'''"''''''"'''''''''"'''"'''''' us - 354 VI Semester B.A.lB.Sc. Examination, May 2017

(Semester Scheme) (Fresh) (CBCS) (2016-17 and Onwards)

MATHEMATICS - VII

Time: 3 Hours -;-::.-~~M.ax. Marks: 70 ~-'t:.;.. .. tiAVE€A v....,~

.z).Y~ .;;;:-- .q~,~ Instruction: Answer a/l Parts. '?" ~ g))

<5> KGF - 563 122 ;-J PART -A ~/ys ~~'

~~ * -~"/ 1. Answerany five questions : ~==:.-"'::='" (5x2=10)

a) Define a vector space over a field.

b) For what value of K the vectors (1, 2, 3), (4, 5, 6) and (7, 8, K) are linearly dependent.

c) Find the matrix of the linear transformation T : R2 ~ R3 defined by

T (x, y) = (3x - y, 2x + 4y, 5x - 6y) w.r.t. the standard basis. d) Find the null space of the linear transformation T : V3 (R) ~ V2(R) defined by

T (x, y, z) = (y - x, y - z).

e) Write scalar factors in cylindrical co-ordinate system.

dx dy dz f) Solve : - = - = -.

yz zx xy

g) Form a partial differential equation by eliminating the arbitrary constants from z = (x + a) (y + b).

h) Solve: pq + p + q = O.

P.T.O.

US-354 -2-

'''''' """" '" "'" "'" "" "" Answer two full questions:

PART -8

2. a) A subset W of a vector space V(F) is a subspace of V(F) if and only if

C1a+ C2~ E W, for a,~ EW.

(2x10=20)

b) Find the basis and dimension of the subspace spanned by (1, - 1, 0), (0, 3, 1), (1, 2, 1) and (2, 4, 2) in V3 (R).

OR

3. a) A set of non-zero vectors (a1, cx2' .... an) of vector space V(F) is linearly

dependent if and only if one of vectors Say a K (2~ K ~ n) is expressed as a linear combination of its preceding ones.

b) Prove that W = {(x, y, z) , x = y = z} is a subspace of R3.

4. a) If T : U ~ V is a linear transformation then prove that

i) T(O) = 0' where 0 and 0' are the zero vectors of U and V respectively. ii) T (- a) = - T( a) , va E U.

b) Find the linear transformation T : R2 ~ R2 such that T (1, 0) = (1, 1) and T (0, 1) = (- 1, 2).

OR

5. a) State and prove rank nullity theorem.

b) Show that the linear transformation T : R3 ~ R3 given by T (e1) = e, + e2, T (e2) = e1 - e2 + e3 and T(e3) = 3e1 + 4e3 is non singular where {e1, e2, e3} is the standard basis of R3.

-- PART -C

Answer two full questions:

6. a) Verify the condition of integrability and solve: 2yzdx + zxdy _ xy (1 + z) dz = O. (2x10=20)

11111111111111111111111111111111111 -3- US-3S4

7. a) Show that cylindrical coordinate system is orthogonal co-ordinate system.

~ A 2A 2A b) Express the vector f = 3y i + x j - z k in cylindrical co-ordinate and find

f , f , f p ¢ z·

b) Solve: -- = y+z dx dy

z+x =

dz x+y

dx dy dz 8 a) Solve' = = . . x(y-z) y(z-x) z(x-y)'

OR

~ A A A 9. a) Express the vector f = 3y i + 2zj + xk in cylindrical co-ordinates and find

f , f , f p o z :

-7

b) Express the vector f = z] - 2xJ + yk in terms of spherical polar co-ordinates

and find fr' fe' f<jJ'

PART - D

Answer two full questions: (2x10=20)

10. a) Form partial differential equation by eliminating arbitrary function f(xy + Z2 , X + Y + z) = O.

b) Solve : p2 - q2 = X _ y.

OR

US-354 -4-

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2 b) Solve: z' (p' + q') = x' + y', by using the transformation u = ~ .

12. a) Find the complete integral of z = pq by using Charpit's method.

222 a z a z a z x-y 2 b) Solve: --7--+12-=e +x ax2 axay ay2

OR

13. a) A tightly stretched string with fixed end points x = 0 and x = l is initially in a

position given by y = Yo Sin3( ~x ). If it is released from rest from this

position, find the displacement y(x, t).

au a2u b) Solve: at = 16 ax2 subjected to the condition:

i) u(O, t) = 0, U(1, t) = 0 for all t

ii) u(x;O) = X2 - x, 0 S; x S; 1.