AS Level · Mathematics 9709 · P1

AS Level Mathematics Paper 1 · Pure Mathematics 1 practice papers

75 marks · 1h 50m. Each generated paper follows the 9709 Paper 1 · Pure Mathematics 1 format, with the same number and style of questions as the real exam, and comes with a Cambridge-style mark scheme.

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Sample Paper 1 · Pure Mathematics 1 questions

Question 1Structured · 9 marksA* challenge
(a)

A geometric progression has a sum to infinity of 24 and a second term of 3.84. Show that 24r2 − 24r + 3.84 = 0 and find the two possible values of the common ratio r.

[5]
(b)

For the larger value of r, find the sum of the first 6 terms.

[2]
(c)

A second progression has first term 1 and common ratio (2x − 1). Find the set of values of x for which it has a sum to infinity.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)r = 15 or 45
ar = 3.84 and a/(1 − r) = 24 B1
eliminate a M1
given quadratic obtained A1
solve M1
r = 15, 45 A1
5
(b)17.7
a = 4.8 M1
17.7 A1
2
(c)0 < x < 1
−1 < 2x − 1 < 1 M1
0 < x < 1 A1
2
Question 2Structured · 8 marksA* challenge

A curve has equation y = 362x − 1 for x > 12. The point P on the curve has x-coordinate 2.

(a)

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

[3]
(b)

Find the equation of the normal to the curve at P.

[2]
(c)

The tangent and the normal meet the y-axis at A and B respectively. Find the area of triangle PAB.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)y − 12 = −8(x − 2)
dydx = −72(2x − 1)−2 B1
gradient at P = −8 M1
y − 12 = −8(x − 2) A1
3
(b)y − 12 = 18(x − 2)
gradient = −1/m M1
y − 12 = 18(x − 2) A1
2
(c)16.3
A (0, 28) B1
B (0, 11.75) B1
area = ½ × AB × 2 = 16.3 A1
3
Question 3Structured · 6 marks
(a)

Express 2x2 − 12x + 12 in the form a(x + b)2 + c.

[3]
(b)

Find the set of values of k for which the line y = kx − 25 does not meet the curve y = x2.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2(x − 3)2 − 6
a = 2 B1
b = −3 B1
c = −6 B1
3
(b)−10 < k < 10
x2 − kx + 25 = 0, b2 − 4ac < 0 M1
k = ±10 A1
−10 < k < 10 A1
3
Question 4Structured · 5 marks

A sector OAB of a circle has radius 10 cm and angle AOB = 1.5 radians. Find (a) the perimeter of the sector, and (b) the area of the segment bounded by the chord AB and the arc AB.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a) 35 cm (b) 25.1 cm²
rθ M1
perimeter 35 A1
½r2θ M1
½r2sin θ subtracted M1
25.1 A1
5
Question 5Structured · 4 marks

Functions f and g are defined by f(x) = 2x + 2 and g(x) = 2x for x ≠ 0.

(a)

Find fg(x) and gf(x).

[2]
(b)

Solve fg(x) = 4.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)fg(x) = 4x + 2; gf(x) = 22x + 2
fg correct B1
gf correct B1
2
(b)x = 2
4x = 42 M1
x = 2 A1
2
Question 6Structured · 5 marks
(a)

A curve passes through (2, 1) and has gradient dydx = 12x2 + 2. Find the equation of the curve.

[3]
(b)

Find the area bounded by y = 12x2, the x-axis and the lines x = 0 and x = 2.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)y = 4x3 + 2x − 35
Integrate M1
Use point to find constant M1
y = 4x3 + 2x − 35 A1
3
(b)32
[4x3] from 0 to 2 M1
32 A1
2

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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