EnrolmentNo,
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-I &II (NEW) EXAMINATION - SUMMER-2019
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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:
- Question No.1 is compulsory. Attempt any four out of remaining six questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1 Objective Question (MCQ)
(a) 07
- The matrix \(\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\) is in the form
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(a) Row echelon. (b) Reduced row echelon. (c) Both(a)and(b). (d) None. - For A= \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) the |AK| =
(a) 1 (b) 2 (c) 2K (d) 2kl - If u and v are vectors in a real inner product space, and ||u||=2, ||v||=3, then |<u,v>| <
(a) 6 (b) 3 (c) 2 (d) 1.5 - Which of the following doesn’t lie in the space spanned by cos²x and sin²x ?
(a) 1 (b) 0 (c) Sin x (d) Cos 2x - Dimension of the subspace { p(x) ? P2 : p(0) =0 } of P2={a+bx+cx²:a,b,c?R} is
(a) 3 (b) 2 (c) 1 (d) 0 - Which of the following subsets of R² is linearly dependent?
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(a) { (1,2),(2,1) } (b) { (1,2),(2,1),(1,1) } (c) { (1,2) } (d) None - Let T: R² ? R² defined by T(x,y) = (x,0) then Ker (T) =
(a) Y-axis (b) X-axis (c) Origin (d) None
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(b) 07
- Which of the following is not an elementary matrix?
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(a) \(\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) (b) \(\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}\) (c) \(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) (d) \(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}\) - For a=(1,-1,2); b =(1, 3, 1) are vectors of R³ with Euclidean inner product then cos ?= , where ? is the angle between the two vectors.
(a) 1 (b) 0 (c) -3 (d) 6 - Which of the following is not true?
(a) (AB)?=B?A? (b) (AB)?¹=B?¹A?¹ (c) A?=A (d) A?=-A - If A is nxn matrix having rank n—1 then A, A², A³ .......... , An,.... have common eigenvalue
(a) 1 (b) -1 (c) 0 (d) 2 - If A is unitary matrix then A?¹=
(a) A (b) A² (c) A? (d) I - The dimension of the solution space of x—y =0 is
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(a) 0 (b) 1 (c) 2 (d) 3 - If f(x,y,z) =xyz then Curl (grad f)=__
(a) 0 (b) X (c) xi+yj+zk (d) Xyz
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Q.2
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(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 \(\begin{pmatrix} -1 & 3 & -4 \\ 2 & 4 & 1 \\ -4 & 2 & -8 \end{pmatrix}\) 07
Q.3
(a) Find the rank of the matrix and basis of the null space of \(\begin{pmatrix} 1 & -1 & 3 \\ 5 & -4 & -4 \\ 7 & -6 & 2 \end{pmatrix}\). 03
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(b) Solve the system of linear equations using Gauss elimination method: 04
x+y+2z=8, —x-2y+3z=1, 3x-7y+4z=10.
(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? 03
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(i) T: P2 ? P2, T(p(x)) = p(x + 1),
(ii) T:P2 ? P2, T(a+bx+cx²)=(a+ 1)+ (b+ 1)x+(c+1)x²
(b) Which of the following sets of vectors of R³ are linearly independent? Justify. 04
(i) { (4, -1,2), (-4, 10, 2)} (ii) {(-3,0,4),(5,-1,2),(1,1,3)}
(c) Find the eigenvalues and bases for the eigenspaces for A?¹, 07
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A= \(\begin{pmatrix} -1 & -2 & -2 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{pmatrix}\)
Q.5
(a) Find basis of kernel and range of T: R²?R² defined by 03
T(x,y)=(2x—y, —8x +4y)
(b) Which of the following are basis of R³? Justify! 04
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(i) {(1,0,0),(2,2,0),(3,3,3) } (ii) { (3, 1,4),(2,5,6),(1,4,8)}
(c) Let T:P2 ? P2, defined by T(p(x)) = p(3x — 5) 07
(i) Find the matrix of T with respect to the basis {1, x, x²}.
(ii) Use the indirect procedure using matrix to compute T(1 + 2x + 3x²).
(iii) Check the result in (b) by computing T(1 + 2x + 3x²) directly.
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Q.6
(a) Show that F = (yl—x²)/(x² +y²) is irrotational. 03
(b) Find the directional derivative of f(x, y, z) = x²z + y²z² —xyz at (1,1,1) in the direction of the vector (—1,0,3). 04
(c) Using Green’s theorem evaluate §C (3x² — 8y²) dx + (4y — 6xy) dy, where C is the boundary of the region bounded by y² = x and y = x². 07
Q.7
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(a) Find the work done by F=(y —x²) i+ (z—y²)j + (x — z²) k over the curve r(t)=ti+tj+tk 0=t=1, 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=38. 04
(c) Verify divergence theorem for F = x i + yj + zk over the sphere x²+ y²+ z² = a² 07
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