AS Level · Mathematics 9709 · Topic

Pure Mathematics 1: AS Level Mathematics questions and answers

10 free AS Level Mathematics Pure Mathematics 1 questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Quadratics, Functions, Coordinate geometry, Circular measure, Trigonometry, Series, Differentiation, Integration. Work through the questions below, then generate a full Pure Mathematics 1 paper with a fresh set every time.

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Pure Mathematics 1 sub-topics

QuadraticsFunctionsCoordinate geometryCircular measureTrigonometrySeriesDifferentiationIntegration

Pure Mathematics 1 practice questions

Question 1Structured · 7 marksA* challenge

The region R is bounded by the curve y = 2x4, the x-axis and the line x = 4, for x ≥ 4, and extends indefinitely in the positive x-direction.

(a)

Show that R has a finite area and find its exact value.

[4]
(b)

R is rotated through 360° about the x-axis. Find the exact volume of the solid formed.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)196
∫ kx−n dx M1
−23x3 A1
limit as upper bound → ∞ is 0 M1
196 A1
4
(b)128672π
π∫ y2 dx M1
correct integral and limits M1
128672π A1
3
Question 2Structured · 9 marksA* challenge

A circle has equation x2 + y2 + 6x + 6y − 7 = 0.

(a)

Find the centre and radius of the circle.

[3]
(b)

Show that the point P (1, 0) lies on the circle.

[1]
(c)

Find the equation of the tangent to the circle at P.

[3]
(d)

The tangent meets the x-axis at T. Find the length PT.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)centre (−3, −3), radius 5
completing the square in x and y M1
centre (−3, −3) A1
radius 5 A1
3
(b)substitution gives 0
substitute and obtain 0 (or distance from centre = 5) B1
1
(c)y = −43(x − 1)
gradient of radius 34 B1
tangent gradient −43 M1
y = −43(x − 1) A1
3
(d)PT = 0
x-coordinate of T = 1 M1
PT = 0 A1
2
Question 3Structured · 5 marksA* challenge

Find the two values of k for which the x-axis is a tangent to the curve y = x3 − 3x2 − 9x + k.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
k = 27, k = −5
dydx = 3x2 − 6x − 9 B1
stationary points x = 3, x = −1 M1
y = 0 at a stationary point M1
k = 27 A1
k = −5 A1
5
Question 4Structured · 9 marksA* challenge

The functions f and g are defined for x ∈ ℝ by f(x) = 3x − 4 and g(x) = x2 + 1.

(a)

Find expressions for fg(x) and gf(x).

[2]
(b)

Solve the equation fg(x) = gf(x).

[3]
(c)

Find the set of values of x for which gf(x) > fg(x).

[2]
(d)

Find f−1(x) and solve f−1(x) = f(x).

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)fg(x) = 3x2 − 1; gf(x) = 9x2 − 24x + 17
fg(x) = 3x2 − 1 B1
gf(x) = (3x − 4)2 + 1 = 9x2 − 24x + 17 B1
2
(b)x = 1, x = 3
6x2 − 24x + 18 = 0 M1
x2 − 4x + 3 = 0 A1
x = 1, 3 A1
3
(c)x < 1 or x > 3
gf(x) − fg(x) = 6(x − α)(x − β) > 0 M1
x < 1, x > 3 A1
2
(d)f−1(x) = x + 43; x = 2
f−1(x) = x + 43 B1
x = 2 (where f(x) = x) B1
2
Question 5Structured · 2 marks

The graph of y = f(x) is transformed by a stretch parallel to the y-axis with factor 2, then a translation of 03. Write down the equation of the new graph in terms of f.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
y = 2f(x) + 3
one transformation correct B1
y = 2f(x) + 3 B1
2
Question 6Structured · 5 marks

The region bounded by the curve y = x2 + 1, the x-axis and the lines x = 0 and x = 4 is rotated through 360° about the x-axis. Find the exact volume.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
377215π
π∫y2 dx M1
(x2 + 1)2 expanded M1
integrate A1
limits M1
exact answer A1
5
Question 7Structured · 5 marks
(a)

Find ∫1∞ 5x2 dx.

[3]
(b)

Find ∫04 1√x dx.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)5
[−5x] M1
limit as upper bound → ∞ M1
5 A1
3
(b)4
2√x M1
4 A1
2
Question 8Structured · 5 marks
(a)

A curve passes through (2, −5) and has gradient dydx = 9x2 + 4. Find the equation of the curve.

[3]
(b)

Find the area bounded by y = 9x2, the x-axis and the lines x = 0 and x = 2.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)y = 3x3 + 4x − 37
Integrate M1
Use point to find constant M1
y = 3x3 + 4x − 37 A1
3
(b)24
[3x3] from 0 to 2 M1
24 A1
2
Question 9Structured · 4 marks

Solve the equation x − 5√x + 4 = 0.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
x = 1, x = 16
substitute u = √x M1
u2 − 5u + 4 = 0 A1
u = 1 or 4 A1
x = 1, 16 A1
4
Question 10Structured · 4 marks

The volume of a spherical balloon is increasing at a constant rate of 40 cm³ s−1. Find the rate of increase of the radius when the radius is 8 cm.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.0497 cm s−1
dVdr = 4πr2 B1
drdt = dVdt ÷ dVdr M1
substitute M1
0.0497 A1
4

Other AS Level Mathematics topics

Pure Mathematics 2MechanicsProbability & Statistics 1

Frequently asked questions

What is in AS Level Mathematics Pure Mathematics 1?

In the 9709 syllabus, Pure Mathematics 1 covers Quadratics, Functions, Coordinate geometry, Circular measure, Trigonometry, Series, Differentiation, Integration.

Where can I get Pure Mathematics 1 questions with mark schemes?

This page has 10 Pure Mathematics 1 questions with answers. For more, the free generator on mentorriyaz.in builds a new Pure Mathematics 1 paper with a full mark scheme each time you click.

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in