Foundation Mathematics

10 March 2015

Examination Paper

Answer ALL questions. Clearly cross out surplus answers.

Time: 2 hours

The maximum mark for this paper is 100.

Any reference material brought into the examination room must be handed to the invigilator before the start of the examination.

Candidates are allowed to use a scientific calculator during this

examination.

Graph paper will be provided by the centre.

You must show your workings. Marks are awarded for these.

Page 2 of 6

Foundation Mathematics © NCC Education Limited 2015

Answer ALL questions

Marks

Question 1 a) Simplify the following. i) (𝑝𝑞2)3 1

ii) 𝑥−3𝑦2 × 𝑥4 𝑦3 1

iii) 𝑚4

𝑚3× 𝑚7

1

b) Simplify the following. i) 𝑝(7 + 𝑞) + 2𝑝(𝑞 − 3) 1 ii) 14𝑎2𝑏𝑐

2ac

1

iii) 15𝑥𝑦

3×

𝑦

𝑥

1

c) Factorise the following. i) 12𝑥𝑦2 − 3𝑦𝑧 2

ii) 𝑥2 + 8𝑥 − 9 2

d) Simplify the following. i) 2𝑥

3×

3

4𝑥2

2

ii) 1

8𝑎+

3

4𝑎

2

e) Transpose the following formula to make 𝑏 the subject.

𝑎 = 5𝑑

𝑏2 + 𝑐

2

f) Solve the following equation and find the value of 𝑥.

4(𝑥 − 3) = 2𝑥 + 2

2

g) Solve the following quadratic equation by factorising.

𝑥2 + 4𝑥 − 21 = 0

2

Total 20 Marks

Marks

Page 3 of 6

Foundation Mathematics © NCC Education Limited 2015

Question 2 a) Solve the following quadratic equation by using the quadratic formula:

5𝑥2 + 7𝑥 − 1 = 0 (You may leave your answer in surd form.)

2

b) Solve the following simultaneous equations and find the value of 𝑥 and 𝑦. i) 3𝑥 + 4𝑦 = 18 and 7𝑥 − 4𝑦 = 2 2 ii) 2𝑥 + 𝑦 = 21 and 4𝑥 + 𝑦 = 37 2 c) A rectangle is 5cm longer than it is wide. If the perimeter of the rectangle is

50cm, find its length and width. 3

d) Calculate the gradient of the following curves at the point where 𝑥 = 2, using

differentiation.

i) 𝑦 = 2𝑥3 + 4𝑥 3

ii)

𝑦 = 2𝑥2 −1

𝑥2

3

e) A particle has a velocity of 𝑣 = 𝑡3 + 12

i) Find the acceleration, 𝑎, after 𝑡 seconds. 2

ii) What is the acceleration at 𝑡 = 1 seconds? 1 iii) At what time, t, is the acceleration 27 m/s2? 2

Total 20 Marks

Marks

Page 4 of 6

Foundation Mathematics © NCC Education Limited 2015

Question 3 a) i) Using differentiation, find the coordinates of the turning point on the curve

𝑦 = 3𝑥2 + 6𝑥 + 2

4

ii) Identify the turning point found in part (i) above as either a maximum or

minimum turning point. 2

b) Integrate the following expression.

8𝑥3 + √𝑥

2

c) The gradient of the curve which passes through the point (1, 0) is given by

2𝑥2 − 1. Find the equation of the curve.

3

d) Evaluate the following definite integrals. i)

∫ (3𝑥2 + 3) 𝑑𝑥2

0

3

ii)

∫ (𝑥

4+ 𝑥3) 𝑑𝑥

1

0

3

e) The part of the curve 𝑦 = 2𝑥 + 3 between the ordinates 𝑥 = 0 and 𝑥 = 1 is

rotated about the 𝑥-axis. Calculate the volume of the solid generated. Leave your answer as a multiple of 𝜋.

3

Total 20 Marks

Marks

Page 5 of 6

Foundation Mathematics © NCC Education Limited 2015

Question 4 a) The acceleration of a moving body at the end of 𝑡 seconds from the

commencement of motion is (8𝑡 − 2) m/s2.

i) Find the velocity at the end of 4 seconds if the initial velocity is 8 m/s. 3 ii) Find the distance travelled by the body at the end of 3 seconds. 3 b) The 8.15am train to Newtown is either on time or late. It is never early.

The probability that 8.15am train to Newtown will be late on Monday is 0.2. The probability that the 8.15am train will be late on Tuesday is 0.1.

i) Draw a probability tree diagram to show all the possible outcomes. 8 ii) Use your tree diagram to find out the probability that the train is late on

ONE (1) of the two days. 2

iii) Use your tree diagram to find the possibility that the train is on time at least

once. 2

c) How many ways can we arrange three letters from the word ‘pencil’ if repetitions

are not allowed and different orders of the same letter count as the same arrangement?

2

Total 20 Marks

Marks

Page 6 of 6

Foundation Mathematics © NCC Education Limited 2015

Question 5 a) A market research company asks 30 consumers which brand of cereal they

prefer: A, B, C, D or E. The responses are recorded below.

A D C B C

B B D B C

A B B C A

C D C D B

B E C D B

C D C E B

i) Summarise this data as a frequency distribution table. 2 ii) Is this data continuous or discrete? 1 iii) Construct a bar chart to illustrate this data. 4 b) The amount of time (in minutes) it takes a group of students to travel to college

on a particular day is recorded in the table below.

Time (minutes) 0 < 20 20 < 40 40 < 60 60 < 80 80 < 100

Frequency 12 20 25 11 8

i) Calculate the mean. You may leave your answer as a fraction. 4 ii) Within which class interval will the modal value lie? 1 iii) Calculate the mode. You may leave your answer as a fraction. 2 c) The following data set is recorded.

21 18 13 10 12 7 9 23 19 24 15

i) Calculate the range of the data. 1 ii) Find the median value. 2 iii) Find the lower quartile and the upper quartile. 2 iv) Calculate the quartile range. 1

Total 20 Marks

End of paper