National Institute of Technology 2011-2nd Sem B.Tech Mechanical Engineering S2 Engineering Mathematics - Winter 010-11 (end semester) . - Question Paper
NATIONAL INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MATHEMATICS
S2 B-TECH END SEMESTER exam APRIL 2011
This is IInd semester Engineering Mathematics ques. paper.Duration of the examination was three hours.maximum marks was 50.syllabus for examination was linear algebra and vector calculus.
NATIONAL INSTITUTE OF TECHNOLOGY CALICUT
DEPARTMENT OF MATHEMATICS S2 B.TECH END SEMESTER EXAMINATION (WINTER SEMESTER) APRIL 2011
Max.marks : 50
Time: 3 Hours
lFind the rate of change of <p = 4x2 + y2 -16z at (2,4,2) in the direction of a normal to the plane x+2y+2z = 9 . What is the maximum rate of change of 0 at (2,4,2)? (3)
2.2. _2\ 1/2
Prove that Vr" = n rn 2 r where r = xi + yj + zk and r = (x+y+z)1'* . (3)
" 2
5Find the work done when a force F =(x - y~ + x)i - (2xy + y) j moves a particle in the XY
2
plane from (0,0) to (1,1) along the parabola y = x. Is the work done different when the path is a
(3)
straight line y = x ? Justify your answer.
Verify Greens theorem for \(xy2 - 2xy)dx + (x2y + 3)dy around the curve C bounded by
c
y2 =8x and x=2 in the XY Plane.
(4)
SrtfProve that every gradient field F , of a scalar function 0 has Curl F = 0.
rffls there a vector field G for which Curl G = xi + yj + zk ? Give reason. (3)
/rTVerify Stokes Theorem for F - { y - z + 4)i + (yz+3)j - xz k and S is an open surface of the cube formed by the planes x=0, x=3, y=0, y=3, and z=3 above the XY plane. (4)
7rTf the eigen values of a square matrix are -1, 2 and 5 find the eigen values of (5A + A2)1 (2)
XDbtain a basis and the dimension of the solution space of the homogenous system of equations
x 1 -X2+2X33x4+ X5 = 0
(3)
X3+ X4+ X5 = 0.
-9r1f u,v and vv are linearly independent vectors show that n + \\ v + w and u + are also linearly independent.
(3)
P.T.O
10 Find k such that the linear transformation 7:/?'-R defined by
T(\ v z) = (kx + y + 2z, x- y- 2z, x + y + 4z) is of Nullity Zero. For k =1 is T invertible? If so,
(4)
find T
1 2. 4 -10 3 3 1 -2
Hamilton
(2)
J2rLet T : R
*R} defined by T(a,b,c) = (a - b + c, 2a + b-c, -a-2b + 2c) be a linear
transformation. Obtain bases for the Range and Null space and hence verify Rank Nullity Theorem.
(4)
Define a real inner product space. ii)Leti7 = (ui,u2) and v = (vi,v2) R and define (ii, v) = 2 uiv u(v2 + u2v,+2 U2V2. Is this
function an inner product on R ? Give reason.
(3)
J4rUsing Gram Schmidt Orthogonalization process find an orthonorma] basis for the space spanned by (1,1,0), (1,0,1), (0,1,1) using dot product as the inner product.
(3)
'tSTDefin.'? Hermitian, Unitary and Orthogonal matrix. Show that eigen-values of a skew
hemiitian matrix are either zero or purely imaginary.
(3)
16. Find the rank, index and signature of the Quadratic form Q = 10 X|"+ 2 x2"+ 15 X3 + 6*7X3 -10X|X24X|X3 .
(3)
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