SAMPLE D
Mathematics C
Term 2 Final Examination Paper
Question 1 (12 marks) Use a SEPARATE book clearly marked Question 1
(i) For the set of scores 3,5,5,5,7,8,9,11,13, find the value of:
(a) the median.
(b) the standard deviation.
(iv) For a given function f (x), f '(x) = 4x and f (-1) = 3. Evaluate f (2) .
(v) In a particular town, 40% of the population are aged under 30 years
and 45% are aged from 30 years to 60 years inclusive. If this information is represented on a sector graph, find the size of the sector angle representing the percentage of the population aged over 60 years.
(vi) A box contains 3 blue and 5 yellow balls. Two balls are withdrawn from the box without
replacement. Find the probability that the two balls will be of the same colour.
(vii) From the frequency histogram below:
(a) calculate the mean.
(b) write down the mode.
(c) write down the median.
Question 2 (12 marks) Use a SEPARATE book clearly marked Question 2
(i) Differentiate each of the following with respect to x giving your answers
in simplest form.
(ii) Find the value of k so that the following is a valid probability density function:
(iii) A game has 3 possible outcomes for a player. The player can win $130,
win $10 or lose $10. The respective probabilities of these outcomes are 5%, 15% and 80%. Determine whether or not the game is fair.
(iv) A factory produces pens in large numbers. It is known that 2% of the pens produced
are defective. Find the probability, in unsimplified form, that in a sample of 40 pens:
(a) no pens are defective.
(b) at least one pen is defective.
(c) exactly seven pens are defective.
(v) In the expansion of (2 + 3x) 14 show that the coefficients of x 8 and x9 are equal.
Question 3 (12 marks) Use a SEPARATE book clearly marked Question 3
(i) Find the indefinite integral of :
(ii) The five digits 1, 2 , 3, 4 and 5 are written down in every possible order
to form. five-digit numbers.
(a) Find how many different five-digit numbers are possible.
(b) Find how many of these five-digit numbers are odd.
(c) If one of these odd numbers is selected at random, find the probability that the
number chosen is 54 321 .
(iii) The rate of population growth of a town at a particular time t is given by dt/dP = kP
where P is the population of the town at time t years and where k is a constant.
(a) Show, by differentiation, that P = P0 ekt satisfies the equation dt/dP = kP
where P = P0 when t = 0 .
(b) Now, the population of the town is 25 000 people and 5 years from now it is expected to be 30 000 people. Calculate the town’s population 10 years from now.
Question 4 (12 marks) Use a SEPARATE book clearly marked Question 4
(i) A cashbox contains 6 ten cent coins and 4 twenty cent coins. Two coins are
removed randomly from the box, one after the other, without replacement. The possible total value of the two coins is recorded as the random variable X.
(a) Construct the probability distribution table for X.
(b) Calculate the expected total value of the two coins.
(ii) In this question, you may use the table of areas under the standard normal curve, which can be found at the end of this examination paper.
The life expectancy of fluorescent light tubes is normally distributed with a mean of 7 000 hours and a standard deviation of 1 000 hours.
(a) Find the probability that a fluorescent light tube will have a life expectancy of less than 5 000 hours.
(b) In a batch of 20 000 fluorescent light tubes, find the number of tubes that
could be expected to last between 6 000 and 8 000 hours. Answer correct to the nearest 100 tubes.
(c) The manufacturer’s guarantee requires them to replace free of charge any
fluorescent light tubes which do not last a specified minimum number of hours (x hours). If 2% of fluorescent light tubes are replaced free of charge because of the guarantee, find the value of x.
(iii) Maintenance costs for an apartment complex generally increase as the complex gets
older. Management determines that the rate of increase in maintenance costs of an
apartment complex is given by dt/dM = 90t2 + 5000 where $M is the total
accumulated maintenance costs t years after the complex was built. Find the total maintenance costs during the seventh year after the apartment complex was built.
Question 5 (12 marks) Use a SEPARATE book clearly marked Question 5
(i) The curves y = 2x 一 x2 and are shown below for the domain 一 0 . 25 ≤ x ≤ 3 . 5.
(a) If the curves intersect at the point R , in the first quadrant, show that R is the point
(b) Find the area bounded by the two curves and the y axis.
(ii) Consider the function f(x) = x + 1+ x2/4.
(a) Find the asymptotes of the curve y = f (x).
(b) Show that f (一 2) = 0 .
(c) Show that the function has only one stationary point and determine its nature.
(d) Find values of x for which the graph of y = f (x) is concave upwards.
(e) Sketch the curve y = f (x) showing its essential features.
(f) If the equation x +1+ x2/4 = k has exactly two solutions, find the value of k.
Question 6 (12 marks) Use a SEPARATE book clearly marked Question 6
(i) Show that
(ii) A circular disc is divided into five sectors identified by the numbers 1, 2, 3 , 4 and
5. A rotating pointer is fixed to the centre of the disc. The table below shows the probability of the rotating pointer landing on a given number.
|
Number
|
1
|
2
|
3
|
4
|
5
|
|
Probability
|
0 . 3
|
0 .1
|
0 . 2
|
0 . 3
|
0 .1
|
A is the event that the pointer lands on an odd number. B is the event that the pointer lands on a prime number.
(a) Find the probability given by P(A B ).
(b) Determine whether or not the events A and B are independent.
(iii) A tutorial group consists of 9 female and 7 male students. A vote is taken to select
a sports committee of 5 from the tutorial group.
(a) Find how many different sports committees are possible if there are no restrictions on who is selected.
(b) Find how many different sports committees are possible if at least one male student must be included.
(c) Given that the sports committee contains at least one male, find the probability that it contains a majority of females.