Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st Sem And 2nd Sem Old 110015 Vector Calculus And Linear Algebra Previous Question Paper
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110015 Date: 01/06/2019
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) (i) Solve the following system by Gauss-Jordan elimination.
3 2 15
5 3 2 0
3 3 11
6 4 2 30
x y z
x y z
x y z
x y z
? ? ? ?
? ? ?
? ? ?
? ? ? ?
(ii) Verify Cauchy-Schwarz inequality for the vectors ( 3,1,0) ? and (2, 1,3) ? .
05
02
(b)
(i) Find the inverse of A=
1 2 3
2 5 3
1 0 8
??
??
??
??
??
.
(ii) For which value of k are ( , ,1) u k k ? and ( ,5,6) vk ? orthogonal?
05
02
Q.2 (a) (i) Use Cramer?s rule to solve the following system.
26
3 4 6 30
2 3 8
xz
x y z
x y z
??
? ? ? ?
? ? ? ?
(ii) Find the rank of the following matrix.
1 2 1
242
3 6 3
A
??
??
?
??
??
??
05
02
(b)
(i) Prove that
n
R is a vector space with the standard operations defined for
n
R .
(ii) Determine whether the set of all matrices of the form
0
ab
c
??
??
??
is a subspace of
22
M or not.
05
02
Q.3 (a) (i) Let
1
(1,2,1) v ? ,
2
(2,9,0) v ? and
3
(3,3,4) v ? . Show that the set
1 2 3
{ , , } S v v v ? is a basis for
3
R .
(ii) Determine whether the vectors
1
( 1,1,1) v ?? ,
2
(2,5,0) v ? and
3
(0,0,0) v ? of
3
R are linearly independent or linearly dependent .
05
02
(b)
(i)Let
3
R have the Euclidean inner product. Transform the basis
{(1,1,1),(0,1,1),(0,0,1)} into an orthogonal basis using gram-Schmidt process.
(ii) Find the eigenvalues of A and
2
A where
20
05
A
??
?
??
??
.
05
02
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?I &II (OLD) EXAMINATION ? SUMMER-2019
Subject Code: 110015 Date: 01/06/2019
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) (i) Solve the following system by Gauss-Jordan elimination.
3 2 15
5 3 2 0
3 3 11
6 4 2 30
x y z
x y z
x y z
x y z
? ? ? ?
? ? ?
? ? ?
? ? ? ?
(ii) Verify Cauchy-Schwarz inequality for the vectors ( 3,1,0) ? and (2, 1,3) ? .
05
02
(b)
(i) Find the inverse of A=
1 2 3
2 5 3
1 0 8
??
??
??
??
??
.
(ii) For which value of k are ( , ,1) u k k ? and ( ,5,6) vk ? orthogonal?
05
02
Q.2 (a) (i) Use Cramer?s rule to solve the following system.
26
3 4 6 30
2 3 8
xz
x y z
x y z
??
? ? ? ?
? ? ? ?
(ii) Find the rank of the following matrix.
1 2 1
242
3 6 3
A
??
??
?
??
??
??
05
02
(b)
(i) Prove that
n
R is a vector space with the standard operations defined for
n
R .
(ii) Determine whether the set of all matrices of the form
0
ab
c
??
??
??
is a subspace of
22
M or not.
05
02
Q.3 (a) (i) Let
1
(1,2,1) v ? ,
2
(2,9,0) v ? and
3
(3,3,4) v ? . Show that the set
1 2 3
{ , , } S v v v ? is a basis for
3
R .
(ii) Determine whether the vectors
1
( 1,1,1) v ?? ,
2
(2,5,0) v ? and
3
(0,0,0) v ? of
3
R are linearly independent or linearly dependent .
05
02
(b)
(i)Let
3
R have the Euclidean inner product. Transform the basis
{(1,1,1),(0,1,1),(0,0,1)} into an orthogonal basis using gram-Schmidt process.
(ii) Find the eigenvalues of A and
2
A where
20
05
A
??
?
??
??
.
05
02
2
Q.4 (a) Determine the algebraic and geometric multiplicity of
0 1 1
1 0 1
1 1 0
A
??
??
?
??
??
??
07
(b)
(i) Let
1, 2
() u u u ? and
1, 2
() v v v ? be vectors in
2
R . Verify that the weighted
Euclidean inner product
1 1 2 2
, 3 2 u v uv u v ? ? ? ? satisfies the four inner product
axioms.
(ii) Let
4
R have the Euclidean inner product. Find the cosine of the angle ?
between the vectors (4,3,1, 2) u?? and ( 2,1, 2,3) v?? .
04
03
Q.5 (a)
(i) Consider the basis
12
{ , } S v v ? for
2
R , where
1
( 2,1) v ?? and
2
(1,3) v ? and let
23
: T R R ? be the linear transformation such that
1
( ) ( 1,2,0) Tv ?? and
2
( ) (0, 3,5) Tv ?? . Find the formula for
12
( , ) T x x . Using it, find (2, 3) T ? .
07
(b)
Let
64
:
A
T R R ? be multiplication by
1 2 0 4 5 3
3 7 2 0 1 4
2 5 2 4 6 1
4 9 2 4 4 7
A
?? ??
??
?
??
?
?? ?
??
? ? ?
??
.
Find the rank and nullity of
A
T .
07
Q.6 (a)
(i) Find the directional derivative of
2 2 2
( , , ) 2 3 f x y z x y z ? ? ? at the point
(2,1,3) P in the direction the vector
? ?
2 a i k ?? .
(ii)Obtain the reduced row echelon form of the matrix
1 1 2 1 1
2 1 2 2 2
1 2 4 1 1
3 0 0 3 3
A
? ? ? ??
??
???
??
?
?? ??
??
??
??
03
04
(b)
(i) Find the gradient of
3 2 2 1
( , , ) 2 3( ) tan ( ) f x y z z x y z xz
?
? ? ? ? at (1,1,1) .
(ii)Find the curlF at the point (2,0,3) where
2
? ??
2 cos ( 2 )
xy
F ze i xy y j x y k ? ? ? ? .
04
03
Q.7 (a)
(i) Prove that
2 3 2
? ??
( cos ) (2 sin 4) 3 F y x z i y x j xz k ? ? ? ? ? is irrotational and find
its scalar potential.
(ii) State Divergence theorem.
05
02
(b) State Green?s theorem and using it, evaluate
?
?? ( 3 ?? 2
? 8 ?? 2
) ?? ?? + ( 4 ?? ? 6 ?? ?? ) ?? ??
where C is the boundary of the region bounded by
2
yx ? and
2
yx ? .
07
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This post was last modified on 20 February 2020