Cambridge International Examinations
Cambridge International Advanced Level
CANDIDATE
NAME
CENTRE CANDIDATE
*0123456789*
NUMBER NUMBER
MATHEMATICS 9709/03
Paper 3 Pure Mathematics 3 (P3) For Examination from 2017
SPECIMEN PAPER 1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
Write your Centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
This document consists of 19 printed pages and 1 blank page.
© UCLES 2016 [Turn over
, 2
1 Solve the inequality 2x − 5 > 32x + 1. [4]
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 3
2 Using the substitution u = 3x , solve the equation 3x + 32x = 33x giving your answer correct to 3 significant
figures. [5]
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 4
3 The angles 1 and & lie between 0Å and 180Å, and are such that
tan 1 − & = 3 and tan 1 + tan & = 1.
Find the possible values of 1 and &. [6]
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 5
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 6
4 The equation x3 − x2 − 6 = 0 has one real root, denoted by !.
(i) Find by calculation the pair of consecutive integers between which ! lies. [2]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(ii) Show that, if a sequence of values given by the iterative formula
_P Q
6
xn+1 = xn +
xn
converges, then it converges to !. [2]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 7
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iii) Use this iterative formula to determine ! correct to 3 decimal places. Give the result of each
iteration to 5 decimal places. [3]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 8
5 The equation of a curve is y = e−2x tan x, for 0 ≤ x < 12 0.
dy
(i) Obtain an expression for and show that it can be written in the form e−2x a + b tan x2 , where
dx
a and b are constants. [5]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 9
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(ii) Explain why the gradient of the curve is never negative. [1]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iii) Find the value of x for which the gradient is least. [1]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 10
6 The polynomial 8x3 + ax2 + bx − 1, where a and b are constants, is denoted by p x. It is given that
x + 1 is a factor of p x and that when p x is divided by 2x + 1 the remainder is 1.
(i) Find the values of a and b. [5]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17
Cambridge International Advanced Level
CANDIDATE
NAME
CENTRE CANDIDATE
*0123456789*
NUMBER NUMBER
MATHEMATICS 9709/03
Paper 3 Pure Mathematics 3 (P3) For Examination from 2017
SPECIMEN PAPER 1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)
READ THESE INSTRUCTIONS FIRST
Write your Centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
This document consists of 19 printed pages and 1 blank page.
© UCLES 2016 [Turn over
, 2
1 Solve the inequality 2x − 5 > 32x + 1. [4]
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 3
2 Using the substitution u = 3x , solve the equation 3x + 32x = 33x giving your answer correct to 3 significant
figures. [5]
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 4
3 The angles 1 and & lie between 0Å and 180Å, and are such that
tan 1 − & = 3 and tan 1 + tan & = 1.
Find the possible values of 1 and &. [6]
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 5
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
................................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 6
4 The equation x3 − x2 − 6 = 0 has one real root, denoted by !.
(i) Find by calculation the pair of consecutive integers between which ! lies. [2]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(ii) Show that, if a sequence of values given by the iterative formula
_P Q
6
xn+1 = xn +
xn
converges, then it converges to !. [2]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 7
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iii) Use this iterative formula to determine ! correct to 3 decimal places. Give the result of each
iteration to 5 decimal places. [3]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 8
5 The equation of a curve is y = e−2x tan x, for 0 ≤ x < 12 0.
dy
(i) Obtain an expression for and show that it can be written in the form e−2x a + b tan x2 , where
dx
a and b are constants. [5]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17
, 9
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(ii) Explain why the gradient of the curve is never negative. [1]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iii) Find the value of x for which the gradient is least. [1]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17 [Turn over
, 10
6 The polynomial 8x3 + ax2 + bx − 1, where a and b are constants, is denoted by p x. It is given that
x + 1 is a factor of p x and that when p x is divided by 2x + 1 the remainder is 1.
(i) Find the values of a and b. [5]
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© UCLES 2016 9709/03/SP/17
