Tuesday 31 July 2012

BCA-3 , Numerical Methods


BCA-3 , Numerical Methods

Q1)   _______is the technique of estimating the value of a function for any intermediate value of the independent variable.
a) Interpolation              b) Extrapolation                        c) Both a and b                         d) neither a and b
Q2) what is the symbol for Central difference?
a)D                               b)                                c)d                                d) None of these
Q3) The population of town in the decimal census was as given below. Estimate the population for year 1895.
              Year: x                 1891            1901                 1911                 1921                 1931
     Population: y        46                        66                     81                     93                     101
(In thousands)                          
a) 56.85                        b) 54.85                        c)57.85                         d)62.65
Q4) What is equation for fitting a straight line?
a) Y=a+bX                    b)Y=a-bX                      c)Y=aX+b                     d)Y=aX-b
Q5) Evaluate  0ò2dx using trapezoidal rule with h=0.1?
a)1.00                           b)10                              c)2.00                           d)5
Q6) Evaluate  0òdx/(1+x) using Simpson’s three eight rule by dividing the interval [0,1] into 6 equal parts?
a)0.693195                    b)0.96358                      c)0.78623                      d)0.36552
Q7) what is relatioship between E and D?
a)E=1+D                       b) E=1-D                       c) E-1=D                       d) E+1=D
Q8) From the following table, estimate the number of student who obtained marks between 40 and 45?
            MARKS                        30-40                40-50                50-60                60-70                70-80
NO. OF STUDENTS       31                     42                     51                     35                     31
a)48                              b)31                              c)17                              d)79
Q9) In the Newton’s divided difference formula what is value of [X0,X1,X2]?
a)([X1,X2]-[X0,X1])/(X2-X0)   b) ([X1,X2]+[X0,X1])/(X2-X0)   c) ([X0,X1]-[X1,X2])/(X2-X0)     d) ([X1,X2]-[X0,X1])/(X0-X2)
Q10) what is equation to fit a parabola?
a)Y=a+bX-cX2               b) Y=a+bX+cX            c) Y=aX+b-cX2                            d) Y=a-bX2+c
Q11) In which following rule’s we can found exact accuracy answer of equation?
a)Trapezoidal rule          b)Simpson’s one third rule         c) Simpson’s three eight rule   d) None of these
Q12) Use Taylor’s series method to solve the initial problem dy/dx=x2+y2  wiyh y(0)=1 for x=0.25?
a)1.44444                      b)1.22222                      c)1.33333                      d)1.55555
Q13) Use Lagrange’s Interpolation formula to fit a polynomial for the date :
                        X:         0                      1                      3                      4         
                        Y:      -12                      0                      6                      12
            Hence estimate y at x=2.
a)6                                b)8                               c)4                                d)2
Q14) In the following which formula is used to unequal intervals?
a) Forward Differences formula                          b) Backward differences formula                                                c) Lagrange’s Interpolation formula                        d)None of these
Q15) In Newton’s divided difference formula [X, X0] equal to?
a)(Y-Y0)/(X-X0)               b) (X-X0)/(Y-Y0)              c) (X+X0)/(Y+Y0)            d) (Y+Y0)/(X+X0)
Q16) Use Taylor’s series method to find y at x=0.1 ( upto fifth derivative term). Given that dy/dx=x-ywith y(0)=1.
a)0.9138                        b)0.6935                       c)0.3695                        d)0.2595
Q17)The digit that are used to express a no. are called?
a)Significant figures      b)Exact figures                         c)Accuracy Numbers     d)None of these
Q18) find the solution for x = 0.1 by Euler’s method for the equation dy/dx = 1-y, y(0) = 0
A) 0.25                          B) 0.10                          C) 0.15                         D) 0.20
Q.19)  given dy/dx = y-x where y(0) = 2 and y (0.1) = 2.2050. so find x2,h?
A) 2.2, 0.1                     B) 0.1,2                        C) 0.1,0.1                      D) 2.2,2
Q.20) Second order of Runge-Kutta method is--?
A) Yi+1 = Y1+1/2[K1+K2]                                       B) Yi+1 = Y1-1/2[K1+K2]
C) Yi+1 = Yi+1/6[K1+2K2+2K3+ K4]                        D) Yi+1 = Yi-1/6[K1+2K2+2K3+ K4]
Q21). Use Runge-Kutta method of order two to estimate y (0.2) of the equation
dy/dx = 3x+y/2, y(0) = 1 by taking h = 0.2
A) 1.165                        B) 1.563                        C) 2.165                        D)2.563
Q.22) Find y at x = 1.02, given dy/dx = xy-1 with y(1) = 2 apply Taylor’s series method.
A) 2.02061                    B) 3.02061                    C) 4.02061                    D) 5.02061
Q23). Use Runge-Kutta method of order four to solve dy/dx = 1/x+y, y(0.4) = 1 at
x = 0.5.
A) 1.963                        B) 1.0674                      C) 2.362                        D)3.221
Q.24) If f(x) = sinx + cosx then find f’’(x) = ?
A) –[sinx - cosx]                        B) [sinx + cosx]             C) –[sinx + cosx]           D) none of these
Q25). What is formula of  Modified Euler’s method?
A) Y1(j+1) = Y0-h/2[f(x0, y0) + f(x1,Y1j)]                         B) Y1(j+1) = Y0+h/2[f(x0, y0) + f(x1,Y1j)]
C) Y1(j+1) = Y0-h/6[f(x0, y0) + f(x1,Y1j)]                         D) Y1(j+1) = Y0-h/6[f(x0, y0) + f(x1,Y1j)]
Q.26) The process of computing the value of the function outside the given range is called-------------------?
A) Interpolation              B) Extrapolation            C) both A and B            D) Neither A and B
Q27). If f(x) = x3-4x2-6x then find f’’(x) =?
A) 8x-6                          B) 6x-8                          C) 3x2-8x-6                   D) 3x2+8x+6
Q28) What is right in forward differences?
A) ∆y0 = y1-y0              B) ∆y0 = y0-y1              C) ∆y0 = y1+y0             D) none of these
Q29) What is right in central differences?
A) dy1/2 = y1+y0           B) dy1/2 = y0-y1            C) dy1/2 = y1-y0            D) none of these
Q30) If X is the pull required to lift a load Y by means of a pulley block, find a linear law of the form X = mY+c connecting X and Y, using the following data:
{X : 12  15   21    25,      Y : 50  70  100  120}
A) X = 2.2785 + 0.1879Y                                    B) X = 3.2785 + 5.1879Y
C) X = 6.366 + 8.364Y                                       D) none of these
Q31) In previous question if X and Y are taken in kilogram then compute X when
Y = 150kg.
A) 32.4556kg                 B) 56.3655kg                 C) 25.3652kg                 D)30.4653kg
Q32)How many significant digits are in 1.0000100
a)8                                b)6                               c)4                                d)3
Q33)Error which are already present in the statement of a problem before its  solution is called as?
a)Truncation error          b)Round off error          c)Inherent error              d)None of these
Q34)Absolute error/True error = ?
a)Absolute error                        b)Relative error              c)Percentage error         d)Inherent error
Q35)Find the sum of 0.123*103  and 0.456*102 and write the result in three significant digits.
a)1.069*102                                       b)1.069*102                   c)1.609*102                    d)None of these
Q36)Calculate the  value of  Ö102  -  Ö101  correct to four significant digits.
a)0.4963                        b)0.04963                      c)4.963                         d)49.63
Q37)  [ 9                       5             7]t is called as
a)Row matrix                 b)Symmetric matrix       c)Column matrix            d)None of these
Q38)     If          é2     x+yù         =                      é2      2x+y-9ù                then find out x, y and z.                                     ëx-y    z   û                                          ëx-y     6       û
a)x=2 ; y=9; z=6            b)x=2 ; y=4; z=6                        c)x=9 ; y=6; z=6                        d)None of these
Q39) If A is (m*n) matrix and B is (n*m) matrix then B*A is
a)m*m                           b)m*n                           c)n*n                             d)n*m
Q40) If  a matrix is skew symmetric matrix A=é 0                         6          -7ù                    then value of a, b, c is
                                                                   |  a             0          5 |
                                                                   ëb               c         0û
a)6,-7,5                         b)5,8,4                          c)-7,6,8                         d)6,7,-5
Section-B
Q41) Determinant of a matrix A  = é 3     -2         4 ù
                                                   |  2     3          4 |
                                                   ë 1     2          1û       
a)-15                             b)15                              c)can’t find out              d)None of these
Q42) If  A =  é1     2         3          2 ù    what is the rank of matrix
                    |  0                0         1        2  |
                   |  0     0          0          1  |   
                   ë  0     0         0           0 û   
a)2                                b)3                               c)4                                d)1
Q43) If  A =  é6              2                      2ù   then power method taking initial vector as [1,1,1]t and find very                | -2             3                     -1 |  first  iteration
                   ë2              -1                     3û        
            a) é 1  ù             b) é 1     ù                      c) é 1     ù            d)None of these
                 |   0  |                | -0.36 |              | -0.46  |
                 ë 0.4û                ë 0.55 û            ë 0.51 û
Q44)We consider the equations 
            3x+20y-z= -18 ;      2x-3y+20z= 25 ;      20x+y-2z= 17 ;
            Using Jacobi iteration method find the second iterative value
a)1.0134, -0.9954, 1.0032           b)0.85, -0.9, 1.25           c)1.0200, -0.965, 1.1515     d)None of these      
Q45)We consider the equations 
            3x+20y-z= -18 ;      2x-3y+20z= 25 ;      20x+y-2z= 17 ;
            Using Gauss siedel iteration method find the second iterative value
a)1.0025, -0.9998, 0.9998           b)0.85, -1.0275, 1.0209              c)1, -1, 1           d)None of these      
Q46) Apply Crammer’s rule to solve the equation:
                        3x+y+2z=3;       2x-3y-z=-3;        x+2y+z=4
a)1,2,-2                         b)1,1,1                          c)1,2,-1                         d)1,-2,-1
Q47) If  A =  é8              -6                     2ù   then by Cayley Hamilton theorem
                   | -6             7                      -4 |   
                   ë2              -4                     3û        
a)λ3 - 18 λ3 + 45 λ =0       b) A3 - 18 A2 + 45 A =0   c) λ3 - 18 λ2 + 45 λ +9=0    d) A3 – 18A2 + 45 A +9=0
Q48)Determine the largest eigen value and the corresponding eigen vector of the matrtix
     If  A =  é6                 2                      2ù   then power method taking initial vector as [1,1,1]t and find the 
                | -2                3                     -1 |   value of second iteration
               ë2                  -1                     3û        
            a)7.94 é 1  ù         b) 7.82é 1     ù          c) 7.34 é 1     ù            d)None of these
                       |   0   |                  | -0.36  |                   | -0.36  |
                       ë 0.4û                    ë 0.55 û                              ë 0.55 û
Q49) What is the value of X2(j+1) of Gauss – Siedel iteration method
            a)(1/a22){a2-a21x1(j+1)-a23x3(j)}                                   b)(1/a22){b2-a21x1(j)-a23x3(j)}                         c)(1/a22){b2-a21x1(j+1)-a23x3(j+1)}                                    d)None of these
Q50) Find a real root of the equation equation X3-4X-9=0 using the bisection method find the fourth iteration
a)2.5                             b)2.6975                       c)2.625                         d)2.75
Q51) Find the real root of the equation X3-2X-5=0 by the method of false position, correct upto three decimal places. Find the fourth iteration.
a)2.0862                        b)2.0813                       c)2.0915                        d)2.0588
Q52)Find root upto four decimal place using the method of false position X sin x + cos x = 0
a)1.524                         b)2.7984                       c)0.5169                        d)2.7474
Q53) Use Newton- Raphson Method to find root to 3 decimal place of Sin x = 1 – x
a)0.6566                        b)0.6071                       c)0.5671                        d)0.5110
Q 54) Name the theorem
If f(x) is countinous in a<=x<=b and if f(a) and f(b) are of opposite signs, then f(c)=0 for at least one numbeer c such that a<c<b.
            A) Rolle’s Theorem                                            b) Lagrange’s Mean value theorem
            c) Intermediat value theorem                              d) Generalised Rolle’s theorem
Q 55) Find the smallest root of the equation f(x)=X3-6X2+11X-6=0 by Ramanujan’s method.
a)0                                b)1                               c)0.5                             d)-1
Q 56) Solve the equation 2X+3Y+Z=9 ; X+2Y+3Z=6 ; 3X+Y+2Z=8 by LU decomposition method.
a)X=29/18,Y=35/18, Z=5/18        b)X=0,Y=1, Z=0            c)X=35/18,Y=29/18, Z=5/18        d)X=1,Y=1, Z=1
Q 57) Use the method of Group averages and find a curve of the form Y=mXn
X:              10                20        30         40         50         60         70         80
            Y:               1.06            1.33      1.52      1.68      1.81      1.91      2.01      2.11
a) Y=0.3305X0.4949                     b) Y=-0.3305X0.4949                   c) Y=0.4949X0.3305          d) none of these
Q 58) Using principle of Least Square, fit an equation of the form  Y=aebx to the data
X:             1          2      3          4
Y:             1.65     2.70   4.50      7.35
a)Y=0                           b) X=e0.4992y                    c) Y=e-0.4992x                                      d) Y=e0.4992x
Q 59) Fit an equation of the form  Y=aX+b to the data by the method of Moments
X:             2          3      4          5
Y:             27        40      55         68
a)Y=12.9375X+2.2188                b)Y=12.9375X-2.2188    c)Y=2.2188X+12.9375    d)Y=2.2188X-12.9375
Q 60) Find maximum and minimum value of Y from the following table.
X:              0                 1         2          3          4          5         
Y:              0                 ¼         0          9/4        16         225/4
a)max=0,min=0.25                     b)max=0,min=2.5          c)max=0.25,min=2.5      d)max=0,min=0
section-C
Q 61)Form  Yn = A 2n +  B (-3)n, derive a differential equation by eliminating the constants.
a) Yn+2 - Yn+1 – 6 Yn = 0                              b) Yn+2 + Yn+1 – 6 Yn = 0     c) Yn+2 - Yn+1 + 6 Yn = 0   d) Yn+2 + Yn+1 + 6 Yn  = 0
Q 62.) A curve passes through the pointa as given in the table.
X:              1                 2          3          4          5          6          7          8          9
            Y:             0.2                0.7        1          1.3        1.5        1.7        1.9        2.1        2.3
Find the area bounded by the curve, the X-axis, X=1 and X=9.
a)10.2                           b)11.2                           c)11.5                           d)10.5




Q 63.) A curve passes through the pointa as given in the table.
X:              1                 2          3          4          5          6          7          8          9
            Y:             0.2                0.7        1          1.3        1.5        1.7        1.9        2.1        2.3
Find the volume of the solid generated by revolving the area about the X-axis.
a)55.6523                      b)59.6588                      c)99.5236                      d)59.1236
Q 64.) A solid of revolution is formed by rotating about the X-axis the area between the X-axis, the lines X=0 and X=1, and a curve through the points with the following coordinates:
X:         0                      0.25                  0.50                  0.75                  1.00                                          Y:                    1.0000              0.9896              0.9589              0.9089              0.8415
Estimate the volume of the solid formed, giving the answer to three decimal places.
a)2.1254                        b)2.3698                       c)2.5874                        d)2.8192
Q 65.) Solve XY’’ + Y = 0, Y(1) = 1, Y(2) = 2 with h = 0.25. (BVP)
            a)Y1=1.3256, Y2=1.6349, Y3=1.8508                    b)Y1=1.3513, Y2=1.5698, Y3=1.8508
            c)Y1=1.3513, Y2=1.6349, Y3=1.8523                    d)Y1=1.3513, Y2=1.6349, Y3=1.8508
Q 66.) Solve X2Y’’ + XY’ + (X2-3)Y = 0, Y(1) = 0, Y(2) = 2 with h = 0.25 . (BVP)
            a)Y1=0.6044, Y2=1.1304, Y3=1.5973                    b)Y1=0.6044, Y2=1.5698, Y3=1.5973
            c)Y1=1.3513, Y2=1.6349, Y3=1.8523                    d)Y1=0.6044, Y2=1.1304, Y3=1.8508
Q 67.)  Solve XY’’ + Y = 0, Y’(1) = 0, Y(2) = 1 with h = 0.5. (BVP)
a)Y0=1.5698, Y1=1.4445    b) Y0=1.6552, Y1=1.4482         c) Y0=1.5698, Y1=1.4482  d) Y0=1.6552, Y1=1.4445
Q 68.) The equation AuXx + 2BuXy + F(X,Y,u,ux,uy) = 0 is ssaid to be elliptical if
a)AC-B2 = 0                   b)AC-B2 > 0                   c)AC-B2 < 0                   d) none of these
Q 69.) Match the following.
            i) Two dimensional Laplace equation                  1)d2u/dx2 + d2u/dy2 = f(x,y)
            ii)Two dimensional Poisson’s equation               2) du/dt = C2 d2u/dx2
            iii)One dimensional heat equation                       3) d2u/dx2 + d2u/dy2 = 0 
            iv)One dimensional wave equation                      4) d2u/dt2 =C2 d2u/dx2
a) i – 3, ii – 2, iii – 1, iv – 4                     b) i – 1, ii – 3, iii – 2, iv – 4
c) i – 3, ii – 1, iii – 2, iv – 4                                 d) i – 4, ii – 2, iii – 1, iv – 3
Q 70.) With the step size h = 1/3 , solve d2u/dx2 + d2u/dy2 = 0   in 0 < x < 1, 0 < y < 1
             u(x,0)=u(0,y)=0,
            u(1,y)=9(y-y2),
            u(x,0)=9(x-x2)
a)a=1/2, b=2, c=3/2       b)a=1/2, b=1, c=2/3                   c)a=1/2, b=1, c=3/2       d)a=2, b=1, c=3/2         
Q 71.) Solve the equation ▼2u = -10(x2 + y2 + 10) over the square with sides x=0, y=0, x=3, y=3 with u=0 on the boundary and mesh length= 1.
            a) a=75, b=82.5, c=75, d=67.5                           b) a=65, b=82.5, c=65, d=67.5
            c) a=75, b=82.5, c=65, d=65.7                           d) a=75, b=85.2, c=75, d=65.7
Q 72.) Use Lagrange’s formula to find the value of f(8) given
X:             4      5          7          10         11         13
Y:            48      100       294       900       1210     2028
a)286                            b)838                            c)448                            d)848
Q 73.)  The curve y=mx+c is a
a)circle              b)straight line                c)parabola                    d)ellipse
Q 74.) The convergence in Gauss Siedel method is __________ than the Jacobi’s method.
a)twice fast                   b)twice slow                  c)thrice fast                   d)thrice slow
Q 75.) The number of non zero rows of an echelon form of a matrix is called ____________
a) Matrix                       b) order                                    c) order                         d) rank  

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