Vector Calculus and Linear Algebra - Dec 2014

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.

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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 3 (a) $$ Find \ A^{-1} \ for \ A=\begin{bmatrix} 1 &0 &1 \\-1 &1 &1 \\0 &1 &0 \end{bmatrix} , \ if \ exits $$(3 marks) 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 marks) 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 marks) 4 (a) $$ If \ A= \begin{bmatrix} 1 &2 &-3 \\0 &2 &3 \\0 &0 &2 \end{bmatrix} $$ then find the eigen values of A^T and 5A(3 marks) 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 marks) 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 marks) 5 (a) Find grad (φ), if φ =log (x²+y²+z²) at the point (1, 0, -2).(3 marks) 5 (b) Find the angle between the surfaces x²+y²+z² and x²+y²-z=3 at the point (2, -1, 2).(4 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 7 (a) Find a basis for the subspace of P₂ spanned by the vectors 1+x, x², -2+2x², -3x(3 marks) 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 marks) 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 marks)