A Level Edexcel Exam Set B Practice Paper 1
Time: 2 hours
Do not write on this paper. Give your answers to 3 significant figures unless otherwise specified.
1.
Calculate the following integrals:
x4
(a) ⌠
 dx
⌡ 5
1
−1
(b) ⌠
 x x 2 − x 2 dx
⌡
(
2.
(a)
(b)
3.
(2)
)
(3)
cot θ
kπ
, k ∈
for θ ≠
2
tan θ + cot θ
cot 2 θ
Hence or otherwise simplify fully the expression
tan 2 θ + cot 2 θ + 2
(4)
Simplify fully the expression
The diagram below shows the graph of y = f(x)
(2)
y
2
y = f(x)
–2
–1
O
x
(a)
(b)
Sketch the graph of y = f(x – 3)
Describe the transformation that transforms the graph of y = f(x) into the graph of
y = f(–x)
The point (3, 2) lies on the graph of y = g(x)
(c) State the coordinates of the image of this point when y = g(x) is transformed to the
following graphs:
(i) y = 6g(x)
(ii) y = g(6x)
4.
5.
The points A, B and C have position vectors 2i – 7j + k, 4i + 5j – 7k and j + 5k respectively.
The point M is the midpoint of the line segment AB.
→
(a) Show that the magnitude of OM is 19
→ →
The point D is such that AB = CD
(b) Show that the position vector of D is 2i + 13j – 3k
(2)
(1)
(1)
(1)
(2)
(3)
Charles investigates the population of finches on an island. He models the population of finches
on the island, F, by the formula F = 80 e0.2t, where t is the time in years since 1st January 2018.
(a) Using Charles’s model:
(i) state the population of finches on the island on 1st January 2018
(1)
(ii) predict the population of finches on the island on 1st January 2028
(1)
(b) According to Charles’s model, at what value of t will the population of finches be equal
to 2000? Give your answer to 1 decimal place.
(2)
(c) Give one reason why Charles’s model is not appropriate for large values of t
(1)
On the same island, Charles determines that the population of ants, A, may be modelled by the
formula A = 2400 e0.1t, where t is the time in years since 1st January 2018.
(d) Determine the year during which the population of finches will first exceed the population
of ants according to Charles’s models.
(3)
A Level Edexcel Mathematics Set B Paper 1
Page 4 of 48
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6.
A triangle includes an obtuse angle θ. The two shorter sides of the triangle have lengths 4 cm
and 7.5 cm; the area of the triangle is 10 cm2.
Find the exact value of tan θ.
(4)
7.
Prove or disprove each of the following statements:
(a) If n is an integer, then 3n 2 − 11n + 13 is a prime number.
(b) If x is a real number, then x 2 − 8 x + 17 is positive.
(c) If p and q are irrational numbers, then pq is irrational.
(2)
(2)
(2)
8.
Prove from first principles that the derivative of sin x with respect to x is cos x, where x is
measured in radians.
cos h − 1
sin h
= 0 and use without proof the addition
= 1 and lim
You may assume that lim
h →0
h →0
h
h
formulae for sine.
(5)
9.
(a)
(b)
10.
Given that
B
C
5x2 + 6 x − 2
A
=+
+
, find the integers A, B and C
x +1 x − 2
( x + 1)( x − 2 )
(4)
5x2 + 6 x − 2
Hence or otherwise expand
in ascending powers of x up to and including
( x + 1)( x − 2 )
the term in x 2
(6)
The shape below is a rectangle attached to a semicircle. It has area A and perimeter P.
y cm
x cm
(a)
(b)
11.
Consider the function f( x) = x 6 − x 2 − 1. The equation f(x) = 0 has a solution α in the interval
(n, n + 1) where n is a positive integer.
(a) Find n, showing detailed reasoning for your answer.
(3)
(b)
(c)
12.
x2 π x2
cm 2
−
(5)
2
8
Find the value of x for which the area is maximal, giving your answer to 1 decimal place.
You must clearly show that your value is a maximum.
(6)
If P = 100 cm, then show that A = 50 x −
Starting with x0 = n, use an iterative formula based on the equation =
x
to 3 decimal places.
Show that α = 1.151 is correct to 3 decimal places.
6
1 + x 2 to find x3
The circle C has equation x 2 − 6 x + y 2 + 16 y =
8
(a) Find the centre and radius of C
(b) PQ is a chord of C whose midpoint is the origin. Show that the length of PQ is k 2
where k is an integer to be found.
A Level Edexcel Mathematics Set B Paper 1
Page 5 of 48
(2)
(3)
(3)
(5)
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13.
Jenny is training for a marathon. In the first week of the year she runs for 10 minutes. Each
week she plans to run for 5 minutes longer than in the previous week. So in the second week
of the year she will run for 15 minutes, in the third week for 20 minutes, and so on.
(a) For how long will Jenny run during the 52nd week of the year?
(2)
(b) How long will Jenny have spent running in total over the 52 weeks? Give your answer
to the nearest hour.
(3)
Jenny’s brother is training for a half marathon. He starts the year by running for 10 minutes,
and he plans to increase the time he spends running each week by 5%.
(c) How long will Jenny’s brother have spent running in total after 52 weeks with this plan?
Give your answer to the nearest hour.
(4)
14.
The diagram to the right shows the curve C with equation
y=
x 2 − 4 x + 11, x ≥ 0
The line L is the tangent to C at the point P(4, 19)
The shaded region R is bounded by C, L and the y-axis
Show that the area of R is 24
(10)
y
C
L
O
R
x
Total 100 Marks
A Level Edexcel Mathematics Set B Paper 1
Page 6 of 48
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