Mathematics Ii Nov2003 Or 210156

Code No. 210156 II B.Tech. I Semester Supplementary Examinations November, 2003 MATHEMATICS-II (Common to all baranches) Time: 3 hours Max. Marks: 70 Answer any Five questions All question carry equal marks --1.a) Find the rank of the matrix

OR

2 3 − 1 − 1  1 − 1 − 2 − 4    3 1 3 − 2   0 − 7 6 3 b)

For what values of k, the equations x+y+z=1, 2x+y+4z=k, 4x+y+10z=k2 have a solution and solve them completely in each case.

2.a) Find the eigen values and eigen vectors of the matrix

 8 −6 2   − 6 7 − 4    2 − 4 3  b) Prove that the sum of the eigen values of a matrix is the sum of the elements of the principal diagonal.

3 1 1   3.a) Verify cayley Hamilton for the matrix A= 1 3 − 3 2 − 4 − 4 b) If a square matrix A of order 3 has 3 linearly independent eigen vectors, prove that a matrix P can be found such that P-1 A P is a diagonal matrix. 4.

If f(x)= πx , 0 ≤ x ≤ 1 = π (2-x), 1 ≤ x ≤ 2 Show that in the internal (0,2) F(x)=

π 4  Cosπx Cos3πx Cos5πx  + + + ........ 2 2 2  2 π  1 3 5 

5.a) Find the half range sine series for the function f(t)=t-t2, 0
Code No. 210156

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OR

6.a) Form the partial differential equation by eliminating arbitrary function from z = xf1(x+t) + f2(x+t) b) Solve (x2-y2-z2) p+ 2xyq = 2xz. 7.a) Solve z2 = 1+ P2 +q2. b) Solve 8.

∂2z ∂2z − 2 = Sinx Cos zy. ∂x∂y ∂x 2

An insulated rod of length l has its ends A and B maintained at 0 oC and 100 oC respectively until steady state conditions prevail. If B is suddenly reduced to 0oC and maintained at 0oC, find the temperature at a distance x from A at time t. !!!!!