# II-B.Tech. I-Semester Supplementary Examinations, May/June-2004

MATHEMATICS – II
(Common to all Branches except Bio-Technology)
Time: 3 Hours Max. Marks: 80
All questions carry equal marks
– – –
1.a) Find the values of a and b for which the equations
x + ay + z = 3, x + 2y + 2z = b, x + 5y + 3z = 9
are consistent. When will these equations have a unique solution?
b) Investigate for what values of ?, ? the simultaneous equations.
x + y + z = 6, x + 2y + 3z = 10, x + 2y + ?z = ?
have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.

2.a) If ? is an eigen value of A then prove that the eigen value of B = a0A2 + a1A + a2I is a0 ?2 + a1? + a2.
b) Find the eigen values and eigen vectors of
3.a) Show that any square matrix A = B + C where B is symmetric and C is skew-symmetric matrices.
b) Determine a, b, c so that A is orthogonal where .
4. Expand as a Fourier series.

5.a) Find the half range sine series for the function f(x) = t-t2, 0 b
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Code No: NR/RR-210101

II-B.Tech. I-Semester Supplementary Examinations, May/June-2004

MATHEMATICS – II
(Common to all Branches except Bio-Technology)
Time: 3 Hours Max. Marks: 80
All questions carry equal marks
– – –
1.a) Solve the equations.
4x + 2y + z + 3? = 0, 6x + 3y + 4z + 7? = 0, 2x + y + ? = 0
b) Show that the system of equations.
2×1 – 2×2 + x3 = ?x1
2×1 – 3×2 + 2×3 = ?x2
-x1 + 2×2 = ?x3
can posses a non-trivial solution only if ? = 1, ? = -3 obtain the general solution in each case.

2.a) Define eigen value and eigen vector of a matrix A. Show that trace of A equals to the sum of the eigen values of A.
b) Find the eigen values and eigen vectors of
3.a) Prove that inverse of a non-singular symmetric matrix A is symmetric.
b) Identify the nature of the quadratic form .

4. Find the Fourier series to represent the function .

5.a) Show that for -?0 and hence deduce the
inversion formula.
x
b) Find the Fourier sine transform of ———-
a2 + x2

1
and Fourier cosine transform of ————
a2 + x2
using the results in (a)

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Code No: NR/RR-210101
II-B.Tech. I-Semester Supplementary Examinations, May/June-2004

MATHEMATICS – II
(Common to all Branches except Bio-Technology)
Time: 3 Hours Max. Marks: 80
All questions carry equal marks
– – –
1.a) Find the inverse of the matrix

A =
by using elementary row operations.
b) Compute the inverse of the matrix

A =
by elementary operations.

2.a) Prove that the eigen values of A-1 are the reciprocals of the eigen values of A
b) Determine the eigen values of A-1 where
3.a) Define an orthogonal transformation.
b) Find the orthogonal transformation which transforms the quadratic form to canonical form.

4. Expand the function as a Fourier series and hence show that .

5.a) Show that in the interval (0,1)
Cos ?x=
b) Find the half range sine series of f(x)=1 in 0 2

is 2 sin s(1 – cos s)/s2
b) Show that Fourier transform of is reciprocal

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Code No: NR/RR-210101

II-B.Tech. I-Semester Supplementary Examinations, May/June-2004

MATHEMATICS – II
(Common to all Branches except Bio-Technology)
Time: 3 Hours Max. Marks: 80
All questions carry equal marks
– – –
1. Find the values of ? for which the equations.
(? – 1) + (3? + 1)y + 2?z = 0
(? – 1) + (4? – 2)y + (? +3)z = 0
2x + (3? + 1)y + 3(? – 1) z = 0
are consistent and find the ratio of x : y : z when ? has the smallest of
there values. What happens when ? has the greater of these values.

2.a) Show that A and AT has same eigen values but different eigen vectors.
b) Determine the eigen values and eigen vectors of B = 2A2 – 1/2 A + 3I
where

3.a) Prove that the eigen values of a Hermitian matrix are real. Deduce the result for real symmetric matrix.
b) Show that is unitary matrix if a2 + b2 + c2 + d2 = 1.
4. Show that for

5. Represent f(x)=sin in 0