Mathematical Methods

Code No. 07A1BS06

UNIT-3 MATHEMATICAL METHODS

1. Show that any square matrix can be writher as sum of a symmetric matrix and a skew-symmetric matrix.  8 6 Express  10 2  as a sum of a symmetric matrix and a skew-symmetric matrix.    1 1 1 1    1  1 1 1 1  A  2. Verify wheter the matrix A = is orthogonal. 2  1 1 1 1     1 1 1 1 3. Define orthogonal matrix.  8 4 1  1  Verify whether the matrix A   1 4 8 is orthogonal 9  4 7 4  4. If A is any square matrix, prove that A+A*, AA*.A*A are all Hermition and A-A* is skew Hermition.  a  ic b  i  2 2 2 2 5. Show that the complex matrix A    is unitary if a b +c +d = 1 b  id a  i   2 3  4i  is Hermition. 2   3  4i Find the eigen values and eigenvectors. 

6. Show that the complex matrix A  

7. Show that the eigen values of a skew – Hermition matrix are purely imaginary or real.  1 2 2 1  8. Define an orthogonal matrix. A   2 1 2  is orthogonal. 3  2 2 1  i   1  2 2  9. Find the eigen values and eigen vectors of the unitary matrix A= A    i 1   2 2  2  1  3i 10. Find the eigen values and the eigen vectors of the complex matrix B     2  i i  11. Reduce the Quadratic form 2x1x2+2x2x3+2x3x1 into conomonical form and classify the quadratic form. 12. Reduce 3x2+3z2+8xz+8yz into canonical form. Give the rank, index and signature of the Quadratic form. 13. Reduce the Quadratic form 2x2+2y2+3x2+2xy-4yz-4xz to conomical form. Find the rank,index and signature. 14. Determine the nature, index and signature of the Quadratic form 2x2+2y2+3z2+2xy-4xz-4yz.

15. Show that the linear transformation Y1=2x1+x2+x3 ; y2= x1+x2+2x3; y3=x1 – 2x3 is regular. Write down the inverse transformation. 16. Find the nature, index and signature of the Quadratic form x12  5 x22  x32  2 x1 , x2  2 x2 x3  6 x3 x1 . 17. Find the nature, index and signature of the Quadratic form 3x12  3 x22  7 x32  6 x1 x2  6 x2 x3  6 x3 x1 . 18. 19. 20.

2 2 Reduce the Quadratic form 2( x1  x1 x2  x2 ) to canonical form. Reduce the Quadratic form 5x26xy+5y2 to sum of squares. 2 2 Reduce the Quadratic form 6 x1  16 x1 x2  6 x2 to sum of squares.

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Code No. 07A1BS06

UNIT-4 MATHEMATICAL METHODS

1. Find a root of the equation x3-4x-9 = 0 using bisection method correct to three decimal places. 2. Find a root of the equation x3-2x2-4 = 0 using bisection method correct to three decimal places. 3. Find a real root equation f(x) = x2+x-3 = 0 correct to three decimal places using Bisection method. 4. Find a real root of the equation cosx = 3x-1, correct to three decimal places using the method of false position. 5. Find a real root of the equation x3-8x-40 = 0 in [4,5] correct to three decimal places using the method of false position. 6. Using Regular falsi method; compute the real root of the equation x ex = 1 in [0,1] correct to three decimal places. 7. Find a real root of the equation x3+x2-1 = 0 by using interative method, correct to three decimal places. 8. Find by the method of interation a real root of the equation x = .21 sin(0.5+x) starting with x = 0.12 xorrect to three decimal places. 9. Using Newton – Raphson method compute the root of equation x sin x + cos x = 0 which lies    between  ,  , correct to three decimal places.  2  10. Find the double root of the equation x3-3x+2 = 0 starting with x0 =1.2 by Newton – Raphson method. 11. Following table gives the weights in pounds of 190 high school students. Weight (in pounds)

30-40

40-50

50-60

60-70

70-80

No.of students 31 42 51 35 31 Estimate the number of students whose weights are between 4 and 45. 12.

Obtain the relations between the operators. (i )   E  1 (ii )   1  E 1 1 1 1 1 1 (iii )   E 2  E 2 (iv)   ( E 2  E 2 ) 2

13.

Estimate f(22) from the following data with the help of an appropriate interpolation formula.

14.

X:

20

25

30

35

40

45

F(x):

354

332

291

260

231

204

Estimate y(3) from the following data, using an appropriate interpolation formula.

X: 15.

4

6

8

10

Y: -14 22 154 430 898 Using an interpolation formula estimate y(4.1) from the following data. X:

16.

2

0

1

2

3

4

Y: 1 1.5 2.2 3.1 4.6 Given that f(45) = 0.7071,f(50) = 0.6427, f(55) =0.5735,f(0) = 0.5,f(65) = 0.4226, find f(63) using Newton’s Backward interpolation formula.

17.

Use stirling’s formula to find y(35), given that y(20) = 512,y(30) = 439, y(40) =346, y(50)= 243.

18.

Given that y(20) = 24, y(24) = 32, y(28) = 35, y(32) = 40. Find y25 central interpolation formula.

19.

The following table gives the viscosity of a lubricant as a function of temperature. Temperature : 100 120 150 170 Viscosity 10.2 .7.9 5.1 4.4 Apply Lagrange’s formula to estimate viscosity of the lubricant at 130 degrees of temperature.

20.

Apply Lagrange’s formula to estimate y (3) from the following deta X: 0 1 2 4 Y: 2 3 12 78.

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Code No. 07A1BS06

UNIT-5 MATHEMATICAL METHODS

1.

2.

3.

4. 5. 6. 7.

8. 9.

Find the stright line of the form y = a + bx that best bits the following data, by method of least sequences. X: 1 2 3 4 5 Y: 12 25 40 50 65. Estimate y (2,5). Find a second degree parabola y = a + bx + cx2 to the given deta, by method of least sequences. X: 1 3 5 7 9 Y: 2 7 10 11 9 In an experiment the measurement of electric resistance R of a meal at various temperatures t0c lirted as. T: 20 24 30 35 42 R: 85 82 80 79 76 Fit a relation of the form R = a +bt, by method of least sequences. Fit a second degree parabola of the form y =a + bx +cx2 to the following data. X: 0 1 2 3 4 Y: 1 1.8 1.3 2.5 6.3. Fit the following deta to an exponential curve of the form y = aebx. X: 1 3 5 7 9 Y: 100 81 73 54 43 For the deta given below find a best flitting curve of the form y = axb. X: 1 2 3 4 5 Y: 2.98 4.26 5.21 6.10 6.8 What is least squares principle ? Fit a stright line y = a + bx to the following deta. X: 0 1 3 6 8 Y: 1 3 2 5 4 Find the best fitting exponential curve y = aebx to the following deta. X: 2 3 4 5 6 Y: 3.72 5.81 7.42 8.91 9.68 Fit a parabola y = ax2 +bx + c which best bits with the observations. X: 2 4 6 8 10 Y: 3.07 12.85 31.47 57.38 91.29.

10.

Fit a least sequence curve y =axb to the following deta X: 1 2 3 4 5 Y: 0.5 2 4.5 8 12.5

11.

Evaluate  (1  e

1

0

1 3

rule.

x

sin 4 x )dx

. Taking h =1 /4 by (i) Trapezoidal (b) simpsous

12.

Find first and second derivation of the function tabulated below, as the point x =1. X: 0.0 0.1 0.2 0.3 0.4 Y: 1.0000 0.9975 0.9900 0.9776 0.9604

13.

Find first and second derivations of the function telruleted below, at the point x =1. X: 0 1 2 3 4 Y: 6.98 7.40 7.78 8.12 8.45.

14.

Explain how the thepezridel rule is obtained from Newton – cote’s general quedreture formula.

15.

Given the following table of values of x and y, first, first and second derivatives at x = 1.25 X: 1.10 1.15 1.20 1.25 Y: 1.05 1.07 1.09 1.12 6

16.

Evaluate

dx

 1  x using Simpson’s 2

0

3 8

1.30. 1.14

the rule.

4

17.

Find  e dx by simpson rule of numerical integration. x

0

18.

Find the first and second derivatives. Of the function tabulated below at the point 1.5. X: 1 2 3 4 5 F(x) 8 15 7 6 2

19.

Evaluate

5.2

 log x dx using (i) simpsous 4

X 4.0 Logx 1.38 

20.

Evaluate

4.2 1.44

4.4 1.48

4.6 1.53

2

 sin xdx by symposiums o

actual value of the integral. -oOo-

1 3

4.8 1.57 1 3

rule (ii)simpsous 5.0 1.61

3 8

5.2 1.65

rule, using 11ordinates and compare with