A-level · Algebra & functions

Simultaneous equations and inequalitiesExam-style questions

Two conditions at once, and conditions that pick out a range. What solving simultaneously means on a graph; a line meeting a curve, with the answers given as pairs; counting the meeting points with the discriminant; linear inequalities, and why dividing by a negative turns the sign round; quadratic inequalities from the critical values and a sketch; and the outside region, set notation and combining two conditions.

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15 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Find the set of values of $x$ satisfying both

$x^{2} - 4x - 5 \leqslant 0 \qquad \text{and} \qquad 2x + 1 > 5$ (4 marks)
Level 3 · 4 marksTry it
2
The equation $x^{2} + (k - 2)x + (k + 1) = 0$ has equal roots.

Find the possible values of $k$ . (4 marks)
Level 3 · 4 marksTry it
3
Solve the inequality $-5 \leqslant 3x + 1 < 10$ . (3 marks)
Level 3 · 3 marksTry it
4
The equation $kx^{2} + 4x + 1 = 0$ , where $k \neq 0$ , has two distinct real roots.

Find the set of possible values of $k$ . (4 marks)
Level 3 · 4 marksTry it
5
The line $y = x + c$ meets the curve $y = x^{2} - 3x + 7$ at two distinct points.

Find the set of possible values of $c$ . (4 marks)
Level 3 · 4 marksTry it

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