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GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
June 2015
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) 1 The angle between $\overline{u}=(-1,1,2,-2)\ and\ \overline{v}=(2,-1,-13)is$_____
1 M
1 (a) 2 Which of the following matrix is orthogonal?
$a)\begin{bmatrix} \sqrt{3/2} &1/2 \\-1/2 &\sqrt{3/2} \end{bmatrix}\ (b) \begin{bmatrix} 1 &-2 \\2 &1 \end{bmatrix}\ (c) \begin{bmatrix} 1/9 &\sqrt{8/9} \\\sqrt{5/9} &2/9 \end{bmatrix}\ (d) \begin{bmatrix} 2 &3 \\4 &1 \end{bmatrix}$
1 M
1 (a) 3 If a square matrix A is involuntary then A2= _______
(a)A
(b)AT (c) I (d) A-1
1 M
1 (a) 4 A homogeneous system of equations have at least _____solutions
(a) 1
(c)3
(b)2
(d)4.
1 M
1 (a) 5 The ranl of $\begin{bmatrix} 1 &3 &6 \\2 &1 &6 \end{bmatrix}$ is
(a) 1
(c)3
(b)2
(d)No rank.
1 M
1 (a) 6 If 3 is the Eigen value of A then the Eigen value of A + 3I is
(a) 9
(c)0
(b)6
(d)27.
1 M
1 (a) 7 Which of the following is a subspace of R 2 under standard operations
a) R3
(b) P2
(c) M22
(d) R.
1 M
Attempt the following MCQ
1 (b) 1 If R is a vector space then which of the following is a trivial subspace of R?
$a)\ \left \{ \overline {0} \right \}\ (b)\ R \ (c)\ \left \{ \overline{0},\overline{1} \right \}\ (d)\ \left \{ \overline{1} \right \}$.
1 M
1 (b) 2 The set S={1,x,x2spans which of the following?
a) R2
(b) M22
(c) P2
(d) R.
1 M
1 (b) 3 The dimension of the solution space of x-3y=0 is___________
(a) 1
(c)3
(b)2
(d)4.
1 M
1 (b) 4 The mapping T:R3→R3 defined by T(V1,V2,V3)=(V1,V2,0) is called as
(a) Reflection
(c)Rotation
(b)Magnification
(d)Projection.
1 M
1 (b) 5 If $\left \langle \overline{u},\overline{v} \right \rangle=9u_{1}v_{1}+4u_{2}v_{2}$ is the product on R2 then is generated by
$a)\ \begin{bmatrix} 2 &0 \\0 &3 \end{bmatrix}\ (b)\ \begin{bmatrix} 3&0 \\0 &2 \end{bmatrix}\ (c)\ \begin{bmatrix} 0 &2 \\3 &0 \end{bmatrix}\ (d)\ \begin{bmatrix} 0 &3 \\2 &0 \end{bmatrix}$.
1 M
1 (b) 6 The divergence of $\overline{F}=xyzi+3x^{2}yj+(xz^{2}-y^{2}z)k \ at(2,-1,1)$ is
a) yz+3x+2xz
(b) yz+xy
(c) yz+3x2+(2xz-y2)
(d) xy-yz
1 M
1 (b) 7 If $overline{F}$ is conservative field then curl$\overline{F}=$_______
a) i
(b) j
(c) k
(d) $overline{0}$
1 M

2 (a) Is $A=\begin{bmatrix} 2 &2 &1 \\-2 &1 &2 \\1 &-2 &2 \end{bmatrix}$ orthogonal? If not, can it be converted into an orthogonal matrix?
3 M
2 (b) Solve the following system: x+y+z=3,x+2y-z=4,x+3y+2z=4.
4 M
2 (c) 1)Find the rank of the matrix $A=\begin{bmatrix} 1 &2 &4 &0 \\-3 &1 &5 &2 \\-2 &3 &9 &2 \end{bmatrix}$
2) Find the inverse of$A=\begin{bmatrix} 3 &-3 &4 \\2 &-3 &4 \\0 &-1 &1 \end{bmatrix}$ using row operations.
7 M

3 (a) Does $W=\left \{ (x,y,z)/x^{2}+y^{2}+z^{2}=1 \right \}$ a subspace of R3 with the standard operations?
3 M
3 (b) Find the projection of $\overline{u}=(1,-2,3)\ along\ \overline{v}=(2,5,4)in \ R^{3}$.
4 M
3 (c) Find the eigen values and eigen vectors of the matrix $A=\begin{bmatrix} 3 &1 &4 \\0 &2 &6 \\0 &0 &5 \end{bmatrix}$.
7 M

4 (a) Find the least square solution for the system 4x1-3x>sub>2=12,2x1+5x2=32,3 x1+x,sub>2=21.
3 M
4 (b) Determine the dimension and basis for the solution space of the system
x1+2x2+x3+3x4=0,2 x1+5x2+2x3+x4=0,x1+3x2+x3-x4=0.
4 M
4 (c) Check whether V=R2 is a vector space with respect to the operations
(u1,u2)+(v1,v2)=(u1+v1-2u2+v2-3)and
a(u1,u2)=a(u1+2a-2,au2-3a+3),aεR.
7 M

5 (a) Consider the inner product space $P_{2}\ Let\ \overline{P_{1}}=a_{2}x^{2}+a_{1}x+a_{0}\ and\ \overline{P_{2}}=b_{2}x^{2}+b_{1}x+b_{0}\ are\ in\ P_{2},\ where \left \langle \overline{P_{1}},\overline{P_{2}} \right \rangle=a_{2}b_{2}+a_{1}b_{1}+a_{0}b_{0}$.
Find the angle between 2x2-3 and 3x+5.
3 M
5 (b) Find the rank and nullity of the matrix$A=\begin{bmatrix} 2 &0 &-1 \\4 &0 &-2 \\0 &0 &0 \end{bmatrix}$ and verify the dimension theorem.
4 M
5 (c) Let T: R2→R3 be the linear transformation defined by T((x1,x2)=(x2-5x1+13x2,-7x1+16x2). Find the matrix for the transformation T with respect to bases B={(3,1),(5,2)} for R2 and B'={(1,0,-1),(-1,2,2),(0,1,2)} for R3.
7 M

6 (a) Find the arc length of the portion of the circular helix $\overline{r}(t)=\cos ti+\sin tj+tk \ from\ t=0\ to=\pi$.
3 M
6 (b) A vector field is given by $\overline{F}=(x^{2}+xy^{2})i+(y^{2}+x^{2}y)j$. show that $\overline{F}$ is irrotational and find its scalar potential.
4 M
6 (c) i) Let R3 have the Euclidean inner product. Transform the basis {(1,1,1),(1,-2,1),(1,2,3)} into an orthogonal basis using Gram-Schmidt process.
ii) Express the following quadratic form in matrix notation:
2x2+5y2-6z2-6z2-2xy-yz+8zy.
7 M

7 (a) If φ=xyz-2y2z+x2z2, find div (gradφ) at the point (2,4,1).
3 M
7 (b) Use Green's theorem to evaluate $\oint _{c}\limits[x^{2}ydx+y^{3}dy]$, where C is the closed path formed by y=x3 from (0,0) to (1,1).
4 M
7 (c) Verify Stoke's theorem for $\overline{F}=(y-z+2)i+(yz+4)j-xzk$ over the surface of the cube x=0,y=0,z=0,x=2,y=2,z=2 above the xy plane.(that is open at bottom)
7 M

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