Test Paper - Applied Maths - Mumbai University - Chemical Engineering -3rd Semester, Study notes of Mathematics

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TY }{/ U7 (REVISED COURSE) C} {3 Hours ) { Total Marks : 100 : (1) Question No. 1 is compulsory. (2) Attempt any four questions out of remaining six questions. (3) Non-programmable calculator is allowed. (a) Prove that : — 5 : ‘i : Lf sinhS tt] = ———____>>"_____ [sent] = aoaye eyes) (b) Find the constant a, b, c, d, e, if 5 f(Z) = (ax? + bxy? + 3x? + cy? + x) + i(dx’y - 2y® + exy + y) is analytic. (c) Prove that : 5 ly (x) = ke sinhx 2+i 2i te (d) IS A= | . a |i matrix ? 5 i zs Ss 4 (a) Prove that : . 6 1 . J x (1-92) dp (@x) dx = =o Jy(a) - 3 Jyfa) 0 (b) Find- 6 (i) L[sin2t-cost-cosh2t] oe Lt s+2 wy Uf pe (c) Find the non-singular matrix P and Q such that PAQ is in normal form. Hence 8 find rank A also find A“ 1 2 ~2 whereA=}7' 3% 0-2 1 Ha) Prove that : 6 i e~?' sint sin ht of i tes (b) Determine the analytic function f(z) given 3u + 2v = y? — x? + 16 xy 6 ic) Prove that — 8 2 ‘ ; _ [2 3-x"., _3 ( "y, (x) = = 2 sin x x cosx 1 Pe |< ly (ax) dx =i y (a) 2 erties Auer