Download GTU BE/B.Tech 2019 Winter 1st And 2nd Sem New And Spfu 2110015 Vector Calculus And Linear Algebra Question Paper

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Seat No.:

GUJARAT TECHNOLOGICAL UNIVERSITY

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E - SEMESTER- I & II (NEW) EXAMINATION - WINTER 2019

Subject Code: 2110015 Date: 01/01/2020

Subject Name: Vector Calculus And Linear Algebra

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

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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 (a) Objective Questions Marks 07

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1. If A = | 0 9 7 | then trace of the matrix A is | 11 9 8 |
   (a) 12 (b) 18 (c) 72 (d) 16
2. If div u = 0 then u is said to be
   (a) Rotational (b) Solenoidal (c) Compressible (d) None of these
3. If A is 3 x 3 invertible matrix then nullity of A is
   

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   (a) 1 (b) 2 (c) 0 (d) 3
4. If matrix A = [ -1 5 -1 ] is having Eigen values 2,3,6 then Eigen [ 1 -1 3 ]
   values of A^-1 are
   (a) 2,3,6 (b) 1/2, 1/3, 1/6 (c) 1, 1, 1 (d) None of these
5. Which set from S₁ = {(x,y,z) ? R³ / z > 0} and
   

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   S₂ = {(x,y,z) ? R³ / x = z = 0} is subspace of R³.
   (a) S₁ (b) S₂ (c) S₁ and S₂ (d) None.
6. If Eigen values of 3 X 3 matrix A are 5,5,5 then Algebraic multiplicity of
   matrix A is
   (a) 3 (b) 1 (c) 5 (d) 0.

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7. For what values of c, the vector (2, -1, c) has norm 3?
   (a) -3 (b) 3 (c) 0 (d) 2

(b) Objective Questions 07

1. If F(x,y,z) = xi + yj + zk then Curl F is
   (a) 1 (b) 3 (c) 2 (d) 0

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2. For what value of k the vectors v₁ = (-1,2,4) , v₂ = (-3,6, k) are Linearly
   Dependent?
   (a) 12 (b) 7 (c) 4 (d) 1
3. Which one is the characteristic equation of A = [ 3 -1 ] ? [ 5 1 ]
   (a) ?² - 8? + 4 = 0 (b) ?² - 4? - 5 = 0

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4. Which matrix represents one to one transformation
   (a) [ 1 0 ] (b) [ 1 2 1 ] (c) [ 1 1 1 ] (d) [ 2 1 ] [ 0 1 ] [ 1 1 1 ] [ 1 1 1 ] [ 1 2 ]
5. Matrix A = [ 2 -1 3 ] is [ 2 -5 1 ]
   (a) Symmetric (b) Skew-symmetric (c) Hermitian (d) None of these
6. If u and v are vectors in an Inner product space then
   

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   (a) (u, v) = ||u|| ||v|| (b) |(u,v)| < ||u|| ||v||
   (c) |(u,v)| = ||u|| ||v|| (d) None of these.
7. Each vector in R² can be rotated in counter clockwise direction with 90° is
   followed by the matrix,
   (a) [ 1 0 ] (b) [ 0 -1 ] (c) [ 0 1 ] (d) [ 1 1 ] [ 0 1 ] [ 1 0 ] [ -1 0 ] [ 0 0 ]

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Q2 (a)

Find the Rank of a Matrix A = [ 1 -1 2 -1 ] 03 [ 2 1 -2 2 ] [ 1 2 -4 1 ] [ 3 0 0 -3 ]

(b) Determine whether the given vectors v₁ = (2,-1,3); v₂ = (4,1,2); v₃ = 04 (8,-1,8) Span R³.

(c) For which values of ‘a’ will the following system have no solutions? 07 Exactly one solution? Infinitely many solutions?

x + 2y - 3z = 4

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3x - y + 5z = 2

ax + y + (a² - 14)z = a + 2

Q-3 (a) Define Singular Matrix. Find the inverse of the matrix A using Gauss Jordan 03 Method if it is invertible A = [ 1 0 1 ] [ -1 1 -1 ] [ 0 1 0 ]

(b) Express the matrix A = [ 1 5 7 ] as the sum of a symmetric and a 04 [ -1 -2 -4 ] [ 8 2 13 ] skew symmetric matrix

(c) Fine a basis for the nullspace, row space and column space of the matrix 07 A= [ -1 2 -1 5 6 ] [ 4 -4 -4 -12 -8 ] [ 2 0 -6 -2 4 ] [ -3 1 7 -2 12 ]

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Also determine rank and nullity of the matrix.

Q-4 (a) Let T₁: R² ? R², T₂: R² ? R³ be transformation given by 03 T₁(x,y) = (x+y, y) and T₂(x, y) = (2x, y, x + y).

Show that T₁ is linear transformation and also find formula for T₂ ° T₁.

(b) Let T: R² ? R² be the linear operator defined by 04 T(x,y) = (2x - y, x + y). Find ker(T) and R(T).

(c) Find a matrix P that diagonalize A, where A = [ 1 -6 -4 ] 07 [ 0 4 2 ] [ 0 -6 -3 ]

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And determine P^-1AP.

Q-5 (a) Find constants a,b,c so that 03 v = (x+2y+az)i + (bx - 3y - z)j + (4x + cy + 2z)k is irrotational.

(b) Let the vector space P₂ have the inner product (p, q) = ?¹_-1 p(x)q(x) dx 04 (i) Find ||p|| for p = x².

(ii) Find d(p,q) of p = 1 and q = x.

(c) Using Gram-Schmidt process orthonormalize the set of linearly independent 07 vectors u₁ =(1, 0, 1, 1), u₂ =(-1, 0,-1, 1) and u₃ = (0,-1, 1, 1) of R⁴ with standard inner product.

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Q-6 (a) The temperature at any point in space is given by T = xy + yz + zx. 03 Determine the derivative of T in the direction of the vector 3i - 4k at the point (1,1,1).

(b) Find the orthogonal projection of u = (2,1,3) on the subspace of R³ 04 spanned by the vectors v₁ = (1,1,0), v₂ = (1,2,1).

(c) Verify Green’s theorem for the field F = (x - y)i + xj and the region R 07 bounded by the unit circle C: r(t) = (cost)i + (sint)j; 0 < t < 2p

Q-7 (a) Find the co ordinate vector of p = 2 - x + x² relative to the basis 03 S = {p₁, p₂, p₃} where p₁ = 1 + x, p₂ = 1 + x², p₃ = x + x²

(b) Let T: R³ ? R³ be multiplication by A determine whether T has inverse. If 04 A = [ 1 4 -1 ] [ 1 2 1 ] [ -1 1 0 ] so find T^-1(x₁, x₂, x₃).

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(c) Determine whether R⁺ of all positive real numbers with operators 07 x + y = xy and kx = x^k as a Vector Space.

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