LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION - MATHEMATICS
SECOND SEMESTER – APRIL 2012
MT 2901 - MATHEMATICAL METHODS
Date : 28-04-2012
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
SECTION A
Answer any five questions
(5 x 4 = 20)
1. Find the equation of the line through the point (4,3) and parallel to the line through the points
(0, −3) and (6,1).
2. Suppose the fixed cost of production for a commodity is $20000; the variable cost is $15 per unit
and the commodity sells for $20 per unit. What is the break-even quantity?
3. If the marginal revenue function is 𝑅 ′ (𝑥) = 12 − 8𝑥 + 𝑥 2 , determine the revenue and demand
functions.
4. Evaluate 𝑦 = ∫(3𝑥 2 + 2𝑥 + 6)𝑑𝑥 if 𝑦 = 5 when 𝑥 = 0.
3
𝑑2 𝑦
5. Find the general solution of the differential equation 𝑑𝑥 2 = 5𝑥 4 + 2𝑥 2 .
6. If 𝑦 = 𝑒 2𝑥 , find ∆2 𝑦 and ∆3 𝑦.
3
𝑥1
1
𝑥3
𝜕
7. If 𝑏 = (−2) and 𝑥 = (𝑥2 ), show that 𝜕𝑥 (𝑏 ′ 𝑥) = 𝑏.
SECTION B
Answer any four questions
(4 x 10 = 40)
8. Find the equation of the line that passes through the intersection of the lines 𝑥 + 𝑦 − 1 = 0 and
2𝑥 − 𝑦 + 1 = 0 and is perpendicular to the line 𝑥 + 3𝑦 − 2 = 0.
9. (i) Find the equation of the curve which has a slope of zero at the point (0,2), has a point of
10
inflection at (−1, 3 ) and for which 𝑦 ′′′ = 4.
2
(ii) Evaluate ∫1 𝑥(𝑥 + 1) 𝑑𝑥
.
(6+4)
(𝑥+3)𝑑𝑥
10. Evaluate ∫ (𝑥 2+3𝑥+2)(𝑥+4) using partial fractions.
𝑑𝑦 2
𝑑𝑦
11. (i) Show that 𝑦 = 2𝑐𝑥 2 + 𝑐 2 is a solution of (𝑑𝑥 ) + 8𝑥 3 𝑑𝑥 = 16𝑥 2 𝑦 where c is an arbitrary
constant and find a particular solution that satisfies the condition 𝑦 = −1 when 𝑥 = 1.
(ii) Solve the equation 𝑦 4 𝑑𝑥 + 𝑥 3 𝑑𝑦 = 0.
(6+4)
12. Solve the difference equation 6𝑦𝑥+1 + 2𝑦𝑥 − 3 = 0, if 𝑦0 = 1 and also calculate 𝑦1 , 𝑦2 , 𝑦3 and 𝑦4 .
1
13. If 𝑥 = (1
1
−1
−3
1 ), 𝑦 = ( 5 ). Find the least squares estimates 𝛽̂ for the regression equation 𝑦 =
2
11
𝑥𝛽 + 𝑢.
SECTION C
Answer any two questions
(2 x 20 = 40)
14. (i) What relation (parallel, perpendicular, coincident or intersecting) does the line 3𝑥 + 4𝑦 − 2 = 0
have to the following lines?
(a) 15𝑥 + 20𝑦 − 10 = 0 (b) 8𝑥 − 6𝑦 + 5 = 0 (c) 9𝑥 + 12𝑦 + 7 = 0
(d) 3𝑥 + 𝑦 − 4 = 0 (e) 12𝑥 − 9𝑦 + 2 = 0 (f) 2𝑥 + 𝑦 − 6 = 0.
(ii) Find the equation of the line which is parallel to the line through the points (5,6) and (7,8) and
also passes through the intersection of the line having slope -2 through the point (-4,-6) and the
line having slope 3 through the point (2,2).
(12+8)
15. (i) The quantity demanded and the corresponding price, under pure competition, are determined
by the demand and supply functions 𝑦 = 20 − 3𝑥 2 and 𝑦 = 2𝑥 2, respectively. Determine the
corresponding consumer’s surplus and producer’s surplus. (ii) Evaluate (a) ∫ 3
(𝑧+1)
√𝑧 2 +2𝑧+2
∫(𝑥 3 + 3𝑥) (3𝑥 2 + 3)𝑒 (𝑥
3 +3𝑥)
𝑑𝑧 (b)
𝑑𝑥
(10+10)
16. (i) Solve the equation
𝑑𝑦
𝑑𝑥
+
1−2𝑥
𝑦 ( 𝑥2 )
= 2.
(ii) State the type, order and degree of the following equations
𝜕3 𝑦
𝜕𝑦
(a) 𝜕𝑥 3 + 𝑥𝑦 𝜕𝑥 − 2 = 0; (b) 18𝑦𝑥+2 − 6𝑦𝑥 = 5𝑥.
(iii) Show that 𝑦𝑥 = 𝑐1 + 𝑐2 2𝑥 + 𝑐3 3𝑥 is a solution of 𝑦𝑥+3 − 6𝑦𝑥+2 + 11𝑦𝑥+1 − 6𝑦𝑥 = 0 and find a
particular solution if 𝑦0 = 1, 𝑦1 = 1, 𝑦2 = −1
(6+4+10)
3
2
17. (i) Write the difference equation ∆ 𝑦𝑥 + ∆ 𝑦𝑥 + ∆𝑦𝑥 + 𝑦𝑥 = 0 in terms of lagged values of 𝑦.
𝑥12 + 3𝑥2
2𝑥1 𝑥2 − 𝑥2
𝜕𝑦
(ii) If 𝑦 =
, find 𝜕𝑥 .
2𝑥1 + 𝑥1 𝑥2 − 3𝑥22
2
2
3
( 3𝑥1 − 𝑥1 𝑥2 − 𝑥2 )
2𝑏1
𝑥1
𝑏
𝑥
(iii) If 𝑥 = [ 2 ], 𝐵 = [ 2 + 2𝑏3
𝑥3
𝑏1 + 𝑏3
𝑏1 + 𝑏2
𝑧
3𝑏2 ] and 𝑧 = [ 1 ] then show that
𝑧
2
4𝑏3 2
(8+4+8)
𝜕
𝜕𝑥
(𝑥 ′ 𝐵𝑧) = 𝐵𝑧.