Higher January 2021 Paper 2R Q18
18 Solve the equation
\(\dfrac{5}{x + 2} + \dfrac{3}{x^2 + 2x} = 2\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
\(\dfrac{5}{x + 2} + \dfrac{3}{x(x + 2)}\;(= 2)\) or \(\dfrac{5x}{x^2 + 2x} + \dfrac{3}{x^2 + 2x}\;(= 2)\) | M1 |
\(\dfrac{5x + 3}{x(x + 2)} = 2\) or \(\dfrac{5x + 3}{x^2 + 2x} = 2\) or \(5x + 3 = 2x(x + 2)\) oe or \(5x + 3 = 2x^2 + 4x\) oe | M1 |
| \(2x^2 - x - 3\;(= 0)\) | M1 |
\((2x - 3)(x + 1)\;(= 0)\) or \(\dfrac{-{-1} \pm \sqrt{(-1)^2 - 4 \times 2 \times (-3)}}{2 \times 2}\) or \(\left(x - \dfrac{1}{4}\right)^2 - \dfrac{1}{16} - \dfrac{3}{2} = 0\) oe | M1ft |
| Working required Answer: 1.5 and –1 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: Factorising \(x^2 + 2x\) in correct expression on LHS or for writing the two fractions over a common denominator.
M1: Correct simplified single fraction = 2
or correct equation with no fractions.
M1: Correct 3 term quadratic
M1ft: independent
For solving their 3 term quadratic equation using any correct method.
If factorising, allow brackets which expanded give 2 out of 3 terms correct (if using formula or completing the square allow one sign error and some simplification – allow as far as eg \(\dfrac{1 \pm \sqrt{1 + 24}}{4}\) or eg \(\left(x - \dfrac{1}{4}\right)^2 = \dfrac{25}{16}\) oe
A1: oe dep on M3
| Scheme | Marks |
|---|---|
| \(\dfrac{5(x^2 + 2x) + 3(x + 2)}{(x^2 + 2x)(x + 2)}\;(= 2)\) oe | M1 |
| eg \(5(x^2 + 2x) + 3(x + 2) = 2(x^2 + 2x)(x + 2)\) oe | M1 |
| \(2x^3 + 3x^2 - 5x - 6\;(= 0)\) | M1 |
| \((x + 1)(2x - 3)(x + 2)\;(= 0)\) | M1 |
| Working required Answer: 1.5 and –1 | A1 |
Notes
M1: Correct fraction over a common denominator (may be 2 separate fractions)
M1: Correct equation with no fractions.
M1: Correct cubic
M1: For product of 3 correct linear factors.
A1: oe dep on M3
Do not award A mark if extra solution (–2) given.