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(Pages : 4)
Reg. No. : ..................................... Name : ........................................
Second Year B.A. Degree Examination, April 2009 Part – III : Subsidiary Subject MATHEMATICS Time : 3 Hours
Max. Marks : 100
Instruction : Half the paper carries full marks. 1. a) Define : i) Intersection of two sets ii) Union of two sets iii) Cartesian product of two sets b) If S1 = {a, b, c}, S2 = {b, c, 1} i) Find S1 – S2 and S2 – S 1 Show that S1 – S2 ≠ S2 – S1 ii) Find (S1 – S2) ∩ (S2 – S1) c) If A = {1, 2, 3} B = {3, 4, 5} and S = {1, 2, 3, 4, 5, 6, 7} Prove that ( A ∩ B)′ = A ′ ∪ B′ ( A ∪ B) ′ = A ′ ∩ B′
(6+6+8=20)
S2 = {1, 2, 3, 4, 5} 2. a) Let S1 = {a, b, c, d} i) Find the Cartesian product S 1× S2 ii) Construct a function f that maps S1 into S2 iii) Show the function graphically by connecting point by straight lines b) Define an equivalence relation with an example. c) Define a real sequence and a cauchy sequence. Give one example each. Show that every convergent sequence is Cauchy. (8+6+6=20) P.T.O.
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(4 x 2 − x )(2 x 2 + 1) 3. a) Find Lim x →1
b) Examine the discontinuity of the function f ( x ) = 5 − x for x ≠ 4 =0
c) Find
for x = 4
dy dx
i) If y = ax
(
) 2 + (x 2 + 1)2
ii) y = x 2 + 1
1
iii) x2y = 6 iv) y = ex
(4+4+12=20)
2
4. a) If y = (sin –1 x) 2 2
show that (1 − x )
d 2y dx
2
−x
dy −2=0 dx
b) Find the radius of curvature at the point (1, 1) on the curve x3 + y3 = 2xy c) The demand law is given by q= 30 – 4p – p2, when p = 3, q = 9 Calculate i) total revenue; ii) the elasticity of demand and iii) the marginal revenue.
(6+6+8=20)
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5. a) If u = Sin (xy), show that x
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∂u ∂u −y =0 ∂x ∂y
b) Given a production function p = k LαC β where p is product, L is labor C is capital and k, α, β are constants. Find dp. c) Given a utility function and budget constraint, u = q1 q2 and M = p1q1 + p2q2 Find d2u. (6+6+8=20) 6. a) Find the maximum and minimum of the function y = 2x2 – x3. b) Find the maximum of u = x2 + 3xy – 5y2, given 2x + 3y = 6. c) Given a cost function C = γ1 x1 + γ 2x2 + F and a production function which serves as a constraint q = f(x1, x2). Find the first and second order conditions for minimum cost.
(6+6+8=20)
7. Integrate with respect to x 2 2 a) 3x − 5x + 6 x − 8
x
b) x2 cos (x3) c) x log x (4×5=20 Marks)
d) eSin x cosx. 8. a) Show that
1.2 2. 3 3.4 + + + ....∞ = 3e 1! 2! 3!
b) Find the sum of the infinite series 1 1. 3 1.3. 5 + + + ............ 3.6 3. 6.9 3. 6.9. 12
c) Expand f ( x ) = x using Taylor’s formula around a = 1.
(8+6+6=20)
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9. a) Form the differential equation given y = x4 + c1x3 + c2x2 + c3x + c4, by eliminating the arbitrary constants c1, c2, c3 and c4. b) Solve the differential equation : i)
dy 1 + y = dx 1 + x x+y
dy
ii) dx = x − y iii)
iv)
dy + 3y = 4 dx d2y dx
2
−7
dy + 12 y = e 3x dx
(4+4×4=20)
10. a) Find the first difference of the function y (x) = x (x – 1). b) If the production function is given by q = f (a, b) where a and b are inputs and let the budget be given by M = Pa .a + Pb .b where Pa and Pb are prices. Derive the equilibrium condition of the producer. c) Solve and check the following equation yx+2 – 7yx+1 + 12yx= 0. ––––––––––––––––––––
(6+6+8=20)