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Code No: C1BS01
I B.Tech (CCC) Supplementary Examinations, May/Jun 2007 MATHEMATICS-I ( Common to Civil Engineering, Electrical & Electronic Engineering, Mechanical Engineering, Electronics & Communication Engineering and Computer Science & Engineering) Time: 3 hours Max Marks: 100 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Test the convergence of the series 4 2 3 1 + 12 + 21 .. 43 + 12 .. 34 .. 56 + . . . .
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(b) Examine whether the following series is absolutely convergent or conditionally convergent 1 √ − 5√1 3 + 5√1 4 − . . . . . + (−1)n 5√1 n + . . . . [6] 5 2 (c) Verify Cauchy mean value theorem for sinx and cosx in (a,b)
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2. (a) Show that the function u = x+y+z , v = x2 +y2 +z2 -2xy-2zx-2yz and w = x3 +y3 +z3 -3xyz are functionally related. Find the relation between them. (b) Find the envelope of the family of circles x2 + y2 – 2ax Cosθ – 2ay Sinθ = c2 . ( θ is a parameter ) [10+10] 3. (a) Trace the Folium of Decartes : x3 + y3 = 3axy. (b) Determine the volume of the solid generated by revolving the limacon r = a + b cosθ (a>b) about the initial line. [10+10] 4. (a) Solve the differential equation: (x+1)y1 – y = e3x (x+1)2 (b) Solve the differential equation: y00 + 4y0 + 4y = 4cosx + 3sinx, y(0)= 1, y0 (0)=0. [10+10] 5. (a) Evaluate L{et (cos2t + 1/2 sinh2t)} (b) Find the inverse Laplace Tranformations of
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4 (s+1)(s+2)
i
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(c) Evaluate the integral ∫ ∫ ∫ xy2 z dxdydz taken through the positive octant of the sphere x2 + y2 + z2 = a2 . [8] 6. Verify Stoke’s theorem for F = xi+z2 j+y2 k over the plane surface x+y+z=1 lying in the first octant. [20] 7. (a) Find the non singular matrices P and Q such that normal form PAQ is in the 1 2 3 −2 of the matrix and find the rank of matrix A = 2 −2 1 3 [10] 3 0 4 1 (b) Show that the only real value of λ for which the following equations have non trivial solution is 6 and solve them, when λ = 6. x +2y + 3z = λx; 3x + y + 2z = λy; 2x + 3y + z = λz. [10] 1 of 2
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Code No: C1BS01 8. (a) Show that eigen values of a Hermitian matrix are all real. (b) Identify the nature, index, and signature of the quadratic form x21 + 4x22 + x23 − 4x1 x2 + 2x3 x2 − 4x2 x3 . ?????
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