2007 Anna University B.E MA1X01 Question paper

Course: B.E

University: Anna University

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 self-inductance 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 L-1 [13432+++sss].
1
PART – B (5 x 16 = 80 marks)
11.(a).(i). Verify Cayley-Hamilton 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(sint-t 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 = 10-4 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 (1-coshnx) + 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+(2ysinx-4)j+3xz2k is irrotational
and find its scalar potential. (8)
(OR)
b.(i). By changing to polar co-ordinates, 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 = 4|f '(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 z-plane onto –1,-i, 1
of the w-plane. Show that the upper half of the z-plane 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 L-1 222)(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)
____________________________
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