STUPIDSID
Answer the following MCQ
1 (a) 1
The angle between \[\overline{u}=(-1,1,2,-2)\ and\ \overline{v}=(2,-1,-13)is\]_____
(a) 3 rad
(b)3.68 rad (c) 2.68 rad
(d)2 rad
(a) 3 rad
(b)3.68 rad (c) 2.68 rad
(d)2 rad
1 M
1 (a) 2
Which of the following matrix is orthogonal?
\[a)\begin{bmatrix} \sqrt{3/2} &1/2 \\-1/2 &\sqrt{3/2} \end{bmatrix}\
(b) \begin{bmatrix} 1 &-2 \\2 &1 \end{bmatrix}\
(c) \begin{bmatrix} 1/9 &\sqrt{8/9} \\\sqrt{5/9} &2/9 \end{bmatrix}\
(d) \begin{bmatrix} 2 &3 \\4 &1 \end{bmatrix}\]
\[a)\begin{bmatrix} \sqrt{3/2} &1/2 \\-1/2 &\sqrt{3/2} \end{bmatrix}\
(b) \begin{bmatrix} 1 &-2 \\2 &1 \end{bmatrix}\
(c) \begin{bmatrix} 1/9 &\sqrt{8/9} \\\sqrt{5/9} &2/9 \end{bmatrix}\
(d) \begin{bmatrix} 2 &3 \\4 &1 \end{bmatrix}\]
1 M
1 (a) 3
If a square matrix A is involuntary then A2= _______
(a)A
(b)AT (c) I (d) A-1
(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.
(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.
(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.
(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.
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?
\[a)\ \left \{ \overline {0} \right \}\
(b)\ R \
(c)\ \left \{ \overline{0},\overline{1} \right \}\
(d)\ \left \{ \overline{1} \right \}\].
\[a)\ \left \{ \overline {0} \right \}\
(b)\ R \
(c)\ \left \{ \overline{0},\overline{1} \right \}\
(d)\ \left \{ \overline{1} \right \}\].
1 M
1 (b) 2
The set S={1,x,x2spans which of the following?
a) R2
(b) M22
(c) P2
(d) R.
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.
(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.
(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
\[a)\ \begin{bmatrix} 2 &0 \\0 &3 \end{bmatrix}\
(b)\ \begin{bmatrix} 3&0 \\0 &2 \end{bmatrix}\
(c)\ \begin{bmatrix} 0 &2 \\3 &0 \end{bmatrix}\
(d)\ \begin{bmatrix} 0 &3 \\2 &0 \end{bmatrix}\].
\[a)\ \begin{bmatrix} 2 &0 \\0 &3 \end{bmatrix}\
(b)\ \begin{bmatrix} 3&0 \\0 &2 \end{bmatrix}\
(c)\ \begin{bmatrix} 0 &2 \\3 &0 \end{bmatrix}\
(d)\ \begin{bmatrix} 0 &3 \\2 &0 \end{bmatrix}\].
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
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) \[overline{0}\]
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.
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.
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.
(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.
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.
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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