AS Level · Mathematics 9709 · P2

AS Level Mathematics Paper 2 · Pure Mathematics 2 practice papers

50 marks · 1h 15m. Each generated paper follows the 9709 Paper 2 · Pure Mathematics 2 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 2 · Pure Mathematics 2 questions

Question 1Structured · 5 marksA* challenge

The variables x and y satisfy the equation y = A b^x, where A and b are constants. The graph of ln y against x is a straight line passing through the points (0.6, 0.91) and (3.5, 3.32).

Find the values of A and b, giving your answers correct to 3 significant figures.

[5]

Show answers and mark scheme

Part	Answer and marking points	Marks
	A = 1.51, b = 2.3 ln y = ln A + x ln b stated B1 gradient from the two points M1 ln b = 0.831 A1 intercept found and exponentiated M1 A = 1.51, b = 2.3 A1	5

Question 2Structured · 6 marksA* challenge

A curve has equation y = ln xx² for x > 0.

(a)

Find the exact coordinates of the stationary point.

[4]

(b)

Determine the nature of the stationary point.

[2]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	(e^1/2, 12e) quotient rule M1 1 − 2ln xx³ A1 ln x = 12 M1 (e^1/2, 12e) A1	4
(b)	maximum: gradient changes from positive to negative valid method (sign of derivative either side or second derivative) M1 maximum A1	2

Question 3Structured · 4 marks

Solve the equation 3^2x − 82(3^x) + 81 = 0.

[4]

Show answers and mark scheme

Part	Answer and marking points	Marks
	x = 0, x = 4 u = 3^x M1 u² − 82u + 81 = 0 A1 u = 1, u = 81 A1 x = 0, x = 4 A1	4

Question 4Structured · 4 marks

Find the gradient of the curve y = x²e^3x at the point where x = 2. Give your answer correct to 3 significant figures.

[4]

Show answers and mark scheme

Part	Answer and marking points	Marks
	6450 Product rule attempted M1 2xe^3x + 3x²e^3x A1 Substitute x = 2 M1 6450 A1	4

Question 5Structured · 3 marks

The equation x³ − 7x − 4 = 0 has a root α. Use the iterative formula x_n+1 = ∛(7x_n + 4) with x₁ = 2 to find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
	α = 2.895 (iterates 2.62074, 2.81662, 2.87309, 2.88896…) Uses iteration correctly at least once M1 Iterates to sufficient accuracy M1 2.895 A1	3

Question 6Structured · 5 marks

The polynomial p(x) is defined by p(x) = 2x³ + ax² − 15x + 18, where a is a constant. It is given that (x − 2) is a factor of p(x).

(a)

Find the value of a.

[2]

(b)

Factorise p(x) completely.

[3]

Show answers and mark scheme

Part	Answer and marking points	Marks
(a)	a = −1 p(2) = 0 M1 a = −1 A1	2
(b)	(x − 2)(2x − 3)(x + 3) divide to find the quadratic factor M1 2x² + 3x − 9 A1 (x − 2)(2x − 3)(x + 3) A1	3

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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