DP1 Mock – Analysis and Approaches

Mathematics: Standard Level

90 minutes

Candidate Name: ____________________________________

INSTRUCTIONS TO CANDIDATES

 Write your session number in the boxes above.

 Do not open this examination paper until instructed to do so.
 You are permitted access to any calculator for this paper.
 Unless otherwise stated in the question, all numerical answers should be given exactly
or correct to three significant figures.
 A clean copy of the formula booklet is required for this paper.
 Section A: Write your answers on the question paper
 Section B: Write your answers on the separate lined paper
 The maximum mark for this examination paper is [80 marks].

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.

Academic Honesty Honour Pledge:

I affirm that this work is entirely my own and that I have not used any dishonest means to answer
this examination, nor have I received or given any unauthorised help on this exam.

Signed: ………………………………………………
*Parent signature: ……………………………………

*Where a parent is available to sign

Section A
Question 1 [4 marks]
The function 𝑔 is defined by 𝑔(𝑥) = e , where 𝑥 ∈ ℝ.
Find 𝑔′(−1). [4 marks]

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Question 2 [Maximum 6]
In an arithmetic sequence, 𝑢 = 5 and 𝑢 = 11.
a) Find the common difference. [2 marks]
b) Find the first term. [2 marks]
c) Find the sum of the first 20 terms. [2 marks]

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Question 3 [Maximum 5]
The following diagram shows triangle ABC, with AB = 6 and AC = 8.

a) Given that cos 𝐴 = find the value of sin 𝐴. [3 marks]

b) Find the area of triangle ABC. [2 marks]

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Question 4 [Maximum 5]
a) Show that (2𝑛 − 1) + (2𝑛 + 1) = 8𝑛 + 2, where 𝑛 ∈ ℤ. [2 marks]
b) Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers
is even. [3 marks]

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Question 5 [Maximum 6]
Let 𝑓(𝑥) = −𝑥 + 4𝑥 + 5 and 𝑔(𝑥) = −𝑓(𝑥) + 𝑘.
Find the values of 𝑘 so that 𝑔(𝑥) = 0 has no real roots. [6 marks]

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Question 6 [Maximum 5]
Consider the function 𝑓(𝑥) = 𝑒 − 0.5, for −2 ≤ 𝑥 ≤ 2.
a) Find the values of 𝑥 for which 𝑓(𝑥) = 0. [2 marks]
b) Sketch the graph of 𝑓 on the following grid. [3 marks]

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Question 7 [Maximum 6]
The following diagram shows part of a circle with centre O and radius 4 cm.

Chord AB has a length of 5 cm and AÔB = θ.

a) Find the value of θ, giving your answer in radians. [3 marks]
b) Find the area of the shaded region. [3 marks]

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Section B
Question 8 [Maximum 14]
The following diagram shows the graph of 𝑦 = −1 − √𝑥 + 3 for 𝑥 ≥ −3.

a) Describe a sequence of transformations that transforms the graph of 𝑦 = √𝑥 for 𝑥 ≥

0 to the graph of 𝑦 = −1 − √𝑥 + 3 for 𝑥 ≥ −3. [3 marks]

A function 𝑓 is defined by 𝑓(𝑥) = −1 − √𝑥 + 3 for 𝑥 ≥ −3.

b) State the range of 𝑓. [1 mark]
c) Find an expression for 𝑓 (𝑥), stating its domain. [5 marks]
d) Find the coordinates of the point(s) where the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑓 (𝑥) intersect.
[5 marks]
Question 9 [15 marks]
Consider the function 𝑓(𝑥) = 𝑎 where 𝑥, 𝑎 ∈ ℝ and 𝑥 > 0, 𝑎 > 1.

The graph of 𝑓 contains the point ,4 .

a) Show that 𝑎 = 8. [2 marks]

b) Write down an expression for 𝑓 (𝑥). [1 mark]

c) Find the value of 𝑓 √32 . [3 marks]

Consider the arithmetic sequence log 27 , log 𝑝 , log 𝑞 , log 125 , where 𝑝 > 1 and 𝑞 > 1.
d) Show that 27, 𝑝, 𝑞 and 125 are four consecutive terms in a geometric sequence.
[4 marks]
e) Find the value of 𝑝 and the value of 𝑞. [5 marks]
Question 10 [Maximum 14]
Let 𝑓(𝑥) = 𝑚𝑥 − 2𝑚𝑥, where 𝑥 ∈ ℝ and 𝑚 ∈ ℝ. The line 𝑦 = 𝑚𝑥 − 9 meets the graph of 𝑓 at
exactly one point.
a) Show that 𝑚 = 4. [6 marks]
The function 𝑓 can be expressed in the form 𝑓(𝑥) = 4(𝑥 − 𝑝)(𝑥 − 𝑞), where 𝑝, 𝑞 ∈ ℝ.
b) Find the value of 𝑝 and the value of 𝑞. [2 marks]
The function 𝑓 can also be expressed in the form 𝑓(𝑥) = 4(𝑥 − ℎ) + 𝑘, where ℎ, 𝑘 ∈ ℝ.
c) Find the value of ℎ and the value of 𝑘. [3 marks]
d) Hence find the values of 𝑥 where the graph of 𝑓 is both negative and increasing.
[3 marks]