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

3

4

x+2

_{y}-3_{z}=4,3

_{x}-y+5_{z}=2,4

_{x}+y+(a^{2}-14)_{z}=a+2
4 M

1 (c)
Let A be the matrix \[ \begin{bmatrix}3 &1 \\2 &1 \end{bmatrix} . \] P

_{1}(x)=x^{2}-9, P_{2}(x)=x+3, and P_{3}(x)=x-3. Show that P_{1}(A) P_{3}(A).
3 M

2 (a) (i)
Verify Gauss divergence theorem for \[ \bar{F}= 7 x \widehat{i} - z \widehat{k} \] over the sphere x

^{2}+y^{2}+z^{2}=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 a

_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}for which a_{0},a_{1},a_{2}and a_{3}are integers, is a subspace of P_{3}.
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 P

_{1}=1+x, P_{2}=1+x^{2}and P_{3}=x+x^{2}. Show that the set S={P_{1}, P_{2}, P_{3}} is a basis for P_{2}. Find the coordinate vector of P=2-x+x^{2}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, sin

i) x, cos x ii) cos 2x, sin

^{2}x, cos^{2}x.
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

x

x

_{1}+3x_{2}+x_{3}=4, 2x_{1}-x_{2}=-2, 4x_{1}-3x_{3}=0
3 M

5 (a) (i)
Let W be the space of P

^{5}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 R

^{2}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 R

^{2}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

^{-I}AP.
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=(u

_{1}, u_{2}), v= (v_{1},v_{2}) be vectors in R^{2}. Verify that the weighted Euclidean inner product ( u, v)= 3u_{1}v_{1}+5u_{2}v_{2}satisfies the four inner product axioms.
2 M

6 (c)
Given the quadratic equation x

^{2}-16y^{2}+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 R

(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.

^{2}:(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:R

^{2}→R^{3}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

^{→}R^{2}defined by the equations \[ w_1=x_1+2x_2 \\ w_2 = -x_1+ x_2 \] is one-to-one, and find T^{-1}(w_{1}, w_{2}).
3 M

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