Vector Calculus and Linear Algebra - May 2016

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)(i) Which of the following is orthogonal to (1, 2, -3)?
(a) (3, 6, 3)   (b) (-3, 6, 3)   (c) (-3, 6, -3)   (d) (-3, -3, 6)(1 marks) 1(a)(ii) If λ = 3 ,2 are eigen values of 2 × 2 matrix A , then one of the eigen value of A⁴ is
(a) 0   (b) 3   (c) 9   (d) 81(1 marks) 1(a)(iii) Which of the following is not a subspace of R²?
(a) {0}   (b) line y = 5x   (c) line y = 3x+2   (d) R²(1 marks) 1(a)(iv) Rank of the matrix $\begin{bmatrix} 5 & -3 & 4\\\\ 0 & 2 & 9\\\\ 0 & 0 & -6 \end{bmatrix}$ is
(a) 0   (b) 1   (c) 2   (d) 3(1 marks) 1(a)(v) If A is an 5 × 6 matrix and rank of A is 4 then nullity of A is
(a) 0   (b) 1   (c) 2   (d) 3(1 marks) 1(a)(vi) If A is any square matrix then, A + A^T
(a) symmetric   (b) skew symmetric   (c) orthogonal   (d) none of these(1 marks) 1(a)(vii) Which of the following is not an elementary matrix?
(a) $ \begin{bmatrix} 1 & 0\\\\ 0 & 1 \end{bmatrix} $   (b) $ \begin{bmatrix} -1 & 0\\\\ 0 & -1 \end{bmatrix} $   (c) $ \begin{bmatrix} -1 & 0\\\\ 0 & 1 \end{bmatrix} $   (d) $ \begin{bmatrix} 0 & 1\\\\ 1 & 0 \end{bmatrix} $(1 marks) 1(b)(i) If r = xi + yj + zk then curl (r) is
(a) 1   (b) 2   (c) 0   (d) none of these(1 marks) 1(b)(ii) If ϕ = xyz, then the value of |gradϕ| at (1, 2, -1) is
(a) 0   (b) 1   (c) 2   (d) 3(1 marks) 1(b)(iii) The set {(0, 0) , (1, 0)} is
(a) linearly independent
(b) linearly dependent
(c) basis of R²
(d) none of these(1 marks) 1(b)(iv) If eigen values of a 3 × 3 matrix A are -1, 0, 1 the trace ( A ) is
(a) 0   (b) 1   (c) -1   (d) none of these(1 marks) 1(b)(v) Dimension of P₃ = {a+bx+cx²+dx³ : a, b, c, d &isin R} is
(a) 1   (b) 2   (c) 3   (d) 4(1 marks) 1(b)(vi) If u and v are nonzero orthogonal vectors in R² with Euclidian inner product then
(a) ||u+v||² = ||u||² + ||v||²
(b) ||u+v||² = 2||u||² + 2||v||²
(c) ||u+v||² = ||u||² + 2||v||²
(d) ||u+v||² = 2||u||² + ||v||²(1 marks) 1(b)(vii) If det A ≠ 0 then
(a) AX = 0 has no solution
(b) AX = 0 has unique solution
(c) AX = 0 has infinitely many solution
(d) none of these(1 marks) 2(a) Express (5, -1, 9) as a linear combination of
v₁ = (2, 9, 0), v₂ = (3, 3, 4), v₃ = (1, 2, 1).(3 marks) 2(b) Let u = (u₁, u₂), v = (v₁, v₂) ∈ R². Check whether (u, v) defined as (u, v) = 4u₁v₁+6u₂v₂ is an inner product on R²?(4 marks) 2(c) Solve
x₁ + x₂ + 2x₃ - 5x₄ = 3
2x₁ + 5x₂ - x₃ - 9x₄ = -3
2x₁ + x₂ - x₃ + 3x₄ = -11
x₁ - 3x₂ + 2x₃ + 7x₄ = -5
Using Gauss Jordan method.(7 marks) 3(a) Find the inverse of the matrix $ A=\begin{bmatrix} 1 & 2 & 3\\\\ 2 & 5 & 3\\\\ 1 & 0 & 8 \end{bmatrix} $(3 marks) 3(b) Find the basis of column space of the matrix $$\begin{bmatrix} 1 & -3 & 4 & -2 & 5 & 4\\ 1 & -6 & 9 & -1 & 8 & 2\\ 2 & -6 & 9 & -1 & 9 & 7\\ -1 & 3 & -4 & 2 & -5 & -4 \end{bmatrix}$$
Hence, find the rank of the matrix(4 marks) 3(c) Determine linear transformation T : R² → R³ such that T(1, 0) = (1, 2, 3) and T(1, 1) = (0, 1, 0). Also find T(2, 3)(7 marks) 4(a) Check whether the function T : R² → R² given by the formula T(x, y) = (x + 2y, 3x ' y) is linear transformation or not.(3 marks) 4(b) Check whether set of following matrices is linearly dependent? $$\left \{ \begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix},\begin{bmatrix} 2 & 3\\ 1 & 2 \end{bmatrix},\begin{bmatrix} 3 & 1\\ 2 & 1 \end{bmatrix}, \begin{bmatrix} 2 & 2\\ 1 & 1 \end{bmatrix}\right \}$$.(4 marks) 4(c) Show that the following set is basis for P₃.
{1 + 4x ' 2x², 2x + x², -3 + x + x², 5 ' 2x ' 3x³(7 marks) 5(a) Find eigen values of $ A=\begin{bmatrix} -5 & 4 & 34\\\\ 0 & 0 & 4\\\\ 0 & 0 & 4 \end{bmatrix}. $ Is A invertible?(3 marks) 5(b) State why the following set are not vector space
(i)       V = R² with the operation
      (x₁, y₁) + (x₂, y₂) = (x₁ + y₁ + 1, x₂ + y₂ + 1)
      k(x, y) = (kx, ky)
(ii)       V = { p ∈ P₂ : p(0) = 1} with the usual operation.(4 marks) 5(c) Find eigenvalues and basis for eigenspace for the matrix $$A=\begin{bmatrix} 3 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 3 \end{bmatrix}.$$(7 marks) 6(a) Find curl F, if F = (y² cos x + z³)i + (2y sin x ' 4)j + 3xz²k. Whether F is irrotational?(3 marks) 6(b) Find the directional derivative of f(x, y, z) = x³ - xy² - z at (1, 1, 0) in the direction of 2i ' 3j + 6k(4 marks) 6(c) For which value of ' a ' will the following system have
(i) No solution?, (ii) Unique solution? (iii) Infinitely many solution.
          x + 2y ' 3z = 4
          3x ' y + 5z = 2
          4x + y + (a² - 14)z = a + 2(7 marks) 7(a) Find the unit normal to the surface z² = 4(x² + y²) at a point (1, 0, 2).(3 marks) 7(b) If F = (2xy + z³)i + x²j + 3xz²k. Show that $ \int _C F.dr $ is independent of path of integration. Hence find the integral when C is many path joining (1, -2, 1) and (3, 1, 4)(4 marks) 7(c) Verify Green's theorem for the function F = (x + y)i + 2xyj and C is the rectangle in the xy ' plane bounded by x = 0, y = 0, x = a, y = b.(7 marks)