STUPIDSID
1(a).1
The number of solutions of the system of equations 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(a).2
Let A be a unitary matrix then 1 A-1 is
(a) A
(b) \(\bar{A}\)
(c) AT
(d)\(\bar{A}\ ^{T}\)
1 M
1(a).3
Let W = span{cos2,sin2x,cos2x}then the dimension of W is
(a) 0
(b) 1
(c)2
(d)3
(a) 0
(b) 1
(c)2
(d)3
1 M
1(a).4
Let P2 be the vector space of all polynomials with d egree less than or equal to two then the dimension of P2 is
(a)1
(b)2
(c) 3
(d)4
(a)1
(b)2
(c) 3
(d)4
1 M
1(a).5
The column vectors of an orthogonal matrix are
(a)Orthogonal
(b) Orthonomal
(c) dependent
(d) none of these
(a)Orthogonal
(b) Orthonomal
(c) dependent
(d) none of these
1 M
1(a).6
Let T:R2 ? R2 be a linear transformation defined by T(x,y) =(y,x) then it is
(a)one to one
(b) onto
(c)both
(d)neither
(a)one to one
(b) onto
(c)both
(d)neither
1 M
1(a).7
Let T:R3 → R3 be a linear transformation defined by T (x,y,z) = (x,z,0)then the dimention of R(T) is
(a)0
(b)1
(c) 2
(d) 3
(a)0
(b)1
(c) 2
(d) 3
1 M
2(a)
Solve the following system of equations using Gauss Elimination method
\[2x_{1}+x_{2}+2x_{3}+x_{4}=6 , 6x_{1}-x_{2}+6x_{3}+12x_{4}=36\]
\[4x_{1}+3x_{2}+3x_{3}-3x_{4}=1 , 2x_{1}+2x_{2}-x_{3}+x_{4}=10\]
\[2x_{1}+x_{2}+2x_{3}+x_{4}=6 , 6x_{1}-x_{2}+6x_{3}+12x_{4}=36\]
\[4x_{1}+3x_{2}+3x_{3}-3x_{4}=1 , 2x_{1}+2x_{2}-x_{3}+x_{4}=10\]
5 M
2(b)
Find the inverse of \[\begin{bmatrix} 1 & 2& 3 &1 \\ 1& 3 & 3 &2 \\ 2& 4 & 3 & 3\\ 1 & 1 & 1 & 1 \end{bmatrix}\] using Gauss Jordan method.
5 M
2(b).1
If\[\left \| u+v \right \|^{2} =\left \| u \right \|^{2}+\left \| v \right \|^{2} \] then u and v are
(a)parallel
(b)perpendicular
(c) dependent
(d)none of these
(a)parallel
(b)perpendicular
(c) dependent
(d)none of these
1 M
2(b).2
\[\left \| u+v \right \|^{2}-\left \| u- v \right \|^{2}\] is
(a)
(b)2
(c) 3
(d)4
(a)
(b)2
(c) 3
(d)4
1 M
2(b).3
Let T:R3rightarrow; R3 be a one to one linear transformation then the dimention of ker(T) is
(a) 0
(b) 1
(c) 2
(d)3
(a) 0
(b) 1
(c) 2
(d)3
1 M
2(b).4
Let A = \[\begin{bmatrix} 2 &1 \\ 2&3 \end{bmatrix}\] then the eigen values of A2 are
(a)1,2
(b) 1,4
(c) 1,6
(d)1,16
(a)1,2
(b) 1,4
(c) 1,6
(d)1,16
1 M
2(b).5
Let A = \[\begin{bmatrix} 2 &1 \\ 2&3 \end{bmatrix}\] then the eigen values of A+3I are
(a) 1,2
(b)2,5
(c) 3,6
(d)4,7
(a) 1,2
(b)2,5
(c) 3,6
(d)4,7
1 M
2(b).6
divr is
(a)0
(b)1
(c) 2
(d) 3
(a)0
(b)1
(c) 2
(d) 3
1 M
2(b).7
If the value of line integral \[\int_{c} \bar{F} .\bar{dr}\] does not depend on path C then \[\bar{F}\] is
(a) solenoidal
(b) incompressible
(c) irrotational
(d) none of these
(a) solenoidal
(b) incompressible
(c) irrotational
(d) none of these
1 M
2(c)
Express \[\begin{bmatrix} 4+2i &7 & 3-i\\ 0& 3i & -2\\ 5+3i&-7+i & 9+6i \end{bmatrix}\] as the sum of a hermitian and a skew-hermitian matrix
4 M
3(a)
Let V be the set of all ordered pairs of real numbers with vector addition defined as \[(x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2}+1,y_{1}+y_{2}+1)\] Show that the first five axioms for vector addition are satisfied. Clearly mention the zero vector and additive inverse.
5 M
3(b)
Find a basis for the subspace of P2 spanned by the vectors \[1+x,x^{2},-2+2x^{2},-3x\]
5 M
3(c)
Express the matrix \[\begin{bmatrix} 5 & 1\\ -1& 9 \end{bmatrix}\] as linear combination of \[\begin{bmatrix} 1& -1\\ 0& 3 \end{bmatrix},\begin{bmatrix} 1&1\\ 0& 2 \end{bmatrix},\begin{bmatrix} 2& 2\\ -1&1 \end{bmatrix}\]
4 M
4(a)
Consider the basis S={v1,v2} for R2 where v1=(1,1)and v2 =(2,3). Let T: R2 → P2 be the linear transformation such that T(v1) =2-3x+x2 and T(v2)=1-x2 then find the formula of T(a,b).
5 M
4(b)
Verify Rank-Nullity theorem for the linear transformation T:R4 → R3 defined by \[x_{1},x_{2},\ x_{3},\ x_{4}=(4x_{1}+x_{2}-2x_{3}-3x_{4},\\ 2x_{1}+x_{2}+x_{3}-4x_{4},\ 6x_{1}-9x_{3}+9x_{4}) \]
5 M
4(c)
Find the algebraic and geometric multiplicity of each of the eigen value of \[\begin{bmatrix} 0 &1 &1 \\ 1& 0 &1 \\ 1&1 &0 \end{bmatrix}\]
4 M
5(a)
For A =\[\begin{bmatrix} a_{1} & b_{1}\\ c_{1}& d_{1} \end{bmatrix}\] and B = \[\begin{bmatrix} a_{2} & b_{2}\\ c_{2}& d_{2} \end{bmatrix}\] Let the inner product on M22 be defined as =a1a2 +b1b2 +c1c2+d1d1.Let A = \[\begin{bmatrix} 2 & 6\\ 1 & -3 \end{bmatrix}\] and B = \[\begin{bmatrix} 3 & 2\\ 1 & 0 \end{bmatrix}\] then verify cauchy-Schwarz inequality and find the angle between A and B.
5 M
5(b)
Let R3 have the inner product defined by <(x1,x2,x3 )y1,y2,y3 )>,=x1y1 + 2x2y2 +3x3y3.Apply the Gram-Schmidt process to transform the vectors (1,1,1),(1,1,0) and (1,0,0) into orthonormal vectors
5 M
5(c)
Find a basis for the orthogonal complement of the subspace spanned by the vectors(2,-1.1.3.0) ,(1,2,0,1,-2),(4,3,1,5,-4),(3,1,2,-1,1) and (2,-1,2,-2,3)
4 M
6(a)
Verify Cayley-Hamilton theorem for A = \[\begin{bmatrix} 6 & -1 & 1\\ -2 & 5& -1\\ 2 & 1 & 7 \end{bmatrix}\] and hence find A4
5 M
6(b)
Show that the vector field \[\sqrt{F} =(ysinz-sinx)I + (xsinz + 2yz)j +(xy cos z+ y^{2})k\] is conservation and find the corresponding scalar potential.
5 M
6(c)
Find the directional derivatives of x2y2z2 at (1,1,-1) along a direction equally inclined with coordinates axes.
4 M
7(a)
Verify Green's Theorem for \[\int_{c} (3x -8y2)dx+(4y-6xy)dy\] where C is the boundary of the triangle with vertices (0, 0) , (1, 0) and (0, 1)
5 M
7(b)
Verify stokes's Theorem for \[\bar{F} = (x+y)I +(y+z)j-xk\] and S is the surface of the plane 2x + y +z =2 which is in the first octant.
5 M
7(c)
Find the work done when a force \[ \bar{F}=(x^2-y^2+x)I-(2xy+y)j \] moves a particle in the XY plane from (0,0) to (1,1) along the parabola y2=x
4 M
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