FP1 Further Pure
Mathematics Advanced
Level 2025 Practice Test
, Paper Reference(s)
6667/01
Edexcel GCE
Further Pure Mathematics FP1
Advanced Level
Friday 30 January 2018 Afternoon
Time: 1 hour 30 minutes
Materials required for examination Items included with question papers
Mathematical Formulae (Orange) Nil
Candidates may use any calculator allowed by the regulations of the Joint Council
for Qualifications. Calculators must not have the facility for symbolic algebra
manipulation, differentiation or integration, or have retrievable mathematical
formulae stored in them.
Instructions to Candidates
In the boxes on the answer book, write the name of the examining body (Edexcel), your
centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper
reference (6667), your surname, initials and signature.
When a calculator is used, the answer should be given to an appropriate degree of
accuracy.
Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 10 questions on this paper. The total mark for this paper is 75.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
N34694A
.
,1. f(x) = 2x3 – 8x2 + 7x – 3.
Given that x = 3 is a solution of the equation f(x) = 0, solve f(x) = 0 completely.
(5)
2. (a) Show, using the formulae for r and r 2 , that
n
(6r 2 4r 1) = n(n + 2)(2n + 1).
r 1
(5)
20
(b) Hence, or otherwise, find the value of (6r 2 4r 1) .
r 11
(2)
5
3. The rectangular hyperbola, H, has parametric equations x = 5t, y = , t ≠ 0.
t
(a) Write the cartesian equation of H in the form xy = c2.
(1)
Points A and B on the hyperbola have parameters t = 1 and t = 5 respectively.
(b) Find the coordinates of the mid-point of AB.
(3)
4. Prove by induction that, for n A+,
n
1
=
n
.
r 1 r(r 1) n 1
(5)
18
5. f(x) = 3x + – 20.
x
(a) Show that the equation f(x) = 0 has a root a in the interval [1.1, 1.2].
(2)
(b) Find f '(x).
(3)
(c) Using x0 = 1.1 as a first approximation to a, apply the Newton-Raphson procedure once
to f(x) to find a second approximation to a, giving your answer to 3 significant figures.
(4)
N34694A 2
, 6. A series of positive integers u1, u2, u3, ... is defined by
u1 = 6 and un + 1 = 6un – 5, for n 1.
Prove by induction that un = 5 × 6n – 1 + 1, for n 1.
(5)
2 a
Given that X = , where a is a constant, and a ≠ 2,
7. 1 1
(a) find X–1 in terms of a.
(3)
Given that X + X–1 = I, where I is the 2 × 2 identity matrix,
(b) find the value of a.
(3)
8. A parabola has equation y2 = 4ax, a > 0. The point Q (aq2, 2aq) lies on the parabola.
(a) Show that an equation of the tangent to the parabola at Q is
yq = x + aq2.
(4)
This tangent meets the y-axis at the point R.
(b) Find an equation of the line l which passes through R and is perpendicular to the tangent
at Q.
(3)
(c) Show that l passes through the focus of the parabola.
(1)
(d) Find the coordinates of the point where l meets the directrix of the parabola.
(2)
N34694A 3
Mathematics Advanced
Level 2025 Practice Test
, Paper Reference(s)
6667/01
Edexcel GCE
Further Pure Mathematics FP1
Advanced Level
Friday 30 January 2018 Afternoon
Time: 1 hour 30 minutes
Materials required for examination Items included with question papers
Mathematical Formulae (Orange) Nil
Candidates may use any calculator allowed by the regulations of the Joint Council
for Qualifications. Calculators must not have the facility for symbolic algebra
manipulation, differentiation or integration, or have retrievable mathematical
formulae stored in them.
Instructions to Candidates
In the boxes on the answer book, write the name of the examining body (Edexcel), your
centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper
reference (6667), your surname, initials and signature.
When a calculator is used, the answer should be given to an appropriate degree of
accuracy.
Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 10 questions on this paper. The total mark for this paper is 75.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
N34694A
.
,1. f(x) = 2x3 – 8x2 + 7x – 3.
Given that x = 3 is a solution of the equation f(x) = 0, solve f(x) = 0 completely.
(5)
2. (a) Show, using the formulae for r and r 2 , that
n
(6r 2 4r 1) = n(n + 2)(2n + 1).
r 1
(5)
20
(b) Hence, or otherwise, find the value of (6r 2 4r 1) .
r 11
(2)
5
3. The rectangular hyperbola, H, has parametric equations x = 5t, y = , t ≠ 0.
t
(a) Write the cartesian equation of H in the form xy = c2.
(1)
Points A and B on the hyperbola have parameters t = 1 and t = 5 respectively.
(b) Find the coordinates of the mid-point of AB.
(3)
4. Prove by induction that, for n A+,
n
1
=
n
.
r 1 r(r 1) n 1
(5)
18
5. f(x) = 3x + – 20.
x
(a) Show that the equation f(x) = 0 has a root a in the interval [1.1, 1.2].
(2)
(b) Find f '(x).
(3)
(c) Using x0 = 1.1 as a first approximation to a, apply the Newton-Raphson procedure once
to f(x) to find a second approximation to a, giving your answer to 3 significant figures.
(4)
N34694A 2
, 6. A series of positive integers u1, u2, u3, ... is defined by
u1 = 6 and un + 1 = 6un – 5, for n 1.
Prove by induction that un = 5 × 6n – 1 + 1, for n 1.
(5)
2 a
Given that X = , where a is a constant, and a ≠ 2,
7. 1 1
(a) find X–1 in terms of a.
(3)
Given that X + X–1 = I, where I is the 2 × 2 identity matrix,
(b) find the value of a.
(3)
8. A parabola has equation y2 = 4ax, a > 0. The point Q (aq2, 2aq) lies on the parabola.
(a) Show that an equation of the tangent to the parabola at Q is
yq = x + aq2.
(4)
This tangent meets the y-axis at the point R.
(b) Find an equation of the line l which passes through R and is perpendicular to the tangent
at Q.
(3)
(c) Show that l passes through the focus of the parabola.
(1)
(d) Find the coordinates of the point where l meets the directrix of the parabola.
(2)
N34694A 3
