AS Level · Mathematics 9709 · Pure Mathematics 1

Quadratics: AS Level Mathematics questions with answers

Free Quadratics questions for AS Level Mathematics (9709), part of Pure Mathematics 1. Each question has the answer and the marking points an examiner would look for.

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Quadratics practice questions

Question 1Structured · 8 marksA* challenge

The function f is defined by f(x) = 3x2 + 12x + 17 for x ∈ ℝ.

(a)

Express f(x) in the form a(x + b)2 + c, where a, b and c are constants.

[3]
(b)

Find the set of values of x for which f(x) < 17.

[3]
(c)

The function g is defined by g(x) = 24f(x) for x ∈ ℝ. Find the range of g.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3(x + 2)2 + 5
3(x + 2)2 B1
completing the square method M1
c = 5 A1
3
(b)−4 < x < 0
(x + 2)2 < 4 M1
critical values −4, 0 A1
−4 < x < 0 A1
3
(c)0 < g(x) ≤ 245
greatest value 245 when f is least M1
0 < g(x) ≤ 245 A1
2
Question 2Structured · 4 marksA* challenge

Find the values of k for which the equation x2 + (k − 2)x + (k + 1) = 0 has equal roots.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
k = 0, k = 8
(k − 2)2 − 4(k + 1) = 0 M1
k2 − 8k = 0 A1
k = 0 A1
k = 8 A1
4
Question 3Structured · 6 marks
(a)

Express 2x2 + 4x − 7 in the form a(x + b)2 + c.

[3]
(b)

Find the set of values of k for which the line y = kx − 4 does not meet the curve y = x2.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2(x + 1)2 − 9
a = 2 B1
b = 1 B1
c = −9 B1
3
(b)−4 < k < 4
x2 − kx + 4 = 0, b2 − 4ac < 0 M1
k = ±4 A1
−4 < k < 4 A1
3
Question 4Structured · 4 marks

Solve the equation 2x − 20√x + 18 = 0.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
x = 1, x = 81
substitute u = √x M1
2u2 − 20u + 18 = 0 A1
u = 1 or 9 A1
x = 1, 81 A1
4
Question 5Structured · 7 marksA* challenge

The line y = kx + 3 and the curve y = x2 − 4x + 12 are given, where k is a constant.

(a)

Find the set of values of k for which the line meets the curve at two distinct points.

[4]
(b)

For the case k = 6, the line meets the curve at A and B. Find the coordinates of the mid-point of AB.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)k < −10 or k > 2
x2 − (4 + k)x + 9 = 0 M1
(4 + k)2 − 36 > 0 M1
critical values −10, 2 A1
k < −10, k > 2 A1
4
(b)(5, 33)
x2 − 10x + 9 = 0 solved or sum of roots M1
x = 1, 9 A1
(5, 33) A1
3
Question 6Structured · 7 marksA* challenge

The equation kx2 + (k − 9)x + k = 0, where k is a constant, is a quadratic equation in x.

(a)

Find the set of values of k for which the equation has two distinct real roots.

[5]
(b)

Find the value of k for which the equation has equal roots and k > 0. Find the root in this case.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)−9 < k < 0 or 0 < k < 3
(k − 9)2 − 4k2 > 0 M1
(k + 9)(3k − 9) < 0 or 3k2 + 18k − 81 < 0 A1
critical values −9 and 3 A1
−9 < k < 3 A1
k ≠ 0 stated (equation must be quadratic) B1
5
(b)k = 3, x = 1
k = 3 B1
x = 1 B1
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