Question Paper: Vector Calculus & Linear Algebra : Question Paper May 2016 - First Year Engineering (Semester 2) | Gujarat Technological University (GTU)
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## 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 + AT
(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)