Download OU B.Sc 2016 April 3rd Year 2019E Mathematics Question Paper

Download OU (Osmania University) B.Sc (Bachelor of Science) 2016 April 3rd Year 2019E Mathematics Previous Question Paper

FACULTEES eF AR
Code No. 2019 I E
HA. I B.Sc; m ?
IS AND SCEENCE
Examination, March [April 2016
Subject: MATHEMATICS
Linear Algebra and Vector Calculus
Year
Paper ? ill :
Time : 3 hours
Max. Marks : 100
Note : An
hoo
swer Six questions from Part-A & Four questions from Part-B.
sing atleast one from ea
ch Unit. Each question in Part-A carries
6 marks and in Part-B carries 16 marks.
Part ? A (6 x 6 = 36 Marks)
Unit = i a ?
1 Prove that the iinear span L(S) of any subset S of a vector 3
of V. v
is a subspace
2 if U(F) and V(F) are two vector spaces and T is a linear trjhnnation from U into V,
then prove that the null space N(T) of T is a su
bs?gkace 9? U.
Unit ? ii ?i'a ?
3 F met the eigen roots and the correspondigf37%%&v?ctors of the matrix
I 4 a 4;?
3 2
?1
4 Prove that S = {[1 "2 ?2 _ 3 7,33?? is an orthonormal set in R3
with standard inner ,1 _-
Unit - iii ?g?
5 Evaluate [(in + yz?x + (By ? 4x)dy around the trian
ck a?
Ao,o,32?ii?, anyczn.
( )fg???t ( )
i {E .7 ?ydxd
6 Evaua #Hy
gie ABC whose vertices are
y overfO, 1 ;0, 1].
Unit =- w ? -- "?
-?> a ?> ?> _ 6f 9
If f =yzi+zxj+xyk then showthat 1x
?) a.) 9
?_ ' ?? kx~=0
7 6X+Jx6y+ 62
00
Show that I [axi+by?+czl?)~l?ids =4?(a+b+c)
S
where S is the sqrface of the '
sphere x2 + y2 + z2 = 1.

Code No. 2019 I E
_ 2 _
. Part - B (4 X 16 = 64 Marks)
Un?-?
9 a) Prove that every non-empty subset of a Linearly Independent set of vectors is
Linearly Independent.
1)) Prove that every Linearly Independent subset of a finitely generated vector space
V(F) is either a basis of V or can be extended to form a basis of V.
10 a) State and prove rank?nullity theorem.
b) Prove that zero transformation is a linear transformation.
Unit ? 11
11 a) F ind characteristic values and characteristic vectors of the matrixfm
1 1 1 a v
1 1 1
1 l 1
12 a) Exptain Gram-Schmidt orthonormalization proggss;
31s nan-u?u for an WV
1)), 1n an inner product?space V(F), prove a
., S
19%
SS
? M. ?352?:
Uh?t ? 111
13 a) Prove that every continuous fulg on is integrable.
b) Prove sufficient condition f ? e tence of the integral.
14 a) Change the order of in
n15: nd hence show that
1 1-):2 I
g;
13 5?%%
b) Evalugt?ti] ??x2 + y?) dx dy.
Unit - w 1%: 57
15 a) If E =C?i2 +6y)??14yzj?+ 20x22 k evaluate jF-dr along the straight line
C .
joining (O, 0, 0) to (1, 0, 0) end 1hen from (1, 1, 0) to (1, 1, 1).
b) If 13 = (x + 3F)?? 2xy_j)+ 23121:. Evaluate jEKIds where's is the surface of
S .
plane 2x+y+22=6 in thefirst octant.
16 a) State and prove Green?s theorem in a plane. .
b) Evaluate ?(cosx siny - xy) dx + sinx cosy dy, by Green?s theorem where C 15
C
the circle x2 + y2 = .
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