Mathematical Methods

Mathematical Methods

Trial Examination 2

2004

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Question 1

The temperature T (in degrees Celsius) at any time t (in hours after midnight) in a particular day could be modelled by the function

a. i. Determine the period of the function .

ii. State the maximum and the minimum temperatures during the particular day.

iii. Calculate the times when the maximum and the minimum temperatures occur during the particular

day.

1 + 1 + 2 = 4 marks

b. Sketch the graph of T versus t from midnight to the next midnight. 2 marks

c. Show that the average rate of increase in temperature from 3.00 am to 3.00 pm is oC per hour.

1 mark

d. At what time(s) in the morning does the rate of increase in temperature the same as the average rate?

Answer in am. 2 marks

The average temperature from to is given by .

e. Show that the average temperature from 9.00 am to 9.00 pm is 17.27 oC. 3 marks

f. For how long (in hours, 2 decimal places) does the temperature stay at or above the average temperature

between 9.00 am and 9.00 pm? 2 marks

Total 14 marks

Question 2

Some psychologists believe that a numerical measure of a child’s learning ability during the early years of life is approximately described by the function

for , where x is the age in years.

a. Use calculus to determine the age when a child’s learning ability is best. Leave answer in exact form.

3 marks

b. Determine the range of a child’s learning ability between the age of 0 and 4. Leave answer in exact

form. 2 marks

A cubic function of the form for can also be used to approximate a child’s learning ability.

c. Show that the maximum value of is when . 4 marks

d. Find the values of a and b to 4 decimal places such that and have the same maximum value

at the same age. 2 marks

e. Find the other two ages (2 decimal places) such that and give the same learning ability of a

child in each case. 2 marks

Total 13 marks

Question 3

Consider function defined by for .

a. i. Use calculus to find . Express in factorised form. 2 marks

ii. Hence find the exact maximum and minimum values of for . 2 marks

iii. Sketch the graph of for , showing coordinates of stationary points and equation of

asymptote. Use the same scale for both axes. 2 marks

iv. On the same set of axes as in part iii, sketch the inverse of for . 1 mark

b. i. Find the exact area bounded by the curve defined by , , and the x-axis.

1 mark

ii. Find such that . 2 marks

iii. State the domain of . 1 mark

iv. Write down the exact area bounded by the curve defined by , , and the x-axis.

1 mark

c. Use calculus to find the exact x-coordinate of the point on the curve where it is steepest.

3 marks

Total 15 marks

Question 4

Suppose the height of an adult female and that of an adult male in a population are both normally distributed with the same standard deviation of 10 cm. 50.0% of the adult females are under 168 cm while 38.2% of the adult males are over 175 cm.

a. What is the mean height of the adult females in the population. 1 mark

b. Show that the mean height of the adult males in the population is 172 cm. 2 marks

c. A particular height h cm exists such that the percentage of adult males over h cm equals the percentage

of adult females under h cm. Find the value of h. 1 mark

d. Find the probability to 3 decimal places that a randomly chosen adult male is taller than 170 cm.

1 mark

The ratio of adult females to adult males in the population is 9 : 11.

e. Show that the probability of a randomly chosen adult taller than 170 cm is 0.508. 2 marks

f. A person is randomly chosen among the adults taller than 170 cm. What is the probability (3 decimal

places) that the person is a female? 2 marks

g. Twenty adults are randomly selected from the population. Find the probability (3 decimal places) that

more than 12 are over 170 cm. 2 marks

h. Five are randomly chosen from the twenty adults consisting of 11 males. Find the probability

(3 decimal places) that three or four are females. 2 marks

Total 13 marks

End of exam II