Mathematics Ii May2004 Or 210156

OR

Code No. 210156

II-B.Tech I-Semester Supplementary Examinations, May/June -2004 MATHEMATICS-II (Common to all branches) Time: 3 hours

1.a)

b)

2.a)

b) 3.a)

b) 4.

Max. Marks: 70 Answer any FIVE questions All questions carry equal marks ---

Find the rank of the matrix 1 2 3 0     2 4 3 2 3 2 1 3   6 8 7 5    Determine the values of λ for which the following equations may possess non-trivial solution 3 x1 + x 2 − λx3 = 0, 4 x1 − 2 x 2 − 3 x3 = 0, 2λx1 + 4 x 2 + λx3 = 0 for each permissible value of λ , determine the general solution. Find the eigen values and eigen vectors of the matrix  6 − 2 2    − 2 3 − 1  2 −1 3   λ If is an eigen value of a matrix A, prove that 1 / λ is the eigen value of A-1. Verify Cayley Hamilton theorem for the matrix  7 2 − 2   A = − 6 −1 2   6 2 −1    and find its inverse. Write down the matrix of the quadratic ax 2 + by 2 + cz 2 + 2 fyz + 2 gzx + 2hxy . 2 Prove that x =

(b)

1

∑ ( 2n − 1)

2

∞ π2 n cos nx + 4∑ ( − 1) , − π < x < π and deduce (a) 3 n2 n =1 π2 = Contd…2 8

1 π2 ∑ n2 = 6

Contd..2.

Code No. 210156

-2-

OR

5.

If f(x) = x, 0 < x < π / 2 = π − x, π / 2 < x < π 4 1 1  Show that f ( x ) =  Sin x − 2 Sin 3x + 2 Sin 5 x + ....... π 3 5 

6.

Form the partial differential equation of all planes which are at a constant distance a from the origin. Solve pyz + qzx = xy ∂2z ∂2z ∂2z − 2 + = Sin x Solve ∂x∂y ∂y 2 ∂x 2

a) b) 7.

A tightly stretched string with fixed end points x = 0 and x = ℓ is initially is a 3 πx position given by y = y o Sin . If it is released from rest from this position, l find the displacement y(x1t). ***

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