Mathematics: analysis and approaches
Standard level
Paper 1
Save My Exams Practice Paper
1 hour 30 minutes
Instructions to candidates
• You are not permitted access to any calculator for this paper.
• Section A: answer all questions.
• Section B: answer all questions.
• Unless otherwise stated in the question, all numerical answers should be given exactly
or correct to three significant figures.
• A 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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–2–
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. 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
1. [Maximum mark: 5]
Let 𝐴 and 𝐵 be events such that P(𝐴) = 0.3, P(𝐵) = 0.75 and P(𝐴 ∪ 𝐵) = 0.9.
Find P(𝐵 | 𝐴).
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–3–
2. [Maximum mark: 5]
d𝑦 𝜋
Given that = 3𝑥 2 cos (3𝑥 3 + 2) and that the graph of 𝑦 passes through the point (0, −1),
d𝑥
find an expression for 𝑦 in terms of 𝑥 .
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–4–
3. [Maximum mark: 5]
𝑥−5
The functions 𝑓 and 𝑔 are defined such that 𝑓(𝑥) = 6𝑥 + 7 and 𝑔(𝑥) = .
3
(a) Show that (𝑓 ∘ 𝑔)(𝑥) = 2𝑥 − 3. [2]
(b) Given that (𝑓 ∘ 𝑔)−1 (𝑎) = 6, find the value of 𝑎. [3]
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–5–
4. [Maximum mark: 5]
(a) (i) Expand (2𝑘 − 1)3 .
(ii) Hence, or otherwise, show that (2𝑘 − 1)3 − (2𝑘 − 1) = 8𝑘 3 − 12𝑘 2 + 4𝑘 . [2]
(b) Thus prove, given 𝑘 > 1, 𝑘 ∈ ℕ, that the difference between an odd natural number
greater than 1 and its cube is always even. [3]
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–6–
5. [Maximum mark: 5]
The following diagram shows triangle ABC, with AB = 5 and BC = 4.
diagram not to scale
5 4
A C
(a) (i) ̂ = 3, find the possible values of cos B
Given that sin B ̂.
5
(ii) ̂ is obtuse, find the precise value of cos B
Given that B ̂. [3]
(b) Find the length of AC. [2]
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–7–
6. [Maximum mark: 8]
(a) Show that log 4 (cos 2𝑥 + 13) = log 2 √cos 2𝑥 + 13 . [3]
𝜋 𝜋
(b) Hence or otherwise solve log 2 (3√2 cos 𝑥) = log 4 (cos 2𝑥 + 13) for − <𝑥<2. [5]
2
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–8–
Section B
7. [Maximum mark: 16]
1 3
Let 𝑓(𝑥) = 𝑥 − 2𝑥 2 − 21𝑥 − 24.
3
(a) Find 𝑓 ′ (𝑥). [2]
The graph of 𝑓 has horizontal tangents at the points where 𝑥 = 𝑎 and 𝑥 = 𝑏 , 𝑎 < 𝑏.
(b) Find the value of 𝑎 and the value of 𝑏. [3]
(c) (i) Find 𝑓 ′′ (𝑥).
(ii) Hence show that the graph of 𝑓 has a local maximum point at 𝑥 = 𝑎. [2]
(d) (i) Sketch the graph of 𝑦 = 𝑓 ′ (𝑥).
(ii) Hence, use your answer to part (d)(i) to explain why the graph of 𝑓 has
a local minimum point at 𝑥 = 𝑏. [4]
The tangent to the graph of 𝑓 at 𝑥 = 𝑎 and the normal to the graph of 𝑓 at 𝑥 = 𝑏 intersect
At the point (𝑝, 𝑞).
(e) Find the value of 𝑝 and the value of 𝑞 . [5]
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8. [Maximum mark: 16]
ln 𝑝𝑥
Let 𝑓(𝑥) = where 𝑥 > 0, 𝑝, 𝑞 ∈ ℝ+ .
𝑞𝑥
1 − ln 𝑝𝑥
(a) Show that 𝑓 ′ (𝑥) = . [3]
𝑞𝑥2
The graph of 𝑓 has exactly one maximum point A.
(b) Find the 𝑥 -coordinate of A. [3]
2 ln 𝑝𝑥 − 3
The second derivative of 𝑓 is given by 𝑓 ′′ (𝑥) = . The graph of 𝑓 has exactly one
𝑞𝑥3
point of inflexion B.
3
𝑒2
(c) Show that the 𝑥 -coordinate of B is . [3]
𝑝
The region 𝑅 is enclosed by the graph of 𝑓 , the 𝑥 -axis, and the vertical lines through the
maximum point A and the point of inflexion B.
(d) Calculate the area of 𝑅 in terms of 𝑞 and show that the value of the area is independent
of 𝑝. [7]
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– 10 –
9. [Maximum mark: 15]
A school surveyed 80 of its final year students to find out how much time they spent reading the
news on a given day. The results of the survey are shown in the following cumulative frequency
diagram.
(This question continues on the following page)
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– 11 –
(Question 9 continued)
(a) Find the median number of minutes spent reading the news. [2]
(b) Find the number of students whose reading time is within 2.5 minutes of the median. [3]
Only 15% of students spent more than 𝑘 minutes reading.
(c) Find the value of 𝑘 . [3]
The results of the survey can also be displayed on the following box-and-whisker diagram.
(d) Write down the value of 𝑏. [1]
(e) (i) Find the value of 𝑎.
(ii) Hence, find the interquartile range. [4]
(f) Determine whether someone who spends 30 minutes reading the news would be an
outlier. [2]
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