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GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
December 2014
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

Choose the appropriate answer for the following MCQs.
1 (a) 1 If $$|\vec{a} \times \vec{b}| \ = \ \vec{a} \cdot \vec{b}$$ then the angle between two vectors $$\vec{a}$$ and $$\vec{b}$$ is
(a) 30° (b) 45° (c) 60° (d) 90°
1 M
1 (a) 2 If $$\bar{a}=2\bar{i}-3\bar{j}+\bar{k}$$ then $$|\bar{a}|=$$
$$(a) \ \sqrt{-4} \ (b) \ \sqrt{4} \ (c) \ \sqrt{13}\ (d) \ \sqrt{14}$$
1 M
1 (a) 3 If $$\bar{F}$$ is conservative then
$$(a) \ \nabla \times \bar{F}=0 \ \ (b) \ \nabla \times \bar{F} \ne 0 \ \ (c) \ \nabla \bar{F} = 0 \ \ (d) \ \nabla \cdot \bar{F}=0$$
1 M
1 (a) 4 $If \ A=\begin{bmatrix} 1 &-2 \\0 &1 \end{bmatrix}$ then the determine of A is
(a) -2 (b) 1 (c) -1 (d) 0
1 M
1 (a) 5 $If \ A=\begin{bmatrix} -5 &-3 \\2 &1 \end{bmatrix}$ then the determine of A is
(a) 0 (b) -1 (c) 1 (d) 2
1 M
1 (a) 6 The characteristic equation for the matrix $A= \begin{bmatrix} 2 &0 \\4 &1 \end{bmatrix}$ $(a) \ (\lambda -2)^2=0 \ \ (b) \ \lambda +2 =0 \ \ (c) \ (\lambda - 2 )(\lambda +2) =0 \ \ (d) \ \lambda -2=0$
1 M
1 (a) 7 If A is a matrix with 5 columns and mulity of A=2 then rank (A) is
(a) 5 (b) 2 (c) 3 (d) 4
1 M
1 (b) 1 $If \ |\bar{a}+ \bar{b}| = |\bar{a}- \bar{b}|$ then the angle between two vectors a and b
(a) 30° (b) 45° (c) 60° (d) 90°
1 M
1 (b) 2 If $$\vec{r}=x\vec{i}+y\vec{j}+ z \vec{k}$$, then the divergence of $$\vec{r}$$ is
(a)  2  (b) -2  (c) 3  (d)  -3
1 M
1 (b) 3 If A and kA have same rank then what can be said about k? (a) zero (b) non-zero (c) positive (d) negative
1 M
1 (b) 4 If V is a vector space having a basis B with n elements then dim(V) =
(a) < n (b) > n (c) n (d) none of these
1 M
1 (b) 5 For a n×n matrix A, Which one of the following statements does not imply the other?
(a) A is not invertible (b) det ( A ) ≠ 0 (c) rank ( A ) =n (d) λ=0 is not an eigen-value of A
1 M
1 (b) 6 If a complex number λ0 is an eigen value of 2×2 real matrix A, then which one of the following is not true?
(a) λ is also an eigen-value of A (b) det (A) ≠ 0 (c) rank (A)=2 (d) A is not invertible
1 M
1 (b) 7 If a 3×3 matrix A is diagonalizable then which one of the following is true?
(a) A has 2 distinct eigen-values.
(b) A has 2 linearly independent eigen-vectors.
(c) A has 3 linearly independent eigen-vectors.
(d) none of these
1 M

2 (a) $Show \ that \ A= \begin{bmatrix} \cos \theta & - \sin \theta &0 \\ \sin \theta &\cos \theta &0 \\0 &0 &1 \end{bmatrix} \ is \ orthogonal$
3 M
2 (b) Is T:R3→R3 defined by T(x,y,z)=(x+3y,y,z+2x) linear? Is it one-to-one, onto or both? Justify.
4 M
2 (c) Define rank of a matrix. Determine the rank of the matrix $A= \begin{bmatrix}3 &4 &5 &6 &7 \\4 &5 &6 &7 &8 \\5 &6 &7 &8 &9 \\10 &1 1&12 &13 &14 \\15 &16 &17 &18 &19 \end{bmatrix}$
7 M

3 (a) $Find \ A^{-1} \ for \ A=\begin{bmatrix} 1 &0 &1 \\-1 &1 &1 \\0 &1 &0 \end{bmatrix} , \ if \ exits$
3 M
3 (b) Obtain the reduced row echelon from of the matrix $A=\begin{bmatrix} 1 &3 &2 &2 \\1 &2 &1 &3 \\2 &4 &3 &4 \\3 &7 &4 & 8 \end{bmatrix}$ and hence find the rank of the matrix A.
4 M
3 (c) State rank-nullity theorem. Also verify it for the linear transformation T: R3 → R2 defined by T ( x , y , z ) = ( x + y + z , x + y ).
7 M

4 (a) $If \ A= \begin{bmatrix} 1 &2 &-3 \\0 &2 &3 \\0 &0 &2 \end{bmatrix}$ then find the eigen values of AT and 5A
3 M
4 (b) Solve the system of linear equations by Cramer's Rule:
x+2y+z=5
3x-y+z=6
x+y+4z=7
4 M
4 (c) Verify Green's Theorem in the plane for $\oint_C(3x^2-8y^2)dx+(4x-6xy)dy$ where C is the boundary of the region defined by y2=x and x2=y
7 M

5 (a) Find grad (φ), if φ =log (x2+y2+z2) at the point (1, 0, -2).
3 M
5 (b) Find the angle between the surfaces x2+y2+z2 and x2+y2-z=3 at the point (2, -1, 2).
4 M
5 (c) 1 Let T: R2 → R3 be the linear transformation defined by T(x,y)=(y, -5x+13y, -7x+16y). Find the matrix for the transformation T with respect to the basic $B= \left \{ (3,1)^T, \ (5,5)^T \right \}$ for R2 and $B= \left \{(1, 0, -1)^T, (-1, 2, 2)^T, (0, 1, 2)^T \right \}$ for R3
5 M
5 (c) 2 Find a basis for the orthogonal complement of the subspace W of R3 defined as W = {(x,y,z) in R3 | -2x + 5y - z = 0}
2 M

6 (a) Show that $\bar{F} = (y^2 -z^2 +3yz - 2x) \bar{i} \\+ (3xz + 2xy) \bar{j} + (3xy - 2xz+2z)\bar{k}$ is both solenoidal and irrotational.
3 M
6 (b) A vector field is given by $\bar{F} = (x^2 +xy^2)\bar{i} + (y^2 + x^2 y) \bar{j}$ Find the scalar potential.
4 M
6 (c) 1 Show that the set of all pairs of real number of the form (1, x) with the operations defined as $(1, x_1)+ (1, x_2)= (1,x_1+x_2) \ and k(1, x)= (1, kx)$
5 M
6 (c) 2 Verify Caylay-Hamilton Theorem for the matrix $A = \begin{bmatrix} 1 &-1 &2 \\0 &2 &1 \\0 &1 &-1 \end{bmatrix}$
2 M

7 (a) Find a basis for the subspace of P2 spanned by the vectors 1+x, x2, -2+2x2, -3x
3 M
7 (b) Let R3 have the Euclidean inner product. Transform the basis = S { ( 1, 0, 0 ) , ( 3, 7, -2 ) , ( 0, 4,1 ) } into an orthonormal basis using the Gram-Schmidt ortho-normalization process
4 M
7 (c) Evaluate $\iint_s \bar{F}\cdot \bar{n}dS$ where $\bar{F} =yz\bar{i}+ xz\bar{j}+ xy \bar{k}$ and S is the surface of the sphere $x^2 +y^2 + z^2=1$ in the first octant.
7 M

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