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

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER-I &II (OLD) EXAMINATION — SUMMER-2019

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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.

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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.

3x+2y—z=-15

5x+3y+2z=0

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3x+y+3z=11

—6x—4y+2z=30

(ii) Verify Cauchy-Schwarz inequality for the vectors (=3,1,0) and (2,-1,3).

(b) (i) Find the inverse of A= 1 2 3 2 5 3 1 0 8

(ii) For which value of k are u = (k,k,1)and v =(k,5,6) orthogonal?

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Q.2 (a) (i) Use Cramer’s rule to solve the following system.

X+2z=6

-3x+4y+6z=30

—x—2y+3z=8

(ii) Find the rank of the following matrix.

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A= 1 2 1 1 4 2 3 6 3

(b) (i) Prove that Rⁿ is a vector space with the standard operations defined for Rⁿ.

(ii) Determine whether the set of all matrices of the form a b c d is a subspace of M_nn or not.

Q.3 (a) (i) Let v₁=(1,2,1), v₂=(2,9,0) and v₃=(3,3,4). Show that the set S ={v₁,v₂,v₃} is a basis for R³.

(ii) Determine whether the vectors v₁ =(-1,1,1), v₂ =(2,5,0) and v₃ =(0,0,0) of R³ are linearly independent or linearly dependent .

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(b) (i)Let R³ 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 A^T where A= 2 0 0 0 5 0 0 5 2

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Q.4 (a) (i) Find a matrix P that orthogonally diagonalizes A= 1 1 1 1 0 1 1 1 0

(b) (i) Let u =(u₁,u₂) and v=(v₁,v₂) be vectors in R². Verify that the weighted Euclidean inner product <u,v>=3u₁v₁ +2u₂v₂ satisfies the four inner product axioms.

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(ii) Let R⁴ have the Euclidean inner product. Find the cosine of the angle ? between the vectors u =(4,3,1,-2) and v=(-2,1,2,3).

Q.5 (a) (i) Consider the basis S ={v₁,v₂}for R², where v₁ =(-2,1) and v₂ =(1,3) and let T:R²—> R³ be the linear transformation such that T(v₁)=(-1,2,0)and T(v₂)=(0,-3,5) . Find the formula for T(x₁,x₂) . Using it, find T(2,-3).

(b) Let T₁:R⁵ > R⁴ be multiplication by A= -1 2 0 4 5 3 2 4 6 1 4 -9 2 -4 4 7 7 7 7 7

Find the rank and nullity of T.

Q.6 (a) (i) Find the directional derivative of f(x,y,z)=2x²+3y²+z² at the point P(2,1,3) in the direction the vector a =i —2k.

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(ii)Obtain the reduced row echelon form of the matrix A= 1 -1 2 -1 -1 2 1 -2 -2 2 -1 2 4 1 1 3 0 0 3 3

(b) (i) Find the gradient of f(x,y,z)=2z² —3(x² + y²)z +tan^-1 (xz) at (1,1,1).

(ii)Find the curl F at the point (2,0,3) where F=ze^xzi+xcosyj+(x+2y)k.

Q.7 (a) (i) Prove that F =(y²cosx+z³)i +(2ysinx—4)j + 3xz²k is irrotational and find its scalar potential.

(ii) State Divergence theorem.

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(b) State Green’s theorem and using it, evaluate ?_c (3x² — 8y²)dx + (4y — 6xy)dy where C is the boundary of the region bounded by y² =x and y=x².

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