Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st And 2nd Sem (New And SPFU) 2110015 Vector Calculus And Linear Algebra Previous Question Paper
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 2110015 Date: 01/06/2019
Subject Name: Vector Calculus & Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Question No.1 is compulsory. Attempt any four out of remaining six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 Objective Question (MCQ)
(a) 07
1.
The matrix is in the form
(a) Row
echelon.
(b) Reduced row
echelon.
(c) Both (a) and (b). (d) None.
2.
For the | A
k
| =_____
(a) 1 (b) 2 (c) 2
k
(d) 2
k-1
3. If u and v are vectors in a real inner product space, and ||u||=2, ||v||=3,
then |
(a) 6 (b) 3 (c) 2 (d) 1.5
4. Which of the following doesn?t lie in the space spanned by cos
2
x and sin
2
x ?
(a) 1 (b) 0 (c) Sin x (d) Cos 2x
5. Dimension of the subspace { p(x) P2 : p(0) = 0 } of P2={a+bx+cx
2
: a, b, c R} is
(a) 3 (b) 2 (c) 1 (d) 0
6. Which of the following subsets of R
2
is linearly dependent?
(a) {(1,2), (2,1)} (b) {(1,2), (2,1), (1,1)} (c) {(1,2)} (d) None
7. Let T: R
2
R
2
defined by T(x,y) = (x,0) then Ker (T) =____
(a) Y-axis (b) X-axis (c) Origin (d) None
(b) 07
1. Which of the following is not an elementary matrix?
(a)
(b)
(c)
(d)
2. For = (1, ?1, 2) , = (1, 3, 1) are vectors of R
3
with Euclidean inner product then
cos =______, where is the angle between the two vectors.
(a) 1 (b) 0 (c) ? 3 (d) 6
3. Which of the following is not true?
(a) (AB)
T
= B
T
A
T
(b) (AB)
?1
= B
?1
A
?1
(c) (A
T
)
T
= A (d) A
T
= ?A
4. If A is n?n matrix having rank n?1 then A, A
2
,A
3
, ???.., A
k
,?. have common
eigenvalue _____
(a) 1 (b) ?1 (c) 0 (d) 2
5. If A is unitary matrix then A
?1
=____
(a) A (b) A
2
(c) A
T
(d) I
6. The dimension of the solution space of x ? y = 0 is ____
(a) 0 (b) 1 (c) 2 (d) 3
7. If f(x,y,z) = xyz then Curl ( grad f ) = ___
(a) 0 (b) x (c) xi+yj+zk (d) xyz
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Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 2110015 Date: 01/06/2019
Subject Name: Vector Calculus & Linear Algebra
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Question No.1 is compulsory. Attempt any four out of remaining six questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 Objective Question (MCQ)
(a) 07
1.
The matrix is in the form
(a) Row
echelon.
(b) Reduced row
echelon.
(c) Both (a) and (b). (d) None.
2.
For the | A
k
| =_____
(a) 1 (b) 2 (c) 2
k
(d) 2
k-1
3. If u and v are vectors in a real inner product space, and ||u||=2, ||v||=3,
then |
(a) 6 (b) 3 (c) 2 (d) 1.5
4. Which of the following doesn?t lie in the space spanned by cos
2
x and sin
2
x ?
(a) 1 (b) 0 (c) Sin x (d) Cos 2x
5. Dimension of the subspace { p(x) P2 : p(0) = 0 } of P2={a+bx+cx
2
: a, b, c R} is
(a) 3 (b) 2 (c) 1 (d) 0
6. Which of the following subsets of R
2
is linearly dependent?
(a) {(1,2), (2,1)} (b) {(1,2), (2,1), (1,1)} (c) {(1,2)} (d) None
7. Let T: R
2
R
2
defined by T(x,y) = (x,0) then Ker (T) =____
(a) Y-axis (b) X-axis (c) Origin (d) None
(b) 07
1. Which of the following is not an elementary matrix?
(a)
(b)
(c)
(d)
2. For = (1, ?1, 2) , = (1, 3, 1) are vectors of R
3
with Euclidean inner product then
cos =______, where is the angle between the two vectors.
(a) 1 (b) 0 (c) ? 3 (d) 6
3. Which of the following is not true?
(a) (AB)
T
= B
T
A
T
(b) (AB)
?1
= B
?1
A
?1
(c) (A
T
)
T
= A (d) A
T
= ?A
4. If A is n?n matrix having rank n?1 then A, A
2
,A
3
, ???.., A
k
,?. have common
eigenvalue _____
(a) 1 (b) ?1 (c) 0 (d) 2
5. If A is unitary matrix then A
?1
=____
(a) A (b) A
2
(c) A
T
(d) I
6. The dimension of the solution space of x ? y = 0 is ____
(a) 0 (b) 1 (c) 2 (d) 3
7. If f(x,y,z) = xyz then Curl ( grad f ) = ___
(a) 0 (b) x (c) xi+yj+zk (d) xyz
Q.2 (a) Which of the following are linear combination of u = (0, ?2, 2) and v = (1, 3, ?1)?
Justify! (i) (2,2,2), (ii) (0, 4, 5)
03
(b) Using Gram-Schmidt orthogonalization process find the corresponding orthonormal
set to { (1, 1, 1), (0, 1, 1), (0, 0, 1)}.
04
(c) Using Gauss- Jordan elimination find the inverse of .
07
Q.3
(a)
Find the rank of the matrix and basis of the null space of .
03
(b)
Solve the system of linear equations using Gauss elimination method:
x + y + 2z = 8, ? x ? 2y + 3z = 1, 3x ? 7y +4z = 10.
04
(c) Show that the set of all real numbers of the form (x, 1) with operations
(x, 1) + (x ?, 1) = (x + x ?, 1) and k(x, 1) = (kx, 1) forms a vector space.
07
Q.4 (a) Determine whether the following are linear transformation or not?
(i) T: P2
P2 , T(p(x)) = p(x + 1),
(ii) T: P2
P2 , T(a + bx + cx
2
) = (a + 1) + (b + 1)x + (c + 1) x
2
.
03
(b)
Which of the following sets of vectors of R
3
are linearly independent? Justify.
(i) { (4, ?1, 2), (?4, 10, 2)} (ii) {(?3, 0, 4), (5, ?1, 2), (1, 1, 3)}
04
(c) Find the eigenvalues and bases for the eigenspaces for A
11
,
07
Q.5 (a) Find basis of kernel and range of T: R
2
R
2
, defined by
T( x, y) = ( 2x ? y, ?8x + 4y)
03
(b)
Which of the following are basis of R
3
? Justify!
(i) { (1, 0, 0), (2, 2, 0), (3, 3, 3) } , (ii) { (3, 1, ?4), (2, 5, 6), (1, 4, 8)}
04
(c) Let T: P2
P2 , defined by T(p(x)) = p(3x ? 5)
(i) Find the matrix of T with respect to the basis {1, x, x
2
}.
(ii) Use the indirect procedure using matrix to compute T(1 + 2x + 3x
2
).
(iii) Check the result in (b) by computing T(1 + 2x + 3x
2
) directly.
07
Q.6
(a)
Show that is irrotational.
03
(b)
Find the directional derivative of f(x, y, z) = x
2
z + y
3
z
2
?xyz at (1,1,1) in the direction
of the vector (?1,0,3).
04
(c)
Using Green?s theorem evaluate (3x
2
? 8y
2
) dx + (4y ? 6xy) dy, where C is the
boundary of the region bounded by y
2
= x and y = x
2
.
07
Q.7 (a) Find the work done by = (y ? x
2
) i + (z ? y
2
) j + (x ? z
2
) k over the curve
r(t) = t i + t
2
j + t
3
k; , from (0,0,0) to (1,1,1).
03
(b)
Use Cramer?s rule to solve: x + 2z = 6, ? x + 4y + 6z = 30, ? x ? 2y + 3z = 8. 04
(c) Verify divergence theorem for = x i + yj + zk over the sphere x
2
+ y
2
+ z
2
= a
2
. 07
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This post was last modified on 20 February 2020