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TIRUMALA ENGINEERING COLLEGE,BOGARAM.

SUB:MATHEMATICAL METHODS

ASSIGNMENT:COMMON TO ALL BRANCHES

I-B.TECH--A.Y-2009-2010

UNIT-III

1.Show that every square real matrix can be uniquely expressed as a sum of a hermitian &

skew-hermitian matrix.

2.The eigen values of a hermitian matrix are all real.

3.The eigen values of a unitary matrix are of unit modulus.

4.Reduce the quadratic form 6x2

+3y2

+3z2

-4xy-2yz+4xz to the sum of squares & find the corresponding

linear transforms. Also find the index,signature.

5.Reduce the quadratic form 9x2+2y2+2z2+6xy+2yz-2zx to a canonical form & find the rank,index &

signature.

6.Reduce the symmetric matrix A = 3 2 -12 2 3 by congruent transform.

-1 3 1

7. Reduce the quadratic form x2+2y2+2z2-2xy-2yz+2zx to a canonical form by orthogonal transform.

8. Reduce the quadratic form 5x2+26y2+10z2+6xy+4yz+14zx to a canonical form by orthogonal transform.

9.Using the legranges reduction transform x2+2y2-7z2-4xy+8zx to canonical form.

-1 1 1 11 -1 1 1

10.Show that A = () 1 1 -1 1 is orthogonal.1 1 1 -1

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UNIT-IV

1.Using Bisection method, find the negative root of x3-4x+9=0.

2.Find a positive root of x3-x2-1=0 by bisection method.

3.Find a positive root of the equation by iteration method: 3x=cosx+1

4.Find the root of the equation 2x-logx=7 which lies between 3.5 & 4 by regula-false method.

5.Find a real root of xex=2 using regula-false method.

6.Show that 10[(1-x)(1-2x2)(1-3x3)(1-4x4)] = 24 x 210 x 10!

7.Find the interpolating polynomial f(x) from the table:

x: 0 1 4 5

f(x): 4 3 24 39

8.Using lagranges formula, find at t = 1.6

T : 1.2 2.0 2.5 3.0t : 1.36 .58 .34 .20

9.Using lagranges formula,find y(1).

X : -1 0 2 3Y : -8 3 1 12

10.Using Newton-Raphson method a) find a square root of a number

b) find a reciprocal of a number.

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UNIT-V

1.Fit a straight line x 1 2 3 4 5 6y 6 4 3 5 4 2

2.Fit a parabola y = a+bx+cx2. x 1 2 3 4 5 6 7y 2.3 5.2 9.3 16.5 29.4 35.5 54.4

3.Find a & b . so that y = abx best fits the following data:

x .2 .3 .4 .5 .6 .7y 3.16 2.38 1.75 1.34 1.00 .74

4.Fit a curve of the form y = aebx to the following data:

x 0 1 2 3y 1.05 2.10 3.85 8.30

5. Fit a curve of the form y = aebx to the following data:

x 3 4 5 6 7y 6 9 10 11 12

6.Find at cos(1.74) from the following data:

x 1.7 1.74 1.78 1.82 1.86y=sinx .9857 .9916 .9781 .9691 .9584

7.A curve passing through the points (0,18),(1,10),(3,-18),(6,90).find the slope of the curve at x = 2

8. sinx dx inbetween o to x by i) trapezoidal ii) simpsons root.Here range in 10 equal parts.

9. Evaluate dx/(1+x) inbetween 0 to 1 by i) trapezoidal ii) simpsons 1/3 rd rule.

10.Find dv/dt at 1.1 t : 1.0 1.1 1.2 1.3 1.4v : 43.1 47.7 52.1 56.4 60.8