Download GTU BE/B.Tech 2019 Summer 1st Sem And 2nd Sem Old 110009 Maths Ii Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

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

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Subject Code: 110009 Date: 01/06/2019

Subject Name: Maths - II

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) Find the value of ? so that the equations
x+y+3z=0 ;
2x+y+2z=0

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

(b) Solve the following equations by Gauss elimination and back substitution,

X+y+2z=9

2x+4y-3z=1

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3x+6y-5z=0

Q-2 (a) (i) Obtain the reduced row echelon form of the matrix
1 & 3 & 2 & 2 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 4 & 8

(ii) Find the rank of the matrix, if A= 1 & 2 & 4 & 0 \\ -3 & 1 & -5 & 2 \\ -2 & 3 & -9 & 2

(b) Use row operation to find A^-1 if A= 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8

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Q-3 (a) (i) Find the eigen values and eigen vectors of the matrix, A= 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5

(ii) Using Cayley-Hamilton theorem, find A^-1,if A= 1 & 2 \\ 2 & 1

(b) Find a matrix P that diagonalizes A= 4 & 1 \\ -3 & -1 and hence find Aⁿ.

Q-4 (a) (i) Reduce S ={(1,0,0),(0,1,-1),(0,4,-3),(0,2,0)} to obtain a basis for R³.

(ii) Determine whether or not the vectors {(1,2,2), (2,1,2), (2,2,1)}in R³ are linearly independent.

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(b) Let V= {(x,y)| x, y ? R, y > 0}, let (a,b),(c,d)?V, a?R.
Define (a,b)+(c,d)=(a+c,b·d) and a·(a,b) =(aa,b^a). Prove that V is a vector space.

Q-5 (a) (i) Show that the transformation T: R² ? R², where T(x,y)=(2x-y,x-y) is a linear transformation

(ii) Express the quadratic form Q(x, y) =2x² +3y² +6xy,in matrix notation.

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(b) Find the rank and nullity of the matrix A= 2 & 0 & -1 \\ 4 & 0 & -2 \\ 0 & 0 & 0

Q-6 (a) (i) Find a basis for the orthogonal complement of the subset of R⁴ spanned by the vectors
v₁ =(1,-1,3),v₂ =(5,-4,-4),v₃ =(7,-6,2).

(ii) Determine whether the linear transformation T: R² ? R³, where T(x,y) = (x,y,x+y) is one-one.

(b) Let R³ have the Euclidean inner product. Use the Gram-Schmidt process to transform the basis S = {(1,1,1), (-1,1,0), (1,2,1)}into an Orthonormal basis.

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Q-7 (a) (i) Prove that A= 1 & 3+4i & -2 \\ 3-4i & 2 & 9-7i \\ -2 & 9+7i & 3 is a Hermitian matrix.

(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).

(b) Find the least square solution of the linear system AX =b and find the orthogonal projection of b onto the column space of A, where A= 2 & -2 \\ 1 & 1 \\ 3 & 1 , b= 2 \\ -1 \\ 1

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