B3A005 Pages:2

ii) zsin (=) (4)
a. Use residue theorem to evaluate [ल-त where C is 2] - 1 (7)
b. Evaluate [ஷி using residue theorem (8)
(1 +x
PART C

Answer any 2 questions

a. Solve the following by Gauss elimination

y+z—-2w=0, 2x—3y—3z2+6w=2, 4x+y+z2-2w=4 (6)

b. Reduce to Echelon form and hence find the rank of the matrix

3 0 2 2
-6 42 24 54
21 -21 0 -15
(6)
2-2 0
c. Find a basis for the null 59302 01 1 0 4 8 (8)
2 0 4
೩.1) Are the vectors (3 -1 4), (6 7 5) 210 (9 6 9) linearly dependent or
independent? Justify your answer. (5)

11) Is all vectors (>, 9, 2) in [९३ with ‏پر‎ - ×+ 42 = 0 form a vector space over the field
of real numbers? Give reasons for your answer. (5)
b. i) Find a matrix © such that 0 = x’ Cx where
@ = ‏2۔3‎ + 422) - a + 2x,x, - 5x} (4)
ii) Obtain the matrix of transformation

37 = cos Ox;—sinOX2, y2= sin Ox) + cos 0202
Prove that it is orthogonal. Obtain the inverse transformation. (6)
a. Find the eigenvalues, eigenvectors and bases and dimensions for each Eigen space

of

2 2 ಇತ
4 =| 2 1 -6
-1 -2 0

(10)
b. Find out what type of conic section, the quadratic form 17x; -30× +177 = 128

and transform it to principal axes. (10)

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