8/8/2019 EE Section D

1/4

SECTION D

Answer FOUR of the six questions from this section.

1. (a) The function f is of the form

f(x) = a + bx +c

x2 + 1,

for some constants a, b and c. Given that f(0) = 5, f(1) = 3 and f(1) = 5,find a, b and c.

(b) If x is measured in radians and is such that 0 x < 2, solve theequation

2sin x + 1

cos2 x + 1= 1.

What are the solutions if x can be any real number?

2. (a) Given that x > 0, explain why the function loga(x) is not definedwhen a 0 or when a = 1.

(b) Show that, if 0 < a < 1 and

loga(x) < 0,

then x > 1. Ifa > 1 andloga(x) < 0,

then what values can x take?

(c) Find all values of x which satisfy the inequality

log2x1(8 + 2x x

2) < 0.

8/8/2019 EE Section D

2/4

8/8/2019 EE Section D

3/4

5. Alchemists Inc. has developed a process which turns lead into gold. This

works by taking a mixture which contains a small amount of gold and alarger amount of lead and then, over time, the process magically turns thelead in the mixture into gold. At any time t during the process, the weightof the lead is xW and the weight of the gold is yW and, at all times, thetotal weight of the mixture of lead and gold is W where x and y depend ont and x + y = 1. The rate at which lead is converted into gold is given bykWxy where k is a positive constant.

Explain why this process is described by the differential equation

dx

dt = kx(1 x).

Assuming that x(0) = a where 0 < a < 1 is a constant, solve this differentialequation and show that your solution can be written in the form

x =1

1 + D ekt,

where D = (1 a)/a. Hence, find a similar expression for y.

Does all the lead ever completely turn into gold? Justify your answer.

6. In the current market, the risk, r(x, y), of buying x shares in company Xand y shares in company Y is given by

r(x, y) = x2 + y2.

Currently, each share in company X costs $5 and each share in company Ycosts $2.

(a) Mr. Risky has at most $20 to spend on shares in the two companies.

(i) Explain, very briefly, why the inequality

5x + 2y 20,

must hold for any portfolio, (x, y), he wishes to purchase. Whatother inequalities must x and y satisfy?

8/8/2019 EE Section D

4/4

(ii) Indicate on a diagram the region, R, of all points in the xy-plane

that satisfy these inequalities.(iii) Mr. Risky wants to purchase a portfolio of shares, (x, y), that

maximises his risk. By considering curves of the form

r(x, y) = constant,

on your diagram, find the portfolio that will enable him to do this.

(b) Miss. Safety has at least $20 to spend on shares in the two companiesand wants to purchase a portfolio of shares, (x, y), that minimises herrisk. By using a similar argument to the one above, and assuming thatfractions of shares can be purchased, find the portfolio that will enableher to do this.

(c) If, instead, Miss. Safety has to buy a whole number of both types ofshares, use your answer from part (b) to deduce the portfolio of shares,(x, y), she should purchase.