2007 Anna University Chennai B.E Computer Science mathematics1 Question paper
ANNA UNIVERSITY CHENNAI :: CHENNAI – 600 025 B.E / B.TECH. DEGREE EXAMINATIONS – I YEAR ANNUAL PATTERN MODEL QUESTION PAPER MA 1X01  ENGINEERING MATHEMATICS  I (Common to all Branches of Engineering and Technology) Regulation 2004 Time : 3 Hrs Maximum: 100 Marks Answer all Questions PART – A (10 x 2 = 20 Marks) 1. Find the sum and product of the eigen values of the matrix ??????????311151113 2. If x = r cos?, y = r sin?, find ),x(),(yr??? 3. Solve (D3+D2+4D+4)y = 0. 4. The differential equation for a circuit in which selfinductance L and capacitance C neutralize each other is L .022=+Cidtid Find the current i as a function of t. 5. Find, by double integration, the area of circle x2+y2 = a2. 6. Prove that curl grad f = o. 7. State the sufficient conditions for a function f(z) to be analytic. 8. State Cauchy’s integral theorem. 9. Find the Laplace transform of unit step function at t = a. 10. Find L1 [13432+++sss]. 1 PART – B (5 x 16 = 80 marks) 11.(a).(i). Verify CayleyHamilton theorem for the matrix A = . ??????????126216227 Hence find its inverse. (8) (ii). Find the radius of curvature at any point ‘t’ on the curve x = a (cost + t sint), y = a(sintt cost) (8) (OR) (b).(i). Diagonalise the matrix by orthogonal transformation. (8). ??????????342476268 (ii). A rectangular box open at the top is to have volume of 32 c.c. Find the dimensions of the box requiring least material for its construction, by Lagrange’s multiplier method. (8). 12(a). (i). Solve (3x+2)2 +22xdyd3(3x+2) =yddy36x3x2+4x+1 (8) (ii). For the electric circuit gover ned by (LD2+RD+C1) q = E where D = dtd if L = 1 henry, R = 100 Ohms, C = 104 farad and E = 100 volts, q = dtdq= 0 when t = 0, find the charge q and the current i. (8) (OR) (b).(i). Solve 032x=++yxdtd, 3x+teydtdy222=+ (8) 2 (ii). The differential equation satisfied by a beam uniformly loaded (w kg/ metre) with one end fixed and the second end subjected to tensile force P is given by 22221wxPydxdEIy= . Show that the elastic curve for the beam with conditions y = 0 = dxdy at x = 0 is given by y = 2Pnw (1coshnx) + Pwx22where EIPn=2 (8) 13. a.(i). Change the order of integration in 220xxaaxxayddy?? and hence evaluate the same. (8). (ii). Prove that F= (y2cosx + z3)i+(2ysinx4)j+3xz2k is irrotational and find its scalar potential. (8) (OR) b.(i). By changing to polar coordinates, evaluate ??+aayydyd0222xxx (8) (ii). Verify Gauss divergence theorem for izFx4=kyzjy+2, taken over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1. (8) 14. (a).(i). If f(z) is an analytic function, prove that ??????????+??2222xy f(z)2 = 4f '(z)2. (8). (ii). Find the Laurent’s series expansion of the function f(z) = )2)(3)(1(162+zzzzz in the region 3 < z+2 < 5. (8). (OR) 3 (b).(i). Find the bilinear map which maps –1, 0, 1 of the zplane onto –1,i, 1 of the wplane. Show that the upper half of the zplane maps onto the interior of the unit circle  w  = 1. (8). (ii). Using contour integration, evaluate ?8++022222)x)(x(xbadx (8). 15.(a) (i). Find the Laplace transform of t sint sinh2t and tatcos1 (8) (ii). Using convolution theorem, find L1 222)(1as+ (8) (OR) (b).(i).Find the Laplace transform of the function ???=+<<<<=)()2(,2,20,)(tftftttttfppppp (8) (ii).Using Laplace transform technique, solve 2225sin0,00tdydyyetdtdtdyywhentdt++==== , (8) ____________________________ 4
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