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 0ò1 2dx using trapezoidal rule with h=0.1?
a)1.00 b)10 c)2.00 d)5
Q6) Evaluate 0ò1 dx/(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+cX2 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-y2 with 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
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
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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