Download GTU BE/B.Tech 2019 Winter 1st And 2nd Sem Old 110015 Vector Calculus And Linear Algebra Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 1st And 2nd Sem Old 110015 Vector Calculus And Linear Algebra Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (OLD) EXAMINATION ? WINTER 2019
Subject Code: 110015 Date: 01/01/2020

Subject Name: Vector Calculus And Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a)
(1) Find the Rank of
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
?? ??
??
??
?? ??
??
??
by row echelon form.
03
(2) Solve the following system of equation by Gauss elimination method.
29
2 4 3 1
3 6 5 0
x y z
x y z
x y z
? ? ?
? ? ?
? ? ?
.
04
(b)
Determine whether the set V of all pairs of real numbers ? ? , xy with the
operations ? ? ? ? ? ?
1 1 2 2 1 2 1 2
, , 1, 1 x y x y x x y y ? ? ? ? ? ? and ? ? ? ? ,, k x y kx ky ? is a
vector space.
07

Q.2 (a)
(1) Find the inverse of
234
4 3 1
1 2 4
A
??
??
?
??
??
??
using Gauss-Jordan method, if exists.
03
(2) Determine whether
3
VR ? is an inner product space under the inner product
1 1 2 2 3 3
, 2 4 u v uv u v u v ? ? ? .
04
(b)
Evaluate ?
S
F ndS ?
??
using Gauss divergence theorem where
2
? ??
43 F xzi xyz j zk ? ? ? over the region bounded by the cone
2 2 2
z x y ?? and
plane 4 z ? , above the xy plane.
07

Q.3 (a)
(1) Find the directional derivative of
23
xy yz ??? at ? ? 2, 1,1 ? in the direction of
the normal to the surface
2
log 4 x z y ? ? ? at ? ? 1,2,1 ? .
03
(2) Show that
? ? ? ? ? ?
2 2 2
? ??
F x yz i y zx j z xy k ? ? ? ? ? ? is conservative. Find its
scalar potential ? .
04
(b)
Let
22
: T M R ? be a linear transformation for which ? ? ? ?
12
1, 2, T v T v ??
? ? ? ?
34
3, 4 T v T v ?? where
1 2 3 4
1 0 1 1 1 1 1 1
, , ,
0 0 0 0 1 0 1 1
v v v v
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
,
Find
ab
T
cd
??
??
??
and
12
34
T
??
??
??
.
07

Q.4 (a)
(1) If
? ??
r xi yj zk ? ? ? , show that
? ? ? ? 3
nn
div r r n r ?? .
03
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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? I & II (OLD) EXAMINATION ? WINTER 2019
Subject Code: 110015 Date: 01/01/2020

Subject Name: Vector Calculus And Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 (a)
(1) Find the Rank of
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
?? ??
??
??
?? ??
??
??
by row echelon form.
03
(2) Solve the following system of equation by Gauss elimination method.
29
2 4 3 1
3 6 5 0
x y z
x y z
x y z
? ? ?
? ? ?
? ? ?
.
04
(b)
Determine whether the set V of all pairs of real numbers ? ? , xy with the
operations ? ? ? ? ? ?
1 1 2 2 1 2 1 2
, , 1, 1 x y x y x x y y ? ? ? ? ? ? and ? ? ? ? ,, k x y kx ky ? is a
vector space.
07

Q.2 (a)
(1) Find the inverse of
234
4 3 1
1 2 4
A
??
??
?
??
??
??
using Gauss-Jordan method, if exists.
03
(2) Determine whether
3
VR ? is an inner product space under the inner product
1 1 2 2 3 3
, 2 4 u v uv u v u v ? ? ? .
04
(b)
Evaluate ?
S
F ndS ?
??
using Gauss divergence theorem where
2
? ??
43 F xzi xyz j zk ? ? ? over the region bounded by the cone
2 2 2
z x y ?? and
plane 4 z ? , above the xy plane.
07

Q.3 (a)
(1) Find the directional derivative of
23
xy yz ??? at ? ? 2, 1,1 ? in the direction of
the normal to the surface
2
log 4 x z y ? ? ? at ? ? 1,2,1 ? .
03
(2) Show that
? ? ? ? ? ?
2 2 2
? ??
F x yz i y zx j z xy k ? ? ? ? ? ? is conservative. Find its
scalar potential ? .
04
(b)
Let
22
: T M R ? be a linear transformation for which ? ? ? ?
12
1, 2, T v T v ??
? ? ? ?
34
3, 4 T v T v ?? where
1 2 3 4
1 0 1 1 1 1 1 1
, , ,
0 0 0 0 1 0 1 1
v v v v
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
,
Find
ab
T
cd
??
??
??
and
12
34
T
??
??
??
.
07

Q.4 (a)
(1) If
? ??
r xi yj zk ? ? ? , show that
? ? ? ? 3
nn
div r r n r ?? .
03
2
(2) Show that
? ? ? ? ? ?
22
? ??
3 2 3 2 3 2 2 F y z yz x i xz xy j xy xz z k ? ? ? ? ? ? ? ? ? is
both solenoidal and irrotational.
04
(b)
Let
3
R have the Euclidean inner product. Use Gram-Schmidt process to
transform the basis ? ?
1 2 3
,, u u u into an orthonormal basis. Where ? ?
1
1,0,0 u ? ,
? ?
2
3,7, 2 u?? and ? ?
3
0,4,1 u ? .
07

Q.5 (a)
(1) Find a basis for the subspace of
2
P spanned by the vectors
2
1, xx ? ,
2
22x ??
, 3x ? .
03
(2) Determine whether the linear transformation
? ? ? ?
2
1
: , , T R P T a b a a b x ? ? ? ? is one-to-one and onto.
04
(b)
Verify Stokes? theorem for ? ? ? ?
? ??
F x y i y z j xk ? ? ? ? ? and S is the surface of
the plane 22 x y z ? ? ? which is in the first octant.
07

Q.6 (a) (1) Find the least-square solution of the linear system Ax b ? given by
12
12
12
7
0
27
xx
xx
xx
??
? ? ?
? ? ? ?
.
03
(2) Determine whether b is in the column space of A , and if so, express b as a
linear combination of the column vectors of A if
1 1 2
1 0 1
2 1 3
A
??
??
?
??
??
??
,
1
0
2
b
? ??
??
?
??
??
??
.
04
(b)
Verify Cayley-Hemilton theorem for the matrix
2 1 1
1 2 1
1 1 2
A
? ??
??
? ? ?
??
?? ?
??
and hence
find
1
A
?
. Also express
6 5 4 3 2
6 9 2 12 23 9 A A A A A A I ? ? ? ? ? ? as a linear
polynomial in A .
07

Q.7 (a)
(1) Evaluate
C
F dr ?
?
along the parabola
2
yx ? between the point ? ? 0,0 and
? ? 1,1 where
2
??
F x i xyj ?? .
03
(2) (i) If ? ?
2 3 2
, , 3 f x y z x y y z ?? , find grad f at the point ? ? 1, 2, 1 ?? .
(ii) Find unit normal vector to the surface
22
28 x y xz ?? at the point ? ? 1,0, 2
.
04
(b)
Find a matrix P that diagonalizes
0 0 2
1 2 1
1 0 3
A
? ??
??
?
??
??
??
.
07

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