Mathematics Ii Jun2003 Or 210156

Code No:210156 II-B.Tech. I-Semester Supplementary Examination - June 2003 MATHEMATICS - II (Common to all branches) Time: 3 hours Max.Marks:70 Answer any five questions All questions carry equal marks --1.a)

For the matrix A=

b)

2.a)

1 1 0

1 2 -1

2 3 -1

find non-singular matrices P and Q such that PAQ is in the normal form and find its rank. Investigate the values of a and b so that the equations: 2x+3y+5z = 9, 7x+3y-2z = 8, 2x+3y+az = b have (i) No solution. (ii) A unique solution. (iii) An infinite number of solutions. Find the eigen values and eigen vectors of the matrix. 1 1 3

b)

OR

1 5 1

3 1 1

Show that the eigen values of a triangular matrix A are equal to the elements of the principal diagonal of A.

3.a) b)

State and prove Cayley-Hamilton theorem. Using Cayley -Hamilton theorem, find A8 if A = 1 2

4.

Find the Fourier series to represent the function f(x) given by f(x) = x, 0≤x≤π = 2 π- x, π≤x≤2π

2 -1

1 1 1 π2 Deduce that + + + ..... = 8 12 32 5 2 5.a) b)

Obtain the half-range Sine series for ex in 0 < x < 1 Obtain the half-range Cosine series for x2 in 0 ≤ x ≤ π. (Contd…2)

Code No: 210156. 6.a) b)

7.a) b)

8.

..2..

OR

Form the partial differential equation by elementary arbitration function from Z = y f(x) + x g(y). Solve: x ( y - z )p + y ( z - x ) q = z ( x - y ) Solve Z = px + qy - 2 pq Solve

∂3z ∂x

3

−3

∂2z 2

∂x ∂y

+4

∂3z ∂y

3

= ex+2 y

A tightly stretched string of length l has its ends fastened at x = 0, x = l. The mid point of the string is then taken to a height h and then released from rest in that position. Find the lateral displacement of a point of the string at time t from the instant of release.

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