## Faculty of Science, Technology, Engineering and Mathematics

MST124 Essential mathematics 1

MST124

TMA 02 2017J

Covers Units 3, 4, 5 and 6 Cut-o↵ date 24 January 2018

You can submit this TMA either by post to your tutor or electronically

as a PDF file by using the University’s online TMA/EMA service.

Before starting work on it, please read the document Student guidance

for preparing and submitting TMAs, available from the ‘Assessment’

area of the MST124 website.

If you have a disability that makes it di!cult for you to attempt any of these

questions, then please contact your Student Support Team or your tutor for

advice.

The work that you submit should include your working as well as your final

answers.

Your solutions should not involve the use of Maxima, except in those parts of

questions where this is explicitly required or suggested. Your solutions

should not involve the use of any other mathematical software.

Your work should be written in a good mathematical style, as described in

Section 6 of Unit 1, and as demonstrated by the example and activity

solutions in the study units. Five marks (referred to as good mathematical

communication, or GMC, marks) on this TMA are allocated for how well

you do this.

Your score out of 5 for GMC will be recorded against Question 10. You do

not have to submit any work for Question 10.

Copyright !c 2017 The Open University WEB 05385 3

8.1

TMA 02 Cut-o↵ date 24 January 2018

## Essential mathematics 1

## Question 1 – 6 marks

You should be able to answer this question after studying Unit 3.

When animals or plants are introduced to a new environment where they

have no predators or competition, their numbers can increase extremely

rapidly. On many isolated islands around the world this has happened with

wild goats, and their population growth can in some circumstances be

modelled by an exponential growth function.

Suppose that g(t) is the number of wild goats on a particular island at time t

(in years), where t = 0 represents the time when the goats were first

introduced onto the island. Assume that the population is modelled by the

exponential growth function

g(t) = Aekt (t ! 0),

where A and k are constants. After 3 years the population was 42, and after

10 years the population was recorded as 1195).

(a) Find the values of the constants A and k, correct to three significant

figures. [4]

(b) Future population surveys will take place every 2 years. By what factor

does the model predict the population will increase every 2 years? Give

your answer to three significant figures. [2]

Question 2 – 5 marks

You should be able to answer this question after studying Unit 3.

Use a table of signs to solve the inequality

“3x2 ” 6x + 9 ! 0.

Give your answer in interval notation. [5]

Question 3 – 15 marks

You should be able to answer this question after studying Unit 3.

(a) This part of the question concerns the graph of the function

f(x) = x2 + 6x + 8.

(i) | Complete the square to rearrange f(x) in the form f(x) = (x + a)2 ” b, |

where a and b are positive integers. [1]

(ii) Hence explain how the graph of f can be obtained from the graph

of y = x2 by using appropriate translations.

(You are not asked to sketch any graphs in this part, but you may

find it helpful to do so.) [2]

page 2 of 5

(b) This part of the question concerns the function

g(x) = x2 + 6x + 8 (!3 x 0).

The function g has the same rule as the function f in part (a), but a

smaller domain.

(i) Sketch the graph of g, using equal scales on the axes. (You should

draw this by hand, rather than using any software.) Mark the

coordinates of the endpoints of the graph. What is the image set

of g? (ii) Find the inverse function g!1, specifying its rule, domain and image set. (iii) Add a sketch of y = g!1(x) to the graph that you produced in part (b)(i). Mark the coordinates of the endpoints of the graph of g!1. | [3] |

[6] | |

[3] |

### Question 4 – 10 marks

You should be able to answer this question after studying Unit 4.

In triangle ABC (with the usual notation), angle A = 36“, side b = 10 and

side c = 12.

(a) Use this information to calculate the area of triangle ABC, giving your

answer to two decimal places. | [2] | |

(b) (i) | Use the cosine rule to find the length of side a, to two decimal places. | [2] |

(ii) Using only the lengths of the sides of the triangle, find the area of | ||

triangle ABC, giving your answer to two decimal places, and check that it is the same as the area that you found in part (a). (Hint: look up Heron’s formula in Unit 4.) | [2] | |

(c) Without using the cosine rule again, find the remaining angles B and C, giving your answers to the nearest degree. | [4] |

Question 5 – 10 marks

You should be able to answer this question after studying Unit 4.

(a) Given that ✓ is an acute angle with

sin ✓ = 17

41,

find the exact value of cot ✓. (b) Using the exact values for the sine and cosine of both 3⇡/4 and ⇡/6, and the angle sum identity for sine, find the exact value of sin(11⇡/12). (c) Use the exact value of cos(11⇡/6) and the half-angle identity for cosine to find the exact value of cos(11⇡/12). | [2] |

[4] | |

[4] |

#### Question 6 – 10 marks

You should be able to answer this question after studying Unit 5 and also

Section 7 of the Computer Algebra Guide.

Use Maxima to plot the parabola y = 4 ! 3x ! 2x2 and the circle

x2 + y2 + 2x ! 2y ! 7 = 0 on the same graph, and to find the coordinates of

the points of intersection between the parabola and the circle.

State the values of the coordinates rounded to two decimal places, but do

not attempt to use Maxima for rounding.

Include a printout or screenshot of your Maxima worksheet with your

solutions. You are not expected to annotate your Maxima worksheet with

explanations. [10]

Question 7 – 15 marks

You should be able to answer this question after studying Unit 5.

A fruit bat has a speed in still air of 10 m s!1. It is pointed in the direction

of the bearing 130“, but there is a wind blowing at a speed of 5 m s!1 from

the south-west.

Take unit vectors i to point east and j to point north.

(a) Express the velocity b of the bat relative to the air and the velocity w

of the wind in component form, giving the numerical values in m s!1 to

two decimal places. [7]

(b) Express the resultant velocity v of the bat in component form, giving

numerical values in m s!1 to two decimal places. [3]

(c) Hence find the magnitude and direction of the resultant velocity v of

the bat, giving the magnitude in m s!1 to two decimal places and the

direction as a bearing to the nearest degree. [5]

## Question 8– | 18 marks |

You should be able to answer this question after studying Unit 6. This question concerns the function | |

f(x) = !x | x2 + 6x + 6. |

3 ! 3 2(a) Find the stationary points of f, and give their exact x– and

y-coordinates. [5]

(b) Use the first derivative test to classify the stationary points that you

found in part (a). [5]

(c) Sketch the graph of f, indicating the y-intercept and the points that

you found in part (a). (You should draw this by hand, rather than using

any software, and you can use di↵erent scales on the two axes if

appropriate.) [5]

(d) Find the greatest and least values taken by f on the interval [!3, 3]. [3]

## Question 9 | – | 6 marks |

You should be able to answer this question after studying Unit 6. An object moves along a straight line. Its displacement s (in metres) from a reference point at time t (in seconds) is given by | ||

s(t) = 4t3 ! 21t2 + 18t + 5 | (t ” 0). | |

(a) Find expressions for the velocity v(t) and the acceleration a(t) of the | ||

object at time t. | [2] | |

(b) (i) | Find the times when the object is momentarily at rest, i.e. the |

velocity v(t) equals zero. (ii) At what time does the object have its minimum velocity? | [2] |

(Hint: think about the value of the acceleration at this point. You

may find it useful to sketch a graph of v(t), but you do not need to

submit this.) [2]

## Question 10 | – | 5 marks |

A score out of 5 marks for good mathematical communication throughout TMA 02 will be recorded under Question 10. | [5] |

Essential mathematics 1