Question Paper: Vector Calculus and Linear Algebra : Question Paper Jun 2014 - First Year Engineering (Semester 2) | Gujarat Technological University (GTU)
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Vector Calculus and Linear Algebra - Jun 2014

First Year Engineering (Semester 2)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
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
(1 marks)
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 marks) 1(a).3 Let W = span{cos2,sin2x,cos2x}then the dimension of W is
(a) 0
(b) 1
(c)2
(d)3
(1 marks)
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
(1 marks)
1(a).5 The column vectors of an orthogonal matrix are
(a)Orthogonal
(b) Orthonomal
(c) dependent
(d) none of these
(1 marks)
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
(1 marks)
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
(1 marks)
2(a) Solve the following system of equations using Gauss Elimination method
$$2x_{1}+x_{2}+2x_{3}+x_{4}=6 , 6x_{1}-x_{2}+6x_{3}+12x_{4}=36$$
$$4x_{1}+3x_{2}+3x_{3}-3x_{4}=1 , 2x_{1}+2x_{2}-x_{3}+x_{4}=10$$
(5 marks)
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 marks) 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
(1 marks)
2(b).2 $$\left \| u+v \right \|^{2}-\left \| u- v \right \|^{2}$$ is
(a)<u,v>
(b)2<u,v>
(c) 3<u,v>
(d)4<u,v> </u,v></u,v></u,v></u,v>
(1 marks)
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
(1 marks)
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
(1 marks)
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
(1 marks)
2(b).6 divr is
(a)0
(b)1
(c) 2
(d) 3
(1 marks)
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
(1 marks)
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 marks) 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 marks) 3(b) Find a basis for the subspace of P2 spanned by the vectors $$1+x,x^{2},-2+2x^{2},-3x$$(5 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 <a,b> =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.</a,b>(5 marks) 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 marks) 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 marks) 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 marks) 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 marks) 6(c) Find the directional derivatives of x2y2z2 at (1,1,-1) along a direction equally inclined with coordinates axes.(4 marks) 7(a) Verify Green's Theorem for $$\int_{c} (3x -8y\ltsup\gt2\lt/sup\gt)dx+(4y-6xy)dy$$ where C is the boundary of the triangle with vertices (0, 0) , (1, 0) and (0, 1)(5 marks) 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 marks) 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 marks)