Anna University Chennai 2004 B.E Electronics
MA 1X01 - ENGINEERING MATHEMATICS - I
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)
3 -1 1 -1 5 -1 1 -1 3
1. Find the sum and product of the eigen values of the matrix
2. If x = r cos9, y = r sin9, find d(r 9)
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 + = 0. 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 = o.
7. State the sufficient conditions for a function f(z) to be analytic.
8. State Cauchys integral theorem.
9. Find the Laplace transform of unit step function at t = a.
10. Find L-1 [ 2 5 + 3 ].
2 + 4s +13
PART - B (5 x 16 = 80 marks)
11.(a).(i). Verify Cayley-Hamilton theorem for the matrix A = Hence find its inverse.
(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)
by orthogonal transformation. (8).
|
8 |
-6 |
2 |
|
6 - |
7 |
-4 |
|
2 |
- 4 |
3 |
(b).(i). Diagonalise the matrix
(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 Lagranges multiplier method. (8).
o d y dy _,
12(a). (i). Solve (3x+2)2 + 3(3x+2) - 36 y = 3x2+4x+1 (8)
dx dx
2 1
(ii). For the electric circuit gover ned by (LD2+RD+) q = E where
C
d 4 D = if L = 1 henry, R = 100 Ohms, C = 10 farad and E = 100 volts, dt
dq
q = = 0 when t = 0, find the charge q and the current i. (8)
dt
(OR)
dx dy 2t
(b).(i). Solve + 2x + 3y = 0, 3x+ + 2y = 2e (8)
dt dt
(ii). The differential equation satisfied by a beam uniformly loaded
(w kg/ metre) with one end fixed and the second end subjected
d2y 1 2 to tensile force P is given by EI = Py--wx . Show that
the elastic curve for the beam with conditions y = 0 = at x = 0 is
w wx2 2 P given by y =-2 (1-coshnx) +-where n = (8)
a 2 a - x
13. a.(i). Change the order of integration in J J xy dx dy and hence evaluate
0 x2 a
the same. (8).
(ii). Prove that F = (y2cosx + z3)i +(2ysinx-4) j +3xz2k is irrotational
and find its scalar potential. (8)
(OR)
a a 2d d
b.(i). By changing to polar co-ordinates, evaluate f {=J== (8)
J J 2 2
0 yV x2 + y 2
(ii). Verify Gauss divergence theorem for F = 4xzi -y2 j + yzk, 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
+
dx dy j
(ii). Find the Laurents series expansion of the function
z2 - 6 z -1
f(z) =- in the region 3 < |z+2| < 5. (8).
(z -1)( z - 3)( z + 2)
(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).
x dx
(ii). Using contour integration, evaluate I 2-2-2-(8).
0 (x + a )(x + b )
1 _ A/Q /"//
15.(a) (i). Find the Laplace transform of t sint sinh2t and--(8)
1
(ii). Using convolution theorem, find L-1 (s 2 + a 2)2 (8)
(b).(i).Find the Laplace transform of the function
f t, 0 < t <n f (t) = \ (8) [2n-1, n< t < 2n, f (t + 2n) = f (t)
(ii).Using Laplace transform technique, solve
d2 y dy -t t
+ 2--+ 5y = e sin t,
dt 2 dt
dy ,
y = 0, = 0 when t = 0 dt
4
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Attachment: |
| Earning: Approval pending. |
