B.Sc., B.Com., DEGREE EXAMINATION – ECO., & COMM.,
FOURTH SEMESTER – APRIL 2012
Date : 19-04-2012
Time : 1:00 - 4:00
MT 4205 / 4202 - BUSINESS MATHEMATICS
Dept. No. Max. : 100 Marks
Part A (Answer ALL questions)
1.
2.
(2 x 10 = 20)
The total cost C for output x is given by 𝐶 =
2
5 𝑥 + 9 . Find the average cost of output of 10 units.
6
If the marginal function for output is given by 𝑅 𝑚
=
(𝑥+2) 2
+ 5 , find the total revenue
3.
4.
5.
6.
7.
8.
9.
function by integration.
Find the differential coefficient of 𝑥 𝑥
2 −1
2 +1
with respect to x.
Find the first order partial derivatives of 𝑢 = 3𝑥 2 + 2𝑥𝑦 + 4𝑦 2
.
Evaluate ∫ 𝑥𝑒 𝑥 𝑑𝑥 . 𝑐
Prove that ∫ 𝑓(𝑥)𝑑𝑥 + 𝑎
If 𝐴 = (
4 1
2 3
) , find A
2
.
1 2 3 𝑐 𝑏 𝑎 𝑏
∫ 𝑓(𝑥)𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 .
If 𝐴 = ( 4 5 6 ) , Can you find the inverse of the matrix of A?
7 8 9 𝑥+1
If
(𝑥−1)(2𝑥+1)
=
𝐴 𝑥−1
+
𝐵
2𝑥+1
then find A and B.
10.
Define Feasible solution.
Part B (Answer any FIVE of the following) (5 x 8 = 40)
11.
The total cost C for output x is given by C

2 35 x
3

2
. Find (i) Cost when output is 4 units (ii)
Average cost when output is 10 units (iii) Marginal cost when output is 3 units.
12.
If y
 x x x ...

then prove that x dy dx y
2

1
 y log x
.
13.
For the following pair of demand functions for two commodities X
1
and X
2
, determine the four partial marginal demands, the nature of relationship (Complementary, Competitive or neither) between X
1
and X
2
and the four partial elasticities of demand 𝑥
14.
Integrate
(𝑥 𝑥
3
2 +1) 3
with respect to x .
1
= 𝑝
1
2
4 𝑝
2
and 𝑥
2
=
16 𝑝
2
2 𝑝
1
.
15.
Prove that 𝑎
∫ 𝑓(𝑥)𝑑𝑥 = {
𝑖𝑓 𝑓(𝑥)𝑖𝑠 𝑒𝑣𝑒𝑛
.
0 𝑖𝑓 𝑓(𝑥)𝑖𝑠 𝑜𝑑𝑑
16.
If A




1 2 1
0 1

1
3

1 1



then show that A 3 
3 A 2  
9 I

0 .

17.
Compute the inverse of the matrix 𝐴 = (
1 0 −4
−2 2 5 ) .
18.
𝑥 2 +1
Resolve into partial fractions:
(𝑥−3)(𝑥−1) 2
.
3 −1 2
Part C (Answer any TWO questions) (2 x 20 = 40)
19.
(a) If AR and MR denote the average and marginal revenue at any output, show that elasticity of demand is equal to
AR
AR

MR
. Verify this for the linear demand law p a bx .
(b) If the marginal revenue function is 𝑀𝑅 = 𝑎𝑏
(𝑥−𝑏) 2
− 𝑐 , show that 𝑝 = 𝑎
(𝑏−𝑥)
− 𝑐 is the demand law.
(10+10)
20.
(a) If 𝑦 = (𝑥 + √1 + 𝑥 2 ) 𝑚
, prove that (1 + 𝑥 2 )𝑦
2
+ 𝑥𝑦
1
= 𝑚 2 𝑦 .
(b) Evaluate ∫
(3𝑥+7)
2𝑥 2 +3𝑥−2 𝑑𝑥 . (10+10)
21.
(a) Find the consumer’s surplus and producer’s surplus for the demand curve 𝐷(𝑥) =
16 − 𝑥 2
and the supply curve 𝑆(𝑥) = 4 + 𝑥 .
(b) Find the maximum and minimum values of the function 𝑥
4
+ 2𝑥
3
− 3𝑥
2
− 4𝑥 + 4 .
(10+10)
22.
(a) Solve the equations 5𝑥 − 6𝑦 + 4𝑧 = 15 ; 7𝑥 + 4𝑦 − 3𝑧 = 19 ; 2𝑥 + 𝑦 + 6𝑧 = 46 by inverse matrix method.
(b) Solve the following LPP by graphical method:
Maximize 𝑍 = 2𝑥
1
+ 5𝑥
2
Subject to 𝑥
1
+ 𝑥
2
≤ 24
3𝑥
1
𝑥
1
+ 𝑥
2
+ 𝑥
2
≤ 21
≤ 9
𝑥
1
, 𝑥
2
≥ 0 . (12+8)

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