A C192001 Pages: 2

Reg No.: Name:

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION(R&S), DECEMBER2019

Course Code: MA201
Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS
Max. Marks: 100 Duration: 3 Hours
PARTA
Answer any two full questions, each carries 15 marks Marks

72
a) Check whether the function f(z) = ۶ 6 [ദ is continuous atz=0. (7)
0, z=0

b) Show that if f(z) = u(x, 9) + iv(x,y) is analytic, then u(x, y)and v(x, y)satisfy

Cauchy- Riemann equations. (8)
2 a) Determine the region in the w —plane into which the triangular region bounded by
x= 1, y=1and x + y= 115 mapped by w = 22, (7)
b) Find the linear fractional transformation that maps (—2, 0, 2)010(0,3,3). Under
this transformation what is the image of the x —axis. (8)
3 8) Find the real part of an analytic function whose imaginary part is
ம =e “(x cosy + 9 siny).Also find the corresponding analytic function. (7)
‏(تا‎ Prove thatw = maps the upper half plane y > 0 into the upper half plane of
w —plane. What is the image of |z| = lunder this mapping? (8)
PART 8
Answer any two full questions, each carries 15 marks
2
4 a) Use Cauchy’s Integral formula to evaluate ಕ್ಲೆ മ്മ counter clock wise around (7)
(൫12൭൦൮ = 1 (൧൮൮2-15 1
0) Bind the Laurent’s series as
(2-1) (2-2) (8)
(i) 1 > |2| < 2 (ii) |z| > 2(॥॥)0 < |2 - 1| < 1
> > Use Cauchy’s Residue theorem 1 inate $ ‏ا‎ here C is th
se Cauchy’s Residue theorem to evaluate © (77) 42, where © is the (7)
ellipse 9x? + y? = 9.
b) 27 dé ⋅ ⋅ ⋅
Evaluate 0 ರಸ್‌ using contour integration. (8)
6 a) Evaluate {(Re 2) dz along the real axis from 0 to 1 and then along a straight line (7)

parallel to imaginary axis from 1 to 1 + 21.

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