QUESTION BANK
PADMASRI DR.B.V.R.I.C.E
LIST OF IMPORTANT QUESTIONS FOR MATHEMATICS PAPERIV
ERRORS IN NUMERICAL COMPUTATIONS
ESSAY QUESTIONS:
1.Derive general error formula of a function of ‘n’ variables.
2.If u=4x2y3/z4 and errors in x, y, z be 0.001, compute the relative
maximum error in u, when x=y=z=1.
3.If u=5xy2/z3,find the relative maximum error in u, where x=y=z=1 and
∆x= ∆y= ∆z=0.001.
SHORT ANSWER QUESTIONS:
1.Define absolute ,relative ,percentage errors.
2.If √(10)=3.162 and e=2.718 correct to four decimal places,find the
percentage error in their difference
3.If u=3x7-6x,find the percentage error in u at x=1,if the error in x is 0.05.
VERY SHORT ANSWER QUESTIONS:
1.define truncation error ,Inherent errors.
2.
SOLUTION FO ALGEBRIAC AND TRANSCENDENTAL EQUATIONS
ESSAY QUESTIONS:
1.Deriv e Regula Falsi Formula to find the root of the equation.
2.Find the cube root of 30 using bisection method.
3.Using Muller’s method ,find the root of the equations cosx=xex,x3-x2-x1=0
4.Find the Newton- Raphson method, a root of the equations x3-5x+3=0,
3x-cosx+1=0.
5.Find the root of a equations x3+x2-100=0, x3+x2-1=0 correct to three
decimal places using Iteration method.
6.Using Newton-Raphson method.find the root of the equation
xsinx+cosx=0 to four decimal places.
7.Using Regula Falsi method compute the root of the equation
2x-logx=6, xex=cosx.
8.Find the double root of the equation f(x)=x3-x2-x+1=0 by generalized
Newton- Raphson method.
9. Find the smallest root of f(x)=x3-6x2+11x-6=0 by Ramanujans method.
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QUESTION BANK
sinx =1-x by Ramanujans method.
10.Find a real root of xex=1,
SHORT ANSWER QUESTIONS:
1.Explain bisection method.
2.Derive Newton-Raphson formula to find a root of a equation.
VERY SHORT ANSWER QUESTIONS:
1.Explain Iteration Method to solve f(x)=0.
2. Explain bisection method.
3.Write the iteration formula to find a root of the equation by NewtonRaphson method.
4.State the condition for the convergence of the root of the equation f(x)=0
by Iteration method.
FINITE DIFFERENCES:
ESSAY QUESTIONS:
1.If f(x) is a polynomial of nth degree in x, then the nth difference of f(x) is a constant
and (n+1)th difference is zero.
SHORT ANSWER QUESTIONS:
1.Find the missing value in the following
x: 0
5
10
5
y: 6
10
_
17
2. Prove that E= ehD .∆=1-e-hD .
3. Obtain the function whose first difference is 2x3+3x2-5x+4=0,9x2+11x+5.
4. Show that ∆=E-1 and =1-E-1.
5.Find the missing values in the following table
x : 45
50
55
60
65
y: 3.0
_
2.0
_
-2.4
6 .If f(0)=-3,f(1)=6,f(2)=8,f(3)=12 find f(6) where third difference is constant.
7 .prove that i) √(1+δ2µ2)=1+δ2/2. Ii)µ2=1+δ2/4 iii)∆=δ2/2 +δ√(1+δ2/4).
8.prove that i) f(4)=f(3)+∆f(2)+∆2f(1)+∆3f(1). Ii)f(4)=f(0)+4∆f(0)+6∆2f(-1)+10∆3f(-1) upto
third differences.
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VERY SHOR ANSWER QUESTIONS
1.Define forward, backward and shift operators.
2.prove i)∆V=∆-V ii)∆logx,∆tan-1x,∆sinx.
INTERPOLATION WITH EQUAL INTERVALS
ESSAY QUESTIONS:
1.State and prove Newtons forward difference interpolation formula.
2. State and prove Newtons backward difference interpolation formula.
3.Using Newtons forward formula to evaluate f(1.6) from the following data.
x:
1
f(x):3.49
1.4
1.8
2.2
4.82
5.96
6.5
4.Using Newtons backward formula find f(0.7) from the following data.
x:
0.1
0.2
f(x): 2.68
3.04
0.3
0.4
0.5
0.6
3.38
3.68
3.96
4.21
5. From the following table, estimate the number of students who obtained marks
between 40 and 45
Marks
: 30-40
No. of students :
31
40-50
50-60
60-70
70-80
42
51
35
31
SHORT ANSWER QUESTIONS:
1. The population of a country in the decennial census where as under .Estimate the
population for the year 1925.
Year(x):
1891
1901
1911
1921
1931
Population(y):
46
66
81
93
101
2.Find the second degree polynomial passes through the points (0,1),(1,3),(2,7)and (3,13).
3. Find the polynomial f(x) from the following table.
x:
0
1
2
3
4
f(x): 3
6
11
18
27
VERY SHOR ANSWER QUESTIONS
1.Define interpolation.
CENTRAL DIFFERENCE INTERPOLATION FORMULAE
ESSAY QUESTIONS:
1. State and prove Gauss forward interpolation formula.
2. State and prove Gauss backward interpolation formula.
3. State and prove Stirling’s interpolation formula.
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4. Use Stirlings formula to evaluate f(25) from the following data.
x: 10
20
30
40
y: 1.1
2
4.4
7.9
5.Using Stirling’s formula to find y35,given y20=512,y30= 439, y40=346,y50=243.
6.Using Gauss forward formula to find the value of when x=3.75 from the following table.
x: 2.5
3
3.5
4
4.5
5
y: 22.145 22.043
20.225
18.644
17.262
16.047
7.Using Gauss backward formula
VERY SHORT ANSWER QUESTIONS
1.Define central difference operator.
2.Define averaging operator.
INTERPOLATION WITH UNEQUAL INTER VALS
ESSAY QUESTIONS
1.State and prove Newtons divided difference formula.
2.State and prove Lagranges interpolation formula.
3.The divided differences are symmetric functions of their arguments.
4.By means of Newtons divided difference formula find the values of f(8) &f(15) from the
following table
x:
4
5
7
10
11
13
f(x): 48
100
294
900
1210
2028
SHORT ANSWER QUESTIONS
1.If f(x)=1/x &f(x)=1/x2 ,find f(a,b) and f(a,b,c) and f(a,b,c,d).
2.Find the cubic polynomial in x from the following table using Newtons divided
difference formula
x:
0
1
2
5
f(x): 2
3
12
147
3.Using Lagranges formula find the value of y when x=10 from the following data
x: 5
6
9
11
y: 12
13
14
16
VERY SHORT ANSWER QUESTIONS
1.Explain divided difference .
2.Write the second and third divided difference.
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QUESTION BANK
CURVE FITTING
ESSAY QUESTIONS:
1.State the principle of least squares .Fit a second degree parabola y=a+bx+cx2 to the
following data.
x:
1
y: 1.1
1.5
2
2.5
3
3.5
4
1.3
1.6
2.0
2.7
3.4
4.1
2.Fit a straight line to the following data
X:
1
2
3
4
5
Y: 14 27
40
55
68
bx
3.Fit a exponential curve y=ae to the following data.
X: 2
4
6
8
Y: 25
38
56
84
4.Fit a curve y=axb by the method of least squares using the following table
X:
61
26
Y: 350
7
400
2.6
500
600
SHORT ANSWER QUESTIONS
1.Write a working procedure to fit a straight line.
2.Write the normal equations to fit a parabola.
VERY SHORT ANSWER QUESTIONS
1.Define curve fitting.
2.Define method of least squares.
NUMERICAL DIFFERENTIATION
ESSAY QUESTIONS
1.find y1(0) from the following data.
X: 0
1
2
3
4
5
Y: 4
8
15
7
6
2
2.From the following values of x &y find the value of y1(4).
X: 1
Y:
0
2
4
8
10
1
5
21
27
3. From the following data ,find x for which y is maximum and find this value of y
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QUESTION BANK
X:
3
Y : 0.205
4
0.240
5
0.259
6
0.262
7
0.250
8
0.224
4.Calculate the first and second derivatives of the function tabulated given below at
x=2.2
X:
1
Y: 2.7183
1.2
1.4
1.6
3.3201
4.0552
4.9530
1.8
2
2.2
6.0496
7.3891
9.0250
5.Find the maximum and minimum values of y =f(x) form the given data
X:
0
1
2
3
4
5
Y:
0
0.25
0
2.25
16.00
56.25
VERY SHORT ANSWER QUESTIONS
1.Define numerical differentiation.
2 .Write the formulas for the derivatives using Newtons forward interpolation formula
3. Write the formulas for the derivatives using Newtons backward interpolation formula
4. Write the formulas for the derivative using stirling’s formula.
NUMERICAL INTEGRATION
ESSAY QUESTIONS
1.State and prove General Quadrature formula.
2. State and prove Trapezoidal rule, Simpson’s 1/3, 3/8 th rules.
3 .State and prove Weddle’s ,Boole’s rule
4.Evaluate
0
.
∫6 dx/(1+x2) by using Weddle’s rule, Simpsons rule
SHORT ANSWER QUESTIONS
1.Evaluate 0 ∫1dx/(1+x) by using Trapezoidal rule, Simpson’s 1/3, 3/8 th rules& also by
Boole’s rule
2.Evaluate0 ∫5dx/(4x+5) by the Trapezoidal rule.
3.Evaluate 4 ∫5.2logexdx by Weddle’s rule.
VERY SHORT ANSWER QUESTIONS
1.Define Numerical integration.
2.Write Quadrature formula.
3.Write Trapezoidal rule, Simpson’s 1/3, 3/8 th rule. Weddle’s ,Boole’s rule.
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NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
ESSAY QUESTIONS
1.Using R-K method compute i) y(0.1)& y(0.2) from dy/dx =( x2+y2 )/10 taking h=0.1
&y(0)=1
ii) y(0.2) from dy/dx =3x+y/2 ,y(0)=1 taking h=0.1.
iii) y(0.4) from dy/dx=(y-x)/(y+x) ,y(0)=1 take h=0.2.
2. Using Taylor series method find y(0.2) from dy/dx=1-2xy,y(0)=0.
3.Find the value of y at x=0.5 from dy/dx=x+y2,y(0)=1 by using Euler’s method.
4.solve dy/dx=x+y2,y(0)=1 ,find Picards second approximation.
SHORT ANSWER QUESTIONS
1.Using Euler’s method compute i) y(0.3) with h=0.1 from the following
dy/dx=x+y2,y(0)=1,
ii)find y(0.4) with h=0.1 from the equation dy/dx=1-2xy,y(0)=0 .
iii)find y(0.3) with h=0.05 from the following dy/dx=x+y,y(0)=1.
2 .Using modified Euler’s method find y(0.1) from dy/dx=x+y+xy, y(0)=1 taking h=0.025.
3.Write about Milnes-predictor formula.
SOLUTIONS OF LINEAR SYSTEM EQUATIONS
1. Solve the following system of equations by Factorization method
2x+3y+z=9,x+2y+3z=6,3x+y+2z=8.
2. Solve the following system of equations by Gauss- Seidal method
.
83x+11y-4z=95,7x+52y+13z=104,3x+8y+29z=71.
3. Solve the following system of equations by Matrix Inversion method.
x+3y+3z=1,x+4y+3z=0,x+3y+4z=2.
4.Solve the following system of equations by Gauss-Elimination method
2x+y+z=10,3x+2y+3z=18,x+4y+9z=16
5 .Solve the following system of equations by Gauss-Jacobi method.
5x+2y+z=12,x+4y+2z=15,x+2y+5z=20.
VERY SHORT ANSWER QUESTIONS
1.Explain about Ill conditioned linear system of equations.
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QUESTION BANK
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