S5/5/MATAA/HP1/ENG/TZ0/FAKE
Mathematics: analysis and approaches
Higher level
Paper 1
Set 1, 2023
Candidate session number
2 hours
Instructions to candidates
• Write your session number in the boxes above.
• Do not open this examination until instructed to do so.
• You are not permitted access to any calculator for this paper.
• Section A:
answer all questions in the 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 [110 marks].
For more goodies: https://adventurousandrewmaths.blogspot.com/
14 pages
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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. 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: 7]
On any given day, the probability of it raining is 0.2.
If it rains, the probability that George is late is 0.4.
If it does not rain, the probability that George is not late is 0.8.
(a)
Represent this information on a tree diagram.
[2]
(b)
Find the probability that George is not late.
[2]
Next week, George works three days a week.
Assume that George’s lateness on one day is independent of his lateness of any other day.
(c)
Find the probability that he is late exactly once or twice next week.
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[3]
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2.
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[Maximum mark: 6]
4 2
𝑥
4
𝑥
Find the real values of 𝑥 for which (𝑥 + ) − 2 (𝑥 + ) − 24 = 0.
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3.
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[Maximum mark: 6]
log4 (36)−1
log4 81
1
(a)
Show that
= 2.
(b)
Hence, or otherwise, solve the equation 2 log 4 𝑥 =
[2]
log4 (36)−1
+ log 4 (2𝑥
log4 81
+ 6).
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[4]
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4.
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[Maximum mark: 7]
(a)
Using implicit differentiation, find the derivative of 𝑦 = arctan(𝑘𝑥).
(b)
Hence, or otherwise, find the value of ∫√3 9+𝑥 2 d𝑥.
3
[4]
1
[3]
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5.
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[Maximum mark: 8]
(a)
Prove the identity cos2(𝜋 + 𝑥) + tan(𝑥 − 𝜋) sin(4𝜋 − 𝑥) cos 𝑥 = cos 2𝑥.
[4]
(b)
Hence, solve the equation
cos2(𝜋 + 𝑥) + tan(𝑥 − 𝜋) sin(4𝜋 − 𝑥) cos 𝑥 = sin 𝑥
for 0 ≤ 𝑥 ≤ 2𝜋.
[4]
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6.
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[Maximum mark: 5]
Consider a normal distribution 𝑋 with mean 𝜇.
Given that 𝑃(𝜇 − 𝑏 < 𝑋 < 𝜇 + 𝑎) = 𝑐, where 𝑎, 𝑏 > 0 and 𝑐 is a constant less than 1.
(a)
show that 𝑃(𝜇 − 𝑎 < 𝑋 < 𝜇 + 𝑏) = 𝑐.
[2]
Suppose that 𝑃(𝑋 < 𝜇 + 𝑎) = 0.7 and 𝑐 = 0.5,
(b)
show that 𝑎 < 𝑏.
[3]
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7.
S5/5/MATAA/HP1/ENG/TZ0/FAKE
[Maximum mark: 6]
Find, in ascending powers of 𝑥, the first three nonzero terms in the expansion of 𝑦 = ln(2 + 2𝑥 2 ).
State the interval in which your expansion is valid.
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8.
S5/5/MATAA/HP1/ENG/TZ0/FAKE
[Maximum mark: 6]
Consider a pyramid 𝑆. 𝐴𝐵𝐶.
The plane (𝑆𝐵𝐶) is perpendicular to the plane (𝐴𝐵𝐶).
Δ𝐴𝐵𝐶 is an equilateral triangle with side length 𝑎.
𝑆𝐵 = 𝑆𝐶 =
𝑎 √13
.
4
Let 𝐻 be the midpoint of 𝐵𝐶.
(a)
Explain why 𝑆𝐻 ⊥ 𝐵𝐶.
[1]
(b)
Find the volume of 𝑆. 𝐴𝐵𝐶 in terms of 𝑎.
[5]
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9.
S5/5/MATAA/HP1/ENG/TZ0/FAKE
[Maximum mark: 7] (Source: some Vietnamese website)
Find the complex numbers 𝑧 that satisfy both
|𝑧 2 | = |𝑧 − 𝑧 ∗ | and |(𝑧 + 2)| ⋅ |(𝑧 ∗ − 2𝑖)| = |𝑧 − 2𝑖|2 .
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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.
10.
[Maximum mark: 19]
Consider the cubic polynomial 𝑓(𝑥) = 𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑.
When 𝑓(𝑥) is divided by (2𝑥 − 4), the remainder is −8.
(a)
Form an equation in terms of 𝑏, 𝑐 and 𝑑.
[1]
Further given that the roots of 𝑓(𝑥) are 𝑥1 , 𝑥2 and 𝑥3 , the sum of the roots is 3 and
𝑥12 + 𝑥22 + 𝑥32 = 21.
(b)
𝑓(𝑥) is translated 𝑎 units to the right to obtain the graph of 𝑔(𝑥).
Find the value of 𝑎 if the sum of the roots is 9.
[2]
(c)
Show that 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 − 6𝑥 + 8.
[5]
(d)
Hence, solve the equation sec 𝑥 tan2 𝑥 + 5 = 3 tan2 𝑥 + 5 sec 𝑥 for 0 ≤ 𝑥 ≤ 𝜋.
[6]
(e)
On separate sets of axes, sketch the graph of
(i)
𝑦 = 𝑓(𝑥),
(ii)
𝑦 = 𝑓(𝑥).
1
Clearly indicate the equations of the asymptotes and the intersections with the coordinate
axes.
[5]
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11.
[Maximum mark: 15]
Plane 𝜋 contains the points 𝐴(2, 1, 5), 𝐵(−1, 2, 10) and 𝐶(0, −1, 3).
(a)
(i)
Show that the normal vector of (𝜋) is < 1, −2, 1 >.
(ii)
Find an equation of plane (𝜋).
[4]
Line (𝑑) has passes the points 𝐷(3, 3, −4) and 𝐸(1, 5, −4).
(b)
Find the acute angle formed by the line (𝑑) and the plane (𝜋).
[4]
Line (Δ) passes through the point 𝑃(3, 1, −2), cuts (𝑑) at 𝑀, cuts (𝜋) at 𝑁 such that 𝐴𝑁 = 2𝐴𝑀.
(c)
12.
Find the equation of the line Δ.
[7]
[Maximum mark: 19]
Finding Nemo functions
Before starting the course, everyone has mastered the art of solving equations, where your
answer is a constant. In the course, you have learned some methods to solve first order
differential equations where you’re finding a function instead of a constant that satisfies the
requirements. This question focuses on finding the unknown function when it is defined implicitly.
Part A
By making a substitution 𝑢 = 𝑦 ′ (𝑡), or otherwise, solve the equation
𝑡𝑦 ′′ − 4𝑦 ′ = 5𝑡 3 with initial conditions 𝑦(0) = 2, 𝑦 ′ (1) = −2.
[8]
Part B
(Source: some other Vietnamese website)
Consider the function 𝑓(𝑥) continuous on ℝ\{±√3} and
1 − 𝑥 3 = 2𝑥 2 𝑓(𝑥) + 𝑥[𝑓(𝑥)]2 − 𝑓′(𝑥)
and 𝑓(1) = 0.
(i)
Show that 1 + 𝑓 ′ (𝑥) = 𝑥[𝑥 + 𝑓(𝑥)]2 .
[1]
(ii)
By making a substitution 𝑢 = 𝑥 + 𝑓(𝑥), solve for 𝑓.
[5]
Part C
d𝑦 2
Find all solutions to the differential equation 𝑦 2 + ( d𝑡 ) = 4.
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