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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+2y-3z=4,
3x-y+5z=2,
4x+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 Wσ.
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 2x-2y=2, x+y=-1, 3x+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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