Vector Calculus and Linear Algebra - Dec 2015

First Year Engineering (Semester 2)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 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 marks) 1(a)2 Rank of 4 × 4 invertible matrix is
(a)1
(b)2
(c)3
(d)4(1 marks) 1(a)3 F is solenoidal vector, If div(F) is
(a) F
(b) 1
(c) 0
(d) -1(1 marks) 1(a)4 Let A be a hermition matrix, then A is
(a) A* (b) A
(c) A^T
(d) -A*(1 marks) 1(a)5 If $$A=\begin{bmatrix} 1 & 0\\ 2 & -1 \end{bmatrix}$$ , then Eigen values of A³ are
(a) 1,-1
(b) 0,2
(c) 1,1
(d) 0,8(1 marks) 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 marks) 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 marks) 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 marks) 1(b)2 The column vector of an orthogonal matrix are
(a) orthogonal
(b) orthonormal
(c) dependent
(d) none of these(1 marks) 1(b)3 If r = xi+yj+zk then div (r) is
(a) r
(b) 0
(c) 1
(d) 3(1 marks) 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 marks) 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 marks) 1(b)6 A Cayley-Hamilton theorem hold for ______ matrices only
(a) singular
(b) all square
(c) null
(d) a few rectangular(1 marks) 1(b)7 If $$A=\begin{bmatrix} 1 & 2\\ 2 & 4 \end{bmatrix}$$, then rank of matrix A is
(a) 1
(b) 0
(c) 2
(d) 4(1 marks) 2(a) Determine whether the vector field $$u=y^2 \hat{i}+2xy\hat{j}-z^2\hat{k}$$ is solenoidal at a point (1, 2, 1).(3 marks) 2(b) Prove that the 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 marks) 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 marks) 3(a) Find the rank of the matrix $$\begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{bmatrix}$$(3 marks) 3(b) Find the inverse of matrix $$\begin{bmatrix} 0 & 1 & 2\\ 1 & 2 & 3\\ 3 & 1 & 1 \end{bmatrix}$$ by Gauss-Jordan method(4 marks) 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 marks) 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 marks) 4(b) Solve the system by Gaussian elimination method
x+y+z = 6
x+2y+3z = 14
2x+4y+7z = 30(4 marks) 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 marks) 5(a) Find the nullity of the matrix $$\begin{bmatrix} 2 & 0 & -1\\ 4 & 0 & -2\\ 0 & 0 & 0 \end{bmatrix}$$(3 marks) 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
$$\begin{bmatrix} 2 & -2\\ 1 & 1\\ 3 & 1 \end{bmatrix}b=\begin{bmatrix} 2\\ -1\\ 1 \end{bmatrix}$$(4 marks) 5(c) Show that $$s=\left \{ \begin{bmatrix} 1 & 2\\ 1 & -2 \end{bmatrix},\begin{bmatrix} 0 & -1\\ -1 & 0 \end{bmatrix},\begin{bmatrix} 0 & 2\\ 3 & 1 \end{bmatrix},\begin{bmatrix} 0 & 0\\ -1 & 2 \end{bmatrix} \right \}$$ is a basis for M₂₂.(7 marks) 6(a) Verify Pythagorean theorem for the vectors u = (3, 0, 1, 0, 4, -1) and V = (-2, 5, 0, 2, -3, -18)(3 marks) 6(b) Find the unit vector normal to surface x² y + 2xz = 4 at the point (2, -2, 3).(4 marks) 6(c) Verify Green's theorem for $$\bar{F}=x^2\hat{i}+xy\hat{j}$$ under the square bounded by x = 0, x = 1, y = 0, y = 1.(7 marks) 7(a) Find curl F at the point (2, 0, 3), if F = $$ze^{2xy}\hat{i}+2xy \cos y\hat{j}+(x+2y)\hat{k}$$(3 marks) 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 marks) 7(c) Use divergence theorem to evaluate $$\int \int _s(x^3dydz+x^2 ydzdx+x^2 zdxdz)$$ where S is the closed surface consisting of the cylinder x² + y² = a and the circular discs z=0and z=b.(7 marks)