Test Paper - Applied Mathematics - IV - Mumbai University - Electronics and Telecommunication Engineering - 4th Semester, Study notes of Mathematics

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" nuur~) LI VLi::Il IVli::Iln.~^.^ I UU

N.S.: (1) Question No, 1 is compulsory.

(2) Attempt any four questions'from question Nos. 2 to 7,

,(3) If in doubt make suitable assumption. Justify your assumption and proceed.

(4) Figures to the right indicate full marks.

1. (^) (a) If the angle between the surfaces x2 + axz + byz = 2 and x2z + xy + y + 1 = z at

(0, 1, 2) is cos-1 ~ then find the constant~.a and b.

(b) Find the eigen values and eigen vectors of the orthogonal matrix :-

[

1 2 2

J

B = ~ 2 1-

2 -2 1

,r--.. ./

(c) Evaluate f f(z) dz along the parabola y = 2x2 from c \ f(z) = x2 - 2ixy. (d) Find unit normal vector to the unit sphere at point -

z = ° to z = 3 + 18i where

\

2.

(

a a a 13'13'13 (^) ) '.

(a) Find the directional derivative of xy2 + yz3 at the point (2, -1, 1) along the tangent to

7t

the curve x = a sin t, y = a cos t, z = at at t = 4'

(b) Verify Cauchy's integral theorem for f(z) = eZ along a circle c : I z I = 1.

(c) Reduce the Quadratic form -

8x2 + 7y2 + 3z2 + 12xy + 4xz - 8yz to sum of squares and find the corresponding Linear

transformation also find the rank, index and signature.

,r 3. (^) (a) Using Caley-Hamilton theorem for - '"'-"

r

-3 -4 -

J

A= 2 2 1

0 1 2

Find A64 + 2A37 - 581.

(b) Prove that V 2 f(r) = f "^ (r) + -1' (r)^2 and hence show that r (c) Find all possible Laurent's expansion of the function :- 7z-

f(z) = z(z- 2) (z + 1) about z = -1.

V4 ef = (1 + ;) ef.

..

4. (^) (a) If A = [

1 2

]

and B = [

2 0

]

° 1 1/2 2 then prove that^ both^ A and B are not^ diagonalizable

but AB is not diagonalizable.

(b) Verify Green's theorem in plane for :-

Hx2 - 2XY)dx + (x2y + 3) dy where c is the boundary of the region defined

c

by y2 = 8x and x = 2. [TURN OVER

5 5 5 5 6 6 8 6 6 8 6 6

~" nuur~) L^ I^ ULCII^ IVICII^ ,,::)^ ;^ I^ UU

N.S.: (1) Question NO.1 is compulsory. (2) Attempt any four questions 'from question Nos. 2 to 7. (3) If in doubt make suitable assumption. Justify your assumption and proceed. (4) Figures to the right indicate full marks.

L (a) If the angle between the surfaces x2 + axz + byz = 2 and x2z + xy + y + 1

(0, 1, 2) is cos-1 ~ then find the constant~ a and b. (b) Find the eigen values and eigen vectors of the orthogonal matrix :-

= z at

l

12 2

1

B = ~ 2 1-

2 -2 1

~ ./

(c) Evaluate f f(z) dz along the parabola y = 2x2 from z = 0 to z = 3 + 18i where c \ \ f(z) = x2 - 2ixy. (d) Find unit normal vector to the unit sphere at point -

\

2.

(

a a a

13'13'13 (^) ) '.

(a) Find the directional derivative of xy2 + yz3 at the point (2, -1, 1) along the tangent to 1t

the curve x = a sin t, y = a cos t, z = at at t = 4'

(b) Verify Cauchy's integral theorem for f(z) = eZ along a circle c : I z I = 1.

(c) Reduce the Quadratic form -

8x2 + 7y2 + 3z2 + 12xy + 4xz - 8yz to sum of squares and find the corresponding Linear

transformation also find the rank, index and signature.

--- 3. (^) (a) Using Caley-Hamilton theorem for -

OJ r

-3 -4 -

1

A= 2 2 1

0 1 2

Find A64 + 2A37 - 581.

(b) Prove that V2 f(r) = t" (r) + ~ f' (r) and hence show that V4 er = (1 + ;) er.

(c) Find all possible Laurent's expansion. of the function :-

f(z) = (^) z(z-2)7z-2 (z + 1) (^) about z -- - 1.

4. (a) If A =

[

1 2

]

and B = [

2 0

]

0 1 1/2 2 then prove that both A and B are not diagonalizable

but AB is not diagonalizable.

(b) Verify Green's theorem in plane for :-

Hx2 - 2XY)dx + (x2y + 3) dy where c is the boundary of the region defined

c by y2 = 8x and x = 2. (^) [TURN OVER

5 5 5 5 6 6 8 6 6 8 6 6

P4/RT -Ex-oe-6SS -^ .. .. Con. 3436-CQ-9712-08.^2

(c)

00 2 (i) Evaluate (^) -00J {)(2 + ~2~ ()(2 + b2) dx, a > 0, b > 0

(ii)

2TC

Evaluate f de

0

5. f

sin6z

(a) Evaluate c (z - 7t / 6)3 dz where c is I z I = 1.

(bj, ()~fine Minimal polynomial and derogatory matrix ar)d Test whether the matrix

[

5 -6 -

J

A = -1 4 2 is derogatory. 3 -6 - (c) Verify Gauss-Divergence theorem for :.,.. F = 4xi + 2y2j + z2k taken over the region of the cylinder bo~nded by x2 .; y2 = 4, z = 0 and z = 3.

6.

[

-1 4

]

(a) If A =. 2 1 then prove that 3 tan A = A tan 3.

(b) Evaluatef(Z - z2) dz where c is the upper half of circle Iz - 2 I = 3. c

(c) Show that F = (yeXYcosz)i + (xexy cosz)j + (-eXY sinz)k is irrotational and

find the scalar potential ~ such that F = V ~.

(i)

(ii) Find div F where F == xi - Yj.

x2 + y

7. (a) State and prove Cauchys-Residue theorem and hence -

Evaluate J

1+ z

c z(2-z)^ dz where c is I z I = 1.

(b) Evaluate JJ F.nds where F = (x + y2) i - 2xj + 2yzk and s is the surface of the pla~e

s

2x + y :+z = 6 in the first octant.

(c) Show that the matrix

[

-9 4 4

J

A = -8 3 4

-16 8 7

is diagonalizable, also find the diagonal form and diagonalizing matrix P.

- - -- (^) -- ..

4

4

6

6

8

.---.

6 6 6 2 ~ 6 6 8