the real root of each of:

    (i) e-x sinx = 0 (ii) logx = cosx (iii) logx x + 3 = 0

  • 5

    10) Using the method of false position, obtain a real root of each of the

    equation (up to 3 iteration)

    (i) x3 + x2 + x + 7 = 0 (ii) x3 4x 9 = 0

    (iii) x3 18 = 0 (iv) x3 x2 1 = 0

    (v) x3 2x 5 = 0 (vi) x3 9x + 1 = 0

    (vii) x3 x 1 = 0 (viii) xlog10x 1.2 = 0

    (ix) cosx = 3x 1 (x) xex = 2

    (xi) x3 x 4 = 0 (xii) x3 x2 2 = 0

    (xiii) xex 3 = 0 (xiv) x2 logex 12 = 0

    11) Using the iterative method, find the real root of each of the equation

    to four significant figures (up to 3 iterations)

    (i) 2x log10x 7 = 0 (ii) e-x = 10x

    (iii) x = cosecx (iv) x = (5 x)1/3

    (v) ex = cotx (vi) 2x = cosx + 3

    (vii) x3 + x2 1 = 0 (viii) cosx = 3x 1

    (ix) sinx = 10(x 1) (x) x3 x2 x 1 = 0

    (xi) tanx = x (xii) x = 0.21sin(0.5+x)

    Unit II

    1 : Questions of 2 marks 1) Define i) forward difference operator ii) backward difference

    operator. Find tan-1x.

    2) Define shift operator E. Prove that E = 1 + .

    3) Define central difference operator and prove that = E-1/2 =

    E1/2.

    4) With usual notations prove that 2 = 41 ( 2 + 4).

  • 6

    5) Prove that u0 u1 + u2 u3 + - - - - = 21 u0 4

    1 u0 + 81 2u0 16

    1 3u0 +

    - - - - .

    6) Given u0 = 3, u1 = 12, u2 = 81, u3 = 200, u4 = 100, u5 = 8. Find 5u0.

    7) Prove that u0 + 1!xu1 +

    2!xu 22 + - - - = ex[u0 + x0 + 2!

    x 2 2u0 + - - - -]

    8) State Gausss forward central difference formula.

    9) State Gausss backward central difference formula.

    10) State Lagranges interpolation formula.

    11) Using Lagranges interpolation formula find u3 if u0 = 580, u1 = 556,

    u2 = 520, u4 = 385.

    12) Define averaging operator . Show that = 2

    EE 1/21/2 + .

    13) Show that 2232

    xx = 3.

    14) Show that = E1/2 E-1/2 .

    15) Using the method of separation of symbols prove that ux+n =

    unxC1un-1 + x+1C2 2un-2 + - - - -

    16) Given that u0 + u8 = 1.9243, u1 + u7 = 1.9590, u2 + u6 = 1.9823, u3 +

    u5 = 1.9956,. Find u4 using 8u0 = 0.

    17) Construct a forward difference table for the following values of x, y :

    x 0 5 10 15 20 25

    y = f(x) 6 10 13 17 23 21

    18) Construct a backward difference table for the following values of x, y

    x 10 20 30 40 50

    y = f(x) 45 65 80 92 100

    19) Prove that (1 + )(1 ) = 1

    20) Find

    E

    2

    (x3).

  • 7

    21) Prove that logf(x) = log

    +

    f(x)f(x)1 .

    22) Prove that u3 = u2 + u1 + 2u0 + 3u0.

    23) Find the difference table for the data given below :

    x 0 1 2 3 4

    f(x) 3 6 11 18 27

    24) Show that nyx = yx+n nC1yx+n-1 + nC2yx+n-2 + - - - - + (1)n yx.

    25) Given u0 = 1, u1 = 11, u2 = 21, u3 = 28, u4 = 29. Show that 4u0 = 0.

    26) Form the difference table for the data :

    x 1 2 3 4

    u 21 15 12 10

    27) Find dxdy at (2 , 2) of a curve passing through the points (0 , 2),

    (2 , 2), (3 , 1) using Lagranges interpolation formula.

    28) Find the value of 4 y2 given below

    x 0 1 2 3 4

    y 1 2 9 28 65

    29 Find the cubic polynomial for y(0) = 1, y(1) = 0, y(2) = 1, y(3) = 10.

    2 : Fill in the blanks/Multiple choice Questions of 1

    marks

    1) h1

    +

    +

    +

    32

    32

    = - - - - .

    2) The value off E-nf(x) = - - - -

  • 8

    3) The value of

    = - - - - -

    4) If in a data six values are given and two values are missing then

    fifth differences are - - - - and sixth differences are - - - -

    5) The value of

    E

    2

    x4 is = - - - -

    6) The value of log

    +