 MORE IN Vector Calculus and Linear Algebra
GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
May 2016
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)(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 M
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 M
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 M
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 M
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 M
1(a)(vi) If A is any square matrix then, A + AT
(a) symmetric   (b) skew symmetric   (c) orthogonal   (d) none of these
1 M
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 M
1(b)(i) If r = xi + yj + zk then curl (r) is
(a) 1   (b) 2   (c) 0   (d) none of these
1 M
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 M
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 M
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 M
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 M
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 M
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 M

2(a) Express (5, -1, 9) as a linear combination of
v1 = (2, 9, 0), v2 = (3, 3, 4), v3 = (1, 2, 1).
3 M
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 M
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 M

3(a) Find the inverse of the matrix $$A=\begin{bmatrix} 1 & 2 & 3\\ 2 & 5 & 3\\ 1 & 0 & 8 \end{bmatrix}$$
3 M
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 M
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 M

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 M
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 M
4(c) Show that the following set is basis for P3.
{1 + 4x ' 2x2, 2x + x2, -3 + x + x2, 5 ' 2x ' 3x3
7 M

5(a) Find eigen values of $$A=\begin{bmatrix} -5 & 4 & 34\\ 0 & 0 & 4\\ 0 & 0 & 4 \end{bmatrix}.$$ Is A invertible?
3 M
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 M
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 M

6(a) Find curl F, if F = (y2 cos x + z3)i + (2y sin x ' 4)j + 3xz2k. Whether F is irrotational?
3 M
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 M
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 M

7(a) Find the unit normal to the surface z2 = 4(x2 + y2) at a point (1, 0, 2).
3 M
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 M
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 M

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