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Posted Date: 13 Nov 2010 Posted By:: Abin Lesly Member Level: Gold Points: 5 (Rs. 1)
2009 University of Kerala General Combined First and Second Semester B.Tech. Degree Examination, May 2009 (2008 Scheme) - MATHEMATICS I University Question paper
Combined First and Second Semester B.Tech. Degree Examination, May 2009
08 -101 : ENGINEERING MATHEMATICS – I
Answer all questions. Each question carries 4 marks.
1. Find the nth derivative of x2 log 3x.
2. Find the radius of curvature at any point of the cycloid x = a( ? + sin ? ),
y = a(1 – cos ? ).
3.Prove that log(1 + ex) = log2 +2x+8x2–192x4+ ...............
4.Find the value of n if rn r is solenoidal, where r = k ˆ z j ˆ y i ˆ x + + and r = | r | .
5.If F = f ? , find f when F = 2xyz3i ˆ + x2z3 j ˆ + 3x2yz2k ˆ and f (1, –2, 2) = 4.
6. Find the particular integral of (D2 + 6D + 9)y = e–2x x3.
7.If ? is an eigen value of a matrix A, then show that 1/? is the eigen value of A–1.
8.State Cayley-Hamilton theorem and use it to find the inverse of the matrix 1 4 .
PART – B
Answer two questions from each Module. Each question carries 10 marks.
Module – I
9. Show that the evolute of the parabola x2 = 4ay is 4(y – 2a)3 = 27ax2.
10.If k ˆ z j ˆ y i ˆ r = x + + and | r |= r, show that
a) ?rn = nrn–2 r
b) ?2rn = n(n + 1) rn–2.
Module – II
11. Solve (D2 – 4D + 4)y = 8(e2x + sin 2x).
a)Find the inverse Laplace transform of log(S+1)/(S-1).
b)Apply convolution theorem to evaluate L–1(s/(s2+a2)2).
Module – III
13. Test for consistency and solve :
x + 2y + z = 3
2x + 3y + 2z = 5
3x – 5y + 5z = 2
3x + 9y – z = 4.
14.Reduce the quadratic form 6x2 + 3y2 + 3z2 – 2yz + 4zx – 4xy to canonical form
by an orthogonal transformation. Find the matrix of transformation.
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