surname names
Number Number
■ ■
Further Mathematics
Advanced
PAPER 2: Core Pure Mathematics 2
Candidates may use any calculator permitted by Pearson regulations. Calculators must
not have the facility for algebraic manipulation, differentiation and integration, or have
retrievable mathematical formulae stored in them.
Instructions
• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are clearly labelled.
Answer the questions in the spaces provided –
there may be more space than you need.
You should show sufficient working to make your methods clear. Answers
without working may not gain full credit.
Inexact answers should be given to three significant figures unless otherwise stated.
• There are 9 questions in this question paper. The total mark for this paper is 75. – use this
as a guide as to how much time to spend on each question.
• Read each question carefully before you start to answer it.
• Check your answers if you have time at the end. Turn over
,1. (a) Using the definition of sinh x in terms of exponentials, prove that
4 sinh3 x 3sinh x sinh 3x
(2)
(b) Hence solve the equation
sinh 3x 19 sinh x
giving your answers as simplified natural logarithms where appropriate.
(5)
2
■■■■
,Question 1 continued
(Total for Question 1 is 7 marks)
3
Turn over
■■■■
, 3 x < 3
2. f(x) = tanh 1
x 2
6 x
(a) Show that
1
f ′(x) = –
2x 3
(4)
(b) Hence determine f ′′(x)
(1)
(c) Hence show that the Maclaurin series for f(x), up to and including the term in x2, is
ln p qx rx2
where p, q and r are constants to be determined.
(3)
4
Number Number
■ ■
Further Mathematics
Advanced
PAPER 2: Core Pure Mathematics 2
Candidates may use any calculator permitted by Pearson regulations. Calculators must
not have the facility for algebraic manipulation, differentiation and integration, or have
retrievable mathematical formulae stored in them.
Instructions
• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are clearly labelled.
Answer the questions in the spaces provided –
there may be more space than you need.
You should show sufficient working to make your methods clear. Answers
without working may not gain full credit.
Inexact answers should be given to three significant figures unless otherwise stated.
• There are 9 questions in this question paper. The total mark for this paper is 75. – use this
as a guide as to how much time to spend on each question.
• Read each question carefully before you start to answer it.
• Check your answers if you have time at the end. Turn over
,1. (a) Using the definition of sinh x in terms of exponentials, prove that
4 sinh3 x 3sinh x sinh 3x
(2)
(b) Hence solve the equation
sinh 3x 19 sinh x
giving your answers as simplified natural logarithms where appropriate.
(5)
2
■■■■
,Question 1 continued
(Total for Question 1 is 7 marks)
3
Turn over
■■■■
, 3 x < 3
2. f(x) = tanh 1
x 2
6 x
(a) Show that
1
f ′(x) = –
2x 3
(4)
(b) Hence determine f ′′(x)
(1)
(c) Hence show that the Maclaurin series for f(x), up to and including the term in x2, is
ln p qx rx2
where p, q and r are constants to be determined.
(3)
4
