Equations and inequalities
Algebra and functions
- AS
- Non-calculator
- 8 questions
- 37 marks
These are original practice questions for the Edexcel A-level Pure content, not a past paper. This site is independent and not affiliated with Pearson.
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Solve the simultaneous equations $2x + y = 7$ and $x - y = 2$.
3 marks
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Solve the simultaneous equations $x + y = 5$ and $x^2 + y^2 = 13$.
3 marks
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Solve the inequalities $2x - 1 < 5$ and $3x + 2 \ge 8$, and write the solution as a single inequality.
4 marks
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Solve $x^2 - 5x + 6 > 0$.
4 marks
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Solve $x^2 < 9$.
5 marks
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Solve the simultaneous equations $y = x + 1$ and $y = x^2 - 3x + 3$.
5 marks
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Solve $(x - 1)(x + 4) \le 6$.
6 marks
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Find the set of values of $k$ for which the line $y = kx - 1$ and the curve $y = x^2 + 2$ meet at two distinct points.
7 marks
Mark scheme
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Answer. $x = 3$, $y = 1$
- Add the equations: $3x = 9$.
- $x = 3$ and $y = 1$.
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Answer. $(x, y) = (2, 3)$ or $(3, 2)$
- Substitute $y = 5 - x$ into the second equation.
- $x^2 - 5x + 6 = 0$, so $x = 2$ or $x = 3$.
- The pairs are $(2, 3)$ and $(3, 2)$.
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Answer. $2 \le x < 3$
- $2x < 6$, so $x < 3$.
- $3x \ge 6$, so $x \ge 2$.
- Both together give $2 \le x < 3$.
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Answer. $x < 2$ or $x > 3$
- $(x - 2)(x - 3) > 0$.
- The product is positive outside the roots.
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Answer. $-3 < x < 3$
- $x^2 - 9 < 0$, so $(x - 3)(x + 3) < 0$.
- The solution is between the roots.
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Answer. $x = 2 \pm \sqrt{2}$ and $y = 3 \pm \sqrt{2}$, with matching signs.
- $x + 1 = x^2 - 3x + 3$.
- $x^2 - 4x + 2 = 0$.
- $x = 2 \pm \sqrt{2}$ and $y = x + 1 = 3 \pm \sqrt{2}$.
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Answer. $-5 \le x \le 2$
- $x^2 + 3x - 4 \le 6$.
- $x^2 + 3x - 10 \le 0$, so $(x + 5)(x - 2) \le 0$.
- The solution is $-5 \le x \le 2$.
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Answer. $k < -2\sqrt{3}$ or $k > 2\sqrt{3}$
- $x^2 - kx + 3 = 0$.
- Two distinct solutions require $k^2 - 12 > 0$.
- $k^2 > 12$, so $k < -2\sqrt{3}$ or $k > 2\sqrt{3}$.