GTU Vector Calculus and Linear Algebra - December 2013 Exam Question Paper

GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
December 2013

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) Show that the differential form under the integral of

$I= \int^{(2,4,0)}_{(0,-1,1)} e^{x-y+z^2} (dx- dy+2zd)$ is exact in space and evaluate the integral.

5 M

1 (a) (ii) A parametric representation of the surface is given, Identity and sketch the surface;

$\bar{r} (u,v) = a \cos u \widehat{i} + a \sin u \widehat{j}+ v \widehat{k},$ where u, v, vary in the rectangle R:0≤u≤2π, -1 ≤v^le;1.

2 M

1 (b) For which values of 'a' will the following system have no solution? Exactly one solution? Infinitely many solutions?
x+2_y-3_z=4,
3_x-y+5_z=2,
4_x+y+(a²-14)_z=a+2

4 M

1 (c) Let A be the matrix

$\begin{bmatrix}3 &1 \\2 &1 \end{bmatrix} .$ P₁(x)=x²-9, P₂(x)=x+3, and P₃(x)=x-3. Show that P₁(A) P₃(A).

3 M

2 (a) (i) Verify Gauss divergence theorem for

$\bar{F}= 7 x \widehat{i} - z \widehat{k}$ over the sphere x²+y²+z²=4

5 M

2 (a) (ii) Find the directional derivative of f(x,y,z)=xyz at the point p:(-1,1,3) in the direction of the vector

$\bar{a}= \widehat{i}-2\widehat{j}+2\widehat{k}$

2 M

2 (b) Find the inverse of the matrix

$A= \begin{bmatrix}1 &2 &3 \\ &5 &3 \\1 &0 &8 \end{bmatrix}$ using Row operations.

4 M

2 (c) Determine whether the set of all polynomials a₀+a₁x+a₂x²+a₃x³ for which a₀,a₁,a₂ and a₃ are integers, is a subspace of P₃.

3 M

3 (a) (i) Show that the set of all 2×2 matrices of the form

$\begin{bmatrix}a &1 \\1 &b \end{bmatrix}$ with addition defined by

$\begin{bmatrix}a &1 \\ 1 &b \end{bmatrix} + \begin{bmatrix}c &1 \\ 1 &d \end{bmatrix} = \begin{bmatrix}a+c &1 \\ 1 &b+d \end{bmatrix}$ and scalar multiplication defined by

$\begin{bmatrix}a&1 \\ 1 &b \end{bmatrix} = \begin{bmatrix}ka&1 \\ 1 &kb \end{bmatrix}$ is a vector space.

5 M

3 (a) (ii) Find the area of the parallelgram determined by the vectors

$\bar{u}=(2,3,0), \bar{v}=(-1, 2,-2).$

2 M

3 (b) using Green's theorem, evaluate the line integral ∮_c(sin y dx + cos x dy) counter clockwise, where C is the boundary of the triangle with vertices (0,0), (π,0), (π,1)

4 M

3 (c) The velocity vector

$\bar{v}= \overrightarrow{r}(t)=x^3 \widehat{k}$ of a fluid motion is given. Is the flow irrotational? Incompressible? Find the path of the particle.

3 M

4 (a) (i) Let P₁=1+x, P₂=1+x² and P₃=x+x². Show that the set S={P₁, P₂, P₃} is a basis for P₂. Find the coordinate vector of P=2-x+x² with respect to S.

5 M

4 (a) (ii) Use appropriate identities, where required to determine which of the following sets of vectors in F(-∞, ∞) are linearly dependent:
i) x, cos x ii) cos 2x, sin² x, cos²x.

2 M

4 (b) Find the rank of the matrix

$A = \begin{bmatrix}1&4&5&2 \\ 2&1&3&0 \\ -1&3&2&2 \end{bmatrix}$

4 M

4 (c) Use Cramer's rule to solve the system
x₁+3x₂+x₃=4, 2x₁-x₂=-2, 4x₁-3x₃=0

3 M

5 (a) (i) Let W be the space of P⁵ spanned by the vectors

$\bar{v}_1 = (1,4,5,6,9), \ \bar{v}_2 = (3,-2,1,4,-1), \\ \bar{v}_3= (-1,0,-1,-2,-1) , \ \bar{v}_4= (2,3,5,7,8).$ Find a basis for the orthogonal complement of W^σ.

5 M

5 (a) (ii) Sketch the unit circle in an x-y coordinate system in R² using the Euclidean inner product

$\left ( \bar{u}, \bar{v} \right ) = \dfrac {1}{4}u_1v_1 + \dfrac {1}{16}u_2v_2$

2 M

5 (b) Find the least squares solution of the linear system AX=b given by 2_x-2_y=2, x+y=-1, 3_x+y=1. Also find the orthogonal projection of b on the column space of A.

4 M

5 (c) Let R² have the Euclidean inner product. Use Gram Schmidt process to transform the basis vectors

$\bar{u}_1=(1,-3), \ \bar{u}_2=(2,2)$ into an orthogonal basis.

3 M

6 (a) Find a matrix P that diagonalize

$A = \begin{bmatrix}-1 &4 &-2 \\-3 &4 &0 \\-3 &1 &3 \end{bmatrix}$ and determine P^-IAP.

7 M

6 (b) (i) Find the geometric and algebraic multiplicity of each eigen values of

$\begin{bmatrix}2&0 \\ 1&2 \end{bmatrix}$

2 M

6 (b) (ii) Let u=(u₁, u₂), v= (v₁,v₂) be vectors in R². Verify that the weighted Euclidean inner product ( u, v)= 3u₁v₁+5u₂v₂ satisfies the four inner product axioms.

2 M

6 (c) Given the quadratic equation x²-16y²+128y = 256. A translation will put the comic in standard position. Name the comic and give its equation in the translated coordinate system.

3 M

7 (a) (i) Find the standard matrix for the stated composition of linear operators on R²:

(a) A rotation of 60°, followed by an orthogonal projection on the x-axis followed by a reflection about the line y=x.

(b) A dilation with factor k=2, followed by a rotation of 45°, followed by a reflection about the y-axis.

5 M

7 (a) (ii) Determine whether the function T:V→R, where V is an inner product space, and T(u)=||u||, is a linear transformation, Justify your answer.

2 M

7 (b) Let T:R²→R³ be the linear transformation defined by

$T \left ( \begin{bmatrix}x_1\\x_2\end{bmatrix} \right )= \begin{bmatrix}x_2\\-5x_1+13x_2 \\-7x_1 + 16x_2 \end{bmatrix} .$ Find the matrix for the transformation T with respect to the bases

$B=\{\bar{u}_1, \bar{u}_2 \}\ for \ R^2 \ and \ B' = \{ \bar{v}_1, \bar{v}_2, \bar{v}_3 \}\ for \ R^3, \\ where \ \bar{u}_1 = \begin{bmatrix}3 \\1 \end{bmatrix} , \ \bar{u}\begin{bmatrix}5 \\2\end{bmatrix}, \ \bar{v}_1 = \begin{bmatrix}1 \\ 0 \\-1 \end{bmatrix} , \ \bar{v}_2 = \begin{bmatrix}-1 \\ 2 \\ 2 \end{bmatrix}, \ \bar{v}_3 = \begin{bmatrix}0\\ 1 \\2\end{bmatrix}$

3 M

7 (c) Show that the linear operator T:R^→ R² defined by the equations

$w_1=x_1+2x_2 \\ w_2 = -x_1+ x_2$ is one-to-one, and find T^-1 (w₁, w₂).

3 M

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