STUPIDSID
1(a)1
Eigen values of A-1 & AT are same if matrix A is
(a) Symmetric
(b) Orthogonal
(c) skew symmetric
(d) None of these
(a) Symmetric
(b) Orthogonal
(c) skew symmetric
(d) None of these
1 M
1(a)2
Rank of 4 × 4 invertible matrix is
(a)1
(b)2
(c)3
(d)4
(a)1
(b)2
(c)3
(d)4
1 M
1(a)3
F is solenoidal vector, If div(F) is
(a) F
(b) 1
(c) 0
(d) -1
(a) F
(b) 1
(c) 0
(d) -1
1 M
1(a)4
Let A be a hermition matrix, then A is
(a) A* (b) A
(c) AT
(d) -A*
(a) A* (b) A
(c) AT
(d) -A*
1 M
1(a)5
If \[A=\begin{bmatrix}
1 & 0\\
2 & -1
\end{bmatrix}\] , then Eigen values of A3 are
(a) 1,-1
(b) 0,2
(c) 1,1
(d) 0,8
(a) 1,-1
(b) 0,2
(c) 1,1
(d) 0,8
1 M
1(a)6
Which set from s1={a0+a1x+a2x2/a0=0} and s2={a0+a1x+a2x2/a0/ ≠0} is subspace of p2?
(a) s2
(b) s1
(c) s1 & s2
(d) none of these
(a) s2
(b) s1
(c) s1 & s2
(d) none of these
1 M
1(a)7
For which value of k vectors u= (2, 1, 3) and v= (1, 7, k) are orthogonal?
(a) -3
(b) -1
(c) 0
(d) 2
(a) -3
(b) -1
(c) 0
(d) 2
1 M
1(b)1
Let T : R3 → R3 be one to one linear transformation then the dimension of ker(T) is
(a)0
(b) 1
(c) 2
(d)3
(a)0
(b) 1
(c) 2
(d)3
1 M
1(b)2
The column vector of an orthogonal matrix are
(a) orthogonal
(b) orthonormal
(c) dependent
(d) none of these
(a) orthogonal
(b) orthonormal
(c) dependent
(d) none of these
1 M
1(b)3
If r = xi+yj+zk then div (r) is
(a) r
(b) 0
(c) 1
(d) 3
(a) r
(b) 0
(c) 1
(d) 3
1 M
1(b)4
The number of solution of the system of equation AX=0 (where A is a singular matrix) is
(a) 0
(b) 1
(c) 2
(d) infinite
(a) 0
(b) 1
(c) 2
(d) infinite
1 M
1(b)5
If the value of line integral does not depend on path C then F is
(a) solenoidal
(b) incompressible
(c) irrotational
(d) none of these
(a) solenoidal
(b) incompressible
(c) irrotational
(d) none of these
1 M
1(b)6
A Cayley-Hamilton theorem hold for ______ matrices only
(a) singular
(b) all square
(c) null
(d) a few rectangular
(a) singular
(b) all square
(c) null
(d) a few rectangular
1 M
1(b)7
If \[A=\begin{bmatrix}
1 & 2\\
2 & 4
\end{bmatrix}\], then rank of matrix A is
(a) 1
(b) 0
(c) 2
(d) 4
(a) 1
(b) 0
(c) 2
(d) 4
1 M
2(a)
Determine whether the vector field \[u=y^2 \hat{i}+2xy\hat{j}-z^2\hat{k}\] is solenoidal at a point (1, 2, 1).
3 M
2(b)
Prove that the matrix \[A=\begin{bmatrix}
-1 & 2+i & 5-3i\\
2-i & 7 & 5i\\
5+3i & -5i & 2
\end{bmatrix}\] is a Hermition and iA is a skew Hermition matrix.
4 M
2(c)
For which value of ? and k the following system have (i) no solution
x + y + z = 6
(ii) unique solution (iii) an infinite no. of solution. x + 2 y + 3 z = 10 , x + 2 y + ? z = k
(ii) unique solution (iii) an infinite no. of solution. x + 2 y + 3 z = 10 , x + 2 y + ? z = k
7 M
3(a)
Find the rank of the matrix \[\begin{bmatrix}
1 & 2 & 3\\
2 & 3 & 4\\
3 & 4 & 5
\end{bmatrix}\]
3 M
3(b)
Find the inverse of matrix \[\begin{bmatrix}
0 & 1 & 2\\
1 & 2 & 3\\
3 & 1 & 1
\end{bmatrix}\] by Gauss-Jordan method
4 M
3(c)
Consider the basis S = {v1, v2} for R2 where v1 = (-2, 1), v2 = (1, 3) and let T : R2 → R3 be the linear transformation such that T(v1)= (-1, 2, 0), T(v2) = (0, 3, 5). Find a formula for T(x1, x2) and use the formula to find T(2, -3).
7 M
4(a)
Express p(x) = 7+8x+9x2 as linear combination of
p1 = 2+x+4x2, p2 = 1-x+3x2, p3 = 2+x+5x2.
p1 = 2+x+4x2, p2 = 1-x+3x2, p3 = 2+x+5x2.
3 M
4(b)
Solve the system by Gaussian elimination method
x+y+z = 6
x+2y+3z = 14
2x+4y+7z = 30
x+y+z = 6
x+2y+3z = 14
2x+4y+7z = 30
4 M
4(c)
Let R3 have standard Euclidean inner product. Transform the basis S = { v1 , v2,
v3 } into an orthonormal basis using Gram-Schmidt Process where v1 = (1,1,1),
v2 = (-1,1,0), v3 = (1,2,1).
7 M
5(a)
Find the nullity of the matrix \[\begin{bmatrix}
2 & 0 & -1\\
4 & 0 & -2\\
0 & 0 & 0
\end{bmatrix}\]
3 M
5(b)
Find the least square solution of the linear system Ax = b and find the orthogonal projection of b onto the column space of A where
\[\begin{bmatrix} 2 & -2\\ 1 & 1\\ 3 & 1 \end{bmatrix}b=\begin{bmatrix} 2\\ -1\\ 1 \end{bmatrix}\]
\[\begin{bmatrix} 2 & -2\\ 1 & 1\\ 3 & 1 \end{bmatrix}b=\begin{bmatrix} 2\\ -1\\ 1 \end{bmatrix}\]
4 M
5(c)
Show that \[s=\left \{ \begin{bmatrix}
1 & 2\\
1 & -2
\end{bmatrix},\begin{bmatrix}
0 & -1\\
-1 & 0
\end{bmatrix},\begin{bmatrix}
0 & 2\\
3 & 1
\end{bmatrix},\begin{bmatrix}
0 & 0\\
-1 & 2
\end{bmatrix} \right \}\] is a basis for M22.
7 M
6(a)
Verify Pythagorean theorem for the vectors u = (3, 0, 1, 0, 4, -1) and
V = (-2, 5, 0, 2, -3, -18)
3 M
6(b)
Find the unit vector normal to surface x2 y + 2xz = 4 at the point (2, -2, 3).
4 M
6(c)
Verify Green's theorem for \[\bar{F}=x^2\hat{i}+xy\hat{j}\] under the square bounded by x = 0, x = 1, y = 0, y = 1.
7 M
7(a)
Find curl F at the point (2, 0, 3), if F = \[ze^{2xy}\hat{i}+2xy \cos y\hat{j}+(x+2y)\hat{k}\]
3 M
7(b)
Show that the set V=R3 with the standard vector addition and scalar
multiplication defined as c(u1, u2, u3 )=(0,0,cu3 ) is not vector space.
4 M
7(c)
Use divergence theorem to evaluate \[\int \int _s(x^3dydz+x^2 ydzdx+x^2 zdxdz)\] where S is the closed surface consisting of the cylinder x2 + y2 = a and the circular discs z=0and z=b.
7 M
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