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

GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
May 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) Is the matrix

[\begin{matrix} 1 & 0 & 4 & 6 \\ 0 & 1 & 5 & 7 \\ 0 & 0 & 0 & 1 \end{matrix}]

$\begin{bmatrix}1 &0 &4 &6 \\0 &1 &5 &7 \\0 &0 &0 &1 \end{bmatrix}$ in now echelon form or reduced row-echelon form?

2 M

1 (a) (ii) Find the rank of the matrix

[\begin{matrix} 8 & 0 & 0 & 16 \\ 0 & 0 & 0 & 6 \\ 0 & 9 & 9 & 9 \end{matrix}]

$\begin{bmatrix}8 &0 &0 &16 \\0 &0 &0 &6 \\0 &9 &9 &9 \end{bmatrix}$

2 M

1 (b) (i) Find the inverse of the matrix

[\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}]

$\begin{bmatrix}1&2 \\ 1 & 3 \end{bmatrix}$

2 M

1 (b) (ii) Find the Eigen values of the matrix A^T. Is A an invertible?

A = [\begin{matrix} 1 & 3 & 4 \\ 0 & 0 & 5 \\ 0 & 0 & 9 \end{matrix}]

$A= \begin{bmatrix}1 & 3 & 4 \\ 0& 0& 5 \\ 0&0&9 \end{bmatrix}$

2 M

1 (c) (i) Express the following quadratic forms in matrix notation.
i) x²-4xy+y²
ii) 2x²+6xy-5y²

3 M

1 (c) (ii) If ϕ=3x²y-y³z², find grad ϕ at the point (1, -2, -1).

3 M

2 (a) (i) Find the length of the arc of the curve y=log sec x from

x = 0 t o x = \frac{π}{3}

$x=0 \ to \ x= \dfrac{\pi}{3}$

3 M

2 (a) (ii) Prove that

\iint_{S} \vec{F} \cdot \hat{n} d S = 3 v, i f \vec{F} = x \hat{i} + y \hat{j} + z \hat{k}

$\iint_S \overrightarrow{F} \cdot \widehat{n}dS = 3v , \ if \ \overrightarrow{F} = x\widehat{i} + y \widehat{j}+ z \widehat{k}$ where S is any closed surface enclosing volume V.

3 M

2 (b)

I f \vec{F} = 3 x y \hat{i} - y^{2} \hat{j}; e v a l u a t e \int_{c} \vec{F} \cdot \vec{d} r .

$If \ \overrightarrow{F}= 3xy \widehat{i} - y^2 \widehat{j}; \ evaluate \ \int_c \overrightarrow{F}\cdot \overrightarrow{d}r.$ Where C is the arc of the Parabola y=2x² from (0,0) to (1,2).

4 M

2 (c) Show that

\vec{F} = (y^{2} - z^{2} + 3 y z - 2 x) \hat{i} + (3 x z + 2 x y) \hat{j} + (3 x y - 2 x z + 2 z) \hat{k}

$\overrightarrow{F} = (y^2 - z^2 + 3yz - 2x)\widehat{i} + (3xz + 2xy)\widehat{j}+ (3xy-2xz + 2z)\widehat{k}$ is both solenoidal and irrotational.

4 M

3 (a) Solve the following homogeneous system of equations by using Gauss-Jorfan elimination
2x₁+2x₂-x₃+x₅=0,
-x₁-x₂+2x₃-3x₄+x₅=0,
x₁+x₂-2x₃-x₅=0,
x₃+x₄+x₅=0

5 M

3 (b) Use Cramer's rule to solve:-
x+5y=2
11_x+y+2_z=3
x+5_y+2_z=1

5 M

3 (c) What conditions must b₁, b₂, b₃ satisfy in order for the system of equations to be consistent?
x₁+2_x+3_x=b₁
2x₁+5x₂+3x₃=b₂
x₁+ 8_x3=b₃

4 M

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

[\begin{matrix} a & 1 \\ 1 & b \end{matrix}]

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

[\begin{matrix} a & 1 \\ 1 & b \end{matrix}] + [\begin{matrix} c & 1 \\ 1 & b \end{matrix}] = [\begin{matrix} a + c & 1 \\ 1 & b + d \end{matrix}]

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

[\begin{matrix} a & 1 \\ 1 & b \end{matrix}] = [\begin{matrix} k a & 1 \\ 1 & k b \end{matrix}]

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

6 M

4 (b) Determine which of the following are subspaces of R³.
(1) all the vector of the form (a,b,c) where b=a+c.
(2) all the vector of the form (a,b,c) where b=a+c+1.

4 M

4 (c) Let v₁=1-3x+2x², v₂=1-x+4x², v₃=1-7x. Show that the set S= {v₁, v₂, v₃} is a basis for P₂.

4 M

5 (a) Use the inner product

(f, g) = \int_{0}^{1} f (x) g (x) d x;

$(f,g)= \int^1_0 f(x) g(x) dx;$ to compute d(f,g) and ||f||. Where f(x) = x, g(x)=e^x in C[0,1].

4 M

5 (b) Let R² have the Euclidean inner product. Use the Gram Schmidt process to transform the basis {u₁, u₂, u₃} into an orthonormal basis u₁=(1,1,1), u₂=(-1,1,0), u₃=(1,2,1)

5 M

5 (c) Verify that the vectors u₁=(1,-1,2,-1) u₂=(-2,2,3,2), u₃=(1,2,0,-1), u₄=(1,0,0,1) form an Orthogonal basis for R⁴ with the Euclidean inner product. Also convert it to an orthonormal set by normalizing the vectors.

5 M

6 (a) Find the eigen values of the eigenspaces of the matrix

[\begin{matrix} 0 & 0 & - 2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{matrix}]

$\begin{bmatrix}0 &0 &-2 \\1 &2 &1 \\1 &0 &3 \end{bmatrix}$

5 M

6 (b) Find matrix P that diagonalizes

A = [\begin{matrix} 1 & 0 \\ 6 & - 1 \end{matrix}] .

$A = \begin{bmatrix}1&0 \\ 6 & -1 \end{bmatrix} .$ Also determine P^-1 AP.

5 M

6 (c) Using Cayley Hamilton theorem, find A^-1;

A = [\begin{matrix} 1 & 4 \\ 2 & 3 \end{matrix}]

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

4 M

7 (a)

L e t T_{A} : R^{6} \to R^{4} b e m u l t i p l i c a t i o n b y A = [\begin{matrix} - 1 & 2 & 0 & 4 & 5 & - 3 \\ 3 & - 7 & 2 & 0 & 1 & 4 \\ 2 & - 5 & 2 & 4 & 6 & 1 \\ 4 & - 9 & 2 & - 4 & - 4 & 7 \end{matrix}]

$Let \ T_A: R^6 \to R^4 \ be \ multiplication \ by \\ A= \begin{bmatrix}-1 &2 &0 &4 &5 &-3 \\3 &-7 &2 &0 &1 &4 \\2 &-5 &2 &4 &6 &1 \\4 &-9 &2 &-4 &-4 &7 \end{bmatrix}$ Find the rank and nullity of T_A

5 M

7 (b) Consider the basis S={v₁, v₂} for R², where v₁=(-2,1) and v₂=(-1,3). Let T:R²→R³, be the linear operator such that T(v₁)= (-1, 2, 0) and T(v₂)=(0, -3, 5) find the formula T(x₁, x₂), and use that formula to find T(2, -3).

5 M

7 (c) Show that the following functions are linear transformations.
i) T:R² → R², where T(x,y)=(x+3y, 3x-y)
ii) T:R² → R², where T(x,y)=(2x-y, x-y)

4 M

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