124 Practice exam papers
Mathematics: analysis and approaches
Higher level
Paper 1 Practice Set B
Candidate session number
2 hours
__________________________________________________________________________________________
Instructions to candidates
• Write your session number in the boxes above.
• Do not open this examination paper until instructed to do so.
• You are not permitted access to any calculator for this paper.
• Section A: answer all questions. Answers must be written within the answer boxes provided.
• Section B: answer all questions in an answer booklet.
• 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 book is required for this paper.
• The maximum mark for this examination paper is [110 marks].
© Hodder Education 2020
Practice exam papers 125
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]
a 4x
Find the value of a > 0 such that ∫0 2 dx = ln16.
x +3
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126 Practice exam papers
2 [Maximum mark: 6]
The box plot summarizes the times taken by a group of 40 children to complete an obstacle course.
7 9.5 12.3 17.5 19.6
5 10 15 20
time (minutes)
Two of the 40 children are selected at random.
a Find the probability that both children completed the course in less than 9.5 minutes. [3]
b Find the probability that one child completed the course in less than 9.5 minutes and the
other in between 9.5 and 17.5 minutes. [3]
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Practice exam papers 127
3 [Maximum mark: 5]
Find the equation of the normal to the graph of y = sinx x at the point where x = π.
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128 Practice exam papers
4 [Maximum mark: 5]
Solve the inequality |x – 3| < |2 x + 1|.
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Practice exam papers 129
5 [Maximum mark: 6]
Given that P(A) = 0.3, P(B|A) = 0.6 and P(A ∪ B) = 0.8, find P(A|B). Give your answer as a simplified fraction.
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130 Practice exam papers
6 [Maximum mark: 6]
The graph in the diagram has equation y = A + Be– .kx
2
x
y = 1
Find the values of A, B and k.
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Practice exam papers 131
7 [Maximum mark: 6]
Use mathematical induction to prove that 7 + 3 – 1 is divisible by 4 for all integers n ù 1.
n n
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132 Practice exam papers
8 [Maximum mark: 7]
Find, in the form z = rei, the roots of the equation z3 = 4 – 4 3 i.
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Practice exam papers 133
9 [Maximum mark: 7]
Find the first two non-zero terms in the Maclaurin series for cos– x2 .
1 x
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134 Practice exam papers
Do not write solutions on this page
Section B
Answer all questions in an answer booklet. Please start each question on a new page.
10 [Maximum mark: 18]
Let f(x) = xe–kx where x ∈ and k > 0.
a Show that f( x) = (1 – kx)e–kx and find f(x) in the form (a + bx)e–kx. [5]
b Find the x-coordinate of the stationary point of f(x) and show that it is a maximum. [5]
c Find the coordinates of the point of inflection of f(x). [3]
d The graph of y = f(x) is shown below. A is the maximum point and B is the point of
inflection. Show that the shaded area equals 2e2–23 . [5]
ke
y
11 [Maximum mark: 15]
The following system of equations does not have a unique solution.
6x + ky + 2z = a
6x – y – z = 7
2 x – 3y + z = 1
a Find the value of k. [6]
Each equation represents a plane.
b Find
i the value of a for which the three planes intersect in a line
ii the equation of the line. [7]
c If the value of a is such that the three planes do not intersect in a line, describe their
geometric configuration, j ustifying your answer. [2]
12 [Maximum mark: 22]
Let f(x) = x2 – 2 x – 3, x ∈ .
a Sketch the graph of y = |f(x)|. [3]
1
b Hence or otherwise, solve the inequality |f(x)| > – x + 4. [6]
2 x–7
2
Let g(x) = .
f(x)
c State the largest possible domain of g. [1]
d Find the coordinates of the turning points of g. [5]
e Sketch the graph of y = g(x), labelling all axis intercepts and asymptotes. [5]
f Hence find the range of g for the domain found in part c. [2]
