b=
19.

20.

21.

22.

23.

24,

25.

26.

210.

28.

29.

30.

32

32.

Let S. denote the set of all symmetric matrices in M(2, 1), the set of all 2 x 2 matrices with ‏لے‎
‎real numbers. Show that 5, is a vector subspace of M(2, 2) over ೩. Determine the dimensii

५, as a vector space over ௩.

Show that the real part of f(2)=—> is harmonic in the xy-plane that doesn’t contain the ॐ
origin (7 Marks)

-1
Expand दिल ವಾ್‌

0, : |z| > 2. ध (7 Marks)

using Laurent series in the domains 0, : 1 < 12] > 2 and also in

Let ‘z’ be a complex variable and f(z)=) ट a 9 Show that Cauchy-Riemann equations are
0 ifz=0
satisfied at (0, 0), but the function is not differentiable at (0, 0). (7 Marks)

Evaluate $ dr’ if F =(x+y)it ‏ز(2- ۱ھ)‎ + )۷۰+ गौ, where C is the boundary of the triangle with vertices

ए
(2, 0, 0), (0, 3, 0) and (0, 0, 6). (10 Marks)
Determine whether the vector field F =(Inx+sec?(x+y)) 10 +y)+ 7 ۷ 5 | 1+
y+z
( त्न ப்‌ न्‌ ] k is conservative and find a potential function for it. (10 Marks)
7 +
8 ⋅ ≘↾∍∁≕⊟↽ ⋅⋅ 8
Let "0" be the permutation ೫0 ‏۔‎ Find the cyclic group generated by ‘p with permutation
e
multiplication as composition. Also determine the inverses of the elements of this cyclic
group. (10 Marks)
Describe the group ೫ . Determine all the subgroups of 218 and draw the subgroup
diagram. (10 Marks)
Evaluate lim (ரூ) ಸ (10 Marks)
உல்‌ x
3
Evaluate [മഃ +1)dx as the limit of sums using a partition of (1, 3]. (10 Marks)
0050 72 ⋅ ⋅↼ ↽⋅ न ⋅
Evaluate | wen) dz, where Cis the circle in the z-plane | द| = 2 described in the anticlockwise
+
direction (10 Marks)
Solve the differential equation :
xy" +x7y"—2xy'+2y=0 (10 Marks)

Find two linearly independent series solutions in powers of ‘x’ of the equation ۷" - 20" +20۷ =0
where ‘p’ 15 a constant. (10 Marks)

Find the Cauchy's Principal Value of the integral :

ட்ப
ணம்‌
ತೇ 324352

(10 Marks)