Mathematics: analysis and approaches

Standard level
Paper 2

Mock Examination: 2024 May Session

1 hour 30 minutes

Instructions to candidates
• Write your session number in the boxes above.
• Do not open this examination paper until instructed to do so.
• A graphic display calculator is required for this paper.
• Section A: answer all questions. Answers must be written within the answer boxes provided.
• Section B: answer all questions in the answer booklet provided. Fill in your session number
on the front of the answer booklet, and attach it to this examination paper and your
cover sheet using the tag provided.
• Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
• A clean copy of the mathematics: analysis and approaches formula booklet is required for
this paper.
• The maximum mark for this examination paper is [80 marks].

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examination paper is error free and that it follows the style, format and material content of the Mathematics: analysis and approaches Examinations. No guarantee or warranty is
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Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. Solutions found from a graphic display calculator should be
supported by suitable working. For example, if graphs are used to find a solution, you should sketch
these as part of your answer. Where an answer is incorrect, some marks may be given for a correct
method, provided this is shown by written working. You are therefore advised to show all working.

Section A

Answer all questions. Answers must be written within the answer boxes provided. Working may be
continued below the lines, if necessary.

1. [Maximum mark: 4]

( x)
6
4 3
Find the value of the coefficient of x in the expansion of 5x + .
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2. [Maximum mark: 6]

The table below lists the average intelligence quotient, Q, and number of questions in
an intelligence test answered, N, of seven people.

Intelligence
92 99 105 107 118 120 126
Quotient (Q)

Questions
7 6 9 11 11 14 15
Answered (N)

The relationship between the variables can be modelled by the regression

equation N = aQ + b.

(a) Find the value of a and the value of b. [2]

(b) Write down the value of the Pearson’s product-moment correlation coefficient, r. [1]

A particular person is found to have an intelligence quotient of 135.

(c) (i) Use the result from part (a) to estimate the number of questions they could
answer in the intelligence test.

(ii) Determine whether this estimate is reliable. Justify your answer. [3]

Turn over
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3. [Maximum mark: 6]

In this question, give all answers correct to two decimal places.

On 1st January 2020, Tom invests P dollars in an account that pays a nominal annual
interest rate of 6 % , compounded quarterly.

(a) Find, in terms of P, the amount that Tom will have in his account on 1st January
2025. [3]

Tom wishes to keep the money in his account until the amount is five times his original
investment.

(b) Find the year in which the amount of money in Tom’s account exceeds five times
his original investment. [3]
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4. [Maximum mark: 5]

Events A and B are independent.

Given that P(A) is three times more likely than P(B), and P(A ∩ B) = 0.15, find
P(A).

Turn over
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5. [Maximum mark: 8]

A building of 110 floors is to be built. The first floor to be built will cost $3 000 000.

The cost of each subsequent floor will be $500 000 more than the floor immediately
below it.

(a) Explain why the cost of building the third floor will be $4 000 000. [2]

(b) Calculate the cost of building the 25th floor. [2]

(c) What is the cost of building the top 10 floors of the building? [4]
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6. [Maximum mark: 6]

π π
Let f (x) = sin(e x ) and g(x) = cos(e x ) for − ≤x≤ .
2 2
(a) Sketch the graphs of y = f (x) and y = g(x) on the axes below. Include any
points of intersection. [3]

(b) Find the area enclosed by the graphs of f (x) and g(x). [3]

Turn over
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Do not write solutions on this page.

Section B

Answer all questions in the answer booklet provided. Please start each question on a new page.

7. [Maximum mark: 13]

A quadrilateral ABCD is inscribed in a circle of radius 9 cm. The points A, B, C

and D lie on the circumference of the circle.
̂ = 75∘.
̂ = 108∘ and DCB
AD = 9 cm, DC = 12 cm, BC = 16 cm, A DC

(a) Find the length of [AC]. [3]

(b) (i) ̂ .
Find the size of DCA

(ii) ̂ .
Hence, find ACB [4]

(c) Find the area of triangle ACD. [2]

(d) Hence, find the total area of the shaded region. [4]
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Do not write solutions on this page.

8. [Maximum mark: 16]

A quality control study finds that the battery life of various portable radios is normally
distributed with a mean of 14 hours and a standard deviation of 0.8 hours. The table
below shows ratings given to a radio’s battery life.

A portable radio that was tested is selected at random.

(a) Find the probability that the radio's battery life is rated as:

(i) poor;

(ii) exceptional. [4]

(b) Find the probability that a radio's battery life is rated exceptional, given that
it is known to be above average. [3]

A manufacturer claims that the battery life of its radios is among the top 30 % .

(c) Find the battery life that the manufacturer's radios must exceed in order to
uphold this claim. [3]

Wildlife rangers use portable radios to maintain lines of communication in the

field. Only radios that have been rated as above average are approved for usage.
For a certain assignment, 12 wildlife rangers are equipped with new portable
radios.

(d) Find the number of these radios that are expected to have a battery life of
16 hours or more. [3]

(e) Find the probability that the battery life of at least 3 of these radios lasts for
more than 16 hours. [3]

Turn over
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Do not write solutions on this page.

9. [Maximum mark: 16]

A piece of gold jewellery is made in the shape of a sector or radius 2r centimetres

and central angle θ radians.

The length of gold wire required is the perimeter, P, of the sector.

(a) (i) Show that the length of gold wire required is given by P = 2r(θ + 2).

(ii) Hence, express the area of the sector, A, in terms of r and P. [6]

There is only 15 cm of gold wire available.

(b) Show that A = 15r − 4r 2. [1]

(c) (i) Find the value of r that will maximise the area of the sector.

(ii) Find the size of θ when the area of the sector is maximised. [5]

(d) Explain why it is only possible to create this piece of jewellery in the shape
of a sector if the radius of the sector lies in the range

15 15
<r < [4]
4π + 4 4
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Answers written on this page

will not be marked.
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Please do not write on this page.

Answers written on this page

will not be marked.