GTU Vector Calculus and Linear Algebra - December 2015 Exam Question Paper

GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a)1 Eigen values of A^-1 & A^T are same if matrix A is
(a) Symmetric
(b) Orthogonal
(c) skew symmetric
(d) None of these

1 M

1(a)2 Rank of 4 × 4 invertible matrix is
(a)1
(b)2
(c)3
(d)4

1 M

1(a)3 F is solenoidal vector, If div(F) is
(a) F
(b) 1
(c) 0
(d) -1

1 M

1(a)4 Let A be a hermition matrix, then A is
(a) A* (b) A
(c) A^T
(d) -A*

1 M

1(a)5 If

A = [\begin{matrix} 1 & 0 \\ 2 & - 1 \end{matrix}]

$A=\begin{bmatrix} 1 & 0\\ 2 & -1 \end{bmatrix}$ , then Eigen values of A³ are
(a) 1,-1
(b) 0,2
(c) 1,1
(d) 0,8

1 M

1(a)6 Which set from s₁={a₀+a₁x+a₂x²/a₀=0} and s₂={a₀+a₁x+a₂x²/a₀/ ≠0} is subspace of p₂?
(a) s₂
(b) s₁
(c) s₁ & s₂
(d) none of these

1 M

1(a)7 For which value of k vectors u= (2, 1, 3) and v= (1, 7, k) are orthogonal?
(a) -3
(b) -1
(c) 0
(d) 2

1 M

1(b)1 Let T : R³ → R³ be one to one linear transformation then the dimension of ker(T) is
(a)0
(b) 1
(c) 2
(d)3

1 M

1(b)2 The column vector of an orthogonal matrix are
(a) orthogonal
(b) orthonormal
(c) dependent
(d) none of these

1 M

1(b)3 If r = xi+yj+zk then div (r) is
(a) r
(b) 0
(c) 1
(d) 3

1 M

1(b)4 The number of solution of the system of equation AX=0 (where A is a singular matrix) is
(a) 0
(b) 1
(c) 2
(d) infinite

1 M

1(b)5 If the value of line integral does not depend on path C then F is
(a) solenoidal
(b) incompressible
(c) irrotational
(d) none of these

1 M

1(b)6 A Cayley-Hamilton theorem hold for ______ matrices only
(a) singular
(b) all square
(c) null
(d) a few rectangular

1 M

1(b)7 If

A = [\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}]

$A=\begin{bmatrix} 1 & 2\\ 2 & 4 \end{bmatrix}$ , then rank of matrix A is
(a) 1
(b) 0
(c) 2
(d) 4

1 M

2(a) Determine whether the vector field

u = y^{2} \hat{i} + 2 x y \hat{j} - z^{2} \hat{k}

$u=y^2 \hat{i}+2xy\hat{j}-z^2\hat{k}$ is solenoidal at a point (1, 2, 1).

3 M

2(b) Prove that the matrix

A = [\begin{matrix} - 1 & 2 + i & 5 - 3 i \\ 2 - i & 7 & 5 i \\ 5 + 3 i & - 5 i & 2 \end{matrix}]

$A=\begin{bmatrix} -1 & 2+i & 5-3i\\ 2-i & 7 & 5i\\ 5+3i & -5i & 2 \end{bmatrix}$ is a Hermition and iA is a skew Hermition matrix.

4 M

2(c) For which value of ? and k the following system have (i) no solution x + y + z = 6
(ii) unique solution (iii) an infinite no. of solution. x + 2 y + 3 z = 10 , x + 2 y + ? z = k

7 M

3(a) Find the rank of the matrix

[\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{matrix}]

$\begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{bmatrix}$

3 M

3(b) Find the inverse of matrix

[\begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix}]

$\begin{bmatrix} 0 & 1 & 2\\ 1 & 2 & 3\\ 3 & 1 & 1 \end{bmatrix}$ by Gauss-Jordan method

4 M

3(c) Consider the basis S = {v₁, v₂} for R² where v_{1_{= (-2, 1), v₂ = (1, 3) and let T : R² → R³ be the linear transformation such that T(v₁)= (-1, 2, 0), T(v₂) = (0, 3, 5). Find a formula for T(x₁, x₂) and use the formula to find T(2, -3).}}

7 M

4(a) Express p(x) = 7+8x+9x² as linear combination of
p₁ = 2+x+4x², p₂ = 1-x+3x², p₃ = 2+x+5x².

3 M

4(b) Solve the system by Gaussian elimination method
x+y+z = 6
x+2y+3z = 14
2x+4y+7z = 30

4 M

4(c) Let R³ have standard Euclidean inner product. Transform the basis S = { v₁ , v₂, v₃ } into an orthonormal basis using Gram-Schmidt Process where v₁ = (1,1,1), v₂ = (-1,1,0), v₃ = (1,2,1).

7 M

5(a) Find the nullity of the matrix

[\begin{matrix} 2 & 0 & - 1 \\ 4 & 0 & - 2 \\ 0 & 0 & 0 \end{matrix}]

$\begin{bmatrix} 2 & 0 & -1\\ 4 & 0 & -2\\ 0 & 0 & 0 \end{bmatrix}$

3 M

5(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

[\begin{matrix} 2 & - 2 \\ 1 & 1 \\ 3 & 1 \end{matrix}] b = [\begin{matrix} 2 \\ - 1 \\ 1 \end{matrix}]

$\begin{bmatrix} 2 & -2\\ 1 & 1\\ 3 & 1 \end{bmatrix}b=\begin{bmatrix} 2\\ -1\\ 1 \end{bmatrix}$

4 M

5(c) Show that

s = {[\begin{matrix} 1 & 2 \\ 1 & - 2 \end{matrix}], [\begin{matrix} 0 & - 1 \\ - 1 & 0 \end{matrix}], [\begin{matrix} 0 & 2 \\ 3 & 1 \end{matrix}], [\begin{matrix} 0 & 0 \\ - 1 & 2 \end{matrix}]}

$s=\left \{ \begin{bmatrix} 1 & 2\\ 1 & -2 \end{bmatrix},\begin{bmatrix} 0 & -1\\ -1 & 0 \end{bmatrix},\begin{bmatrix} 0 & 2\\ 3 & 1 \end{bmatrix},\begin{bmatrix} 0 & 0\\ -1 & 2 \end{bmatrix} \right \}$ is a basis for M₂₂.

7 M

6(a) Verify Pythagorean theorem for the vectors u = (3, 0, 1, 0, 4, -1) and V = (-2, 5, 0, 2, -3, -18)

3 M

6(b) Find the unit vector normal to surface x² y + 2xz = 4 at the point (2, -2, 3).

4 M

6(c) Verify Green's theorem for

\bar{F} = x^{2} \hat{i} + x y \hat{j}

$\bar{F}=x^2\hat{i}+xy\hat{j}$ under the square bounded by x = 0, x = 1, y = 0, y = 1.

7 M

7(a) Find curl F at the point (2, 0, 3), if F =

z e^{2 x y} \hat{i} + 2 x y \cos y \hat{j} + (x + 2 y) \hat{k}

$ze^{2xy}\hat{i}+2xy \cos y\hat{j}+(x+2y)\hat{k}$

3 M

7(b) Show that the set V=R³ with the standard vector addition and scalar multiplication defined as c(u₁, u₂, u₃ )=(0,0,cu₃ ) is not vector space.

4 M

7(c) Use divergence theorem to evaluate

\int \int_{s} (x^{3} d y d z + x^{2} y d z d x + x^{2} z d x d z)

$\int \int _s(x^3dydz+x^2 ydzdx+x^2 zdxdz)$ where S is the closed surface consisting of the cylinder x² + y² = a and the circular discs z=0and z=b.

7 M

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