| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Complete the square technique |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise followed by a routine inequality. Part (i) is pure algebraic manipulation with clear guidance on the form required. Part (ii) is a standard application requiring only solving a quadratic inequality after rearrangement. Both parts are below average difficulty as they involve well-practiced techniques with no problem-solving insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((2x - 3)^2 - 9\) | B1B1 [2] | For \(-3\) and \(-9\) |
| (ii) \(2x - 3 > 4\) or \(2x - 3 < -4\) | M1 A1 [2] | At least one of these statements; Allow 'and' \(3\frac{1}{2}, -\frac{1}{2}\) soi scores first M1 |
| OR \(x > 3\frac{1}{2}\) (or) \(x < -\frac{1}{2}\) cao | M1 A1 [2] | Attempt to solve 3-term quadratic; Allow 'and' \(3\frac{1}{2}, -\frac{1}{2}\) soi scores first M1 |
| Answer | Marks |
|---|---|
| OR \(4x^2 - 12x - 7 \to (2x - 7)(2x + 1)\) | M1 A1 [2] |
(i) $(2x - 3)^2 - 9$ | B1B1 [2] | For $-3$ and $-9$
(ii) $2x - 3 > 4$ or $2x - 3 < -4$ | M1 A1 [2] | At least one of these statements; Allow 'and' $3\frac{1}{2}, -\frac{1}{2}$ soi scores first M1
OR $x > 3\frac{1}{2}$ (or) $x < -\frac{1}{2}$ cao | M1 A1 [2] | Attempt to solve 3-term quadratic; Allow 'and' $3\frac{1}{2}, -\frac{1}{2}$ soi scores first M1
Allow $-\frac{1}{2} > x > 3\frac{1}{2}$
OR $4x^2 - 12x - 7 \to (2x - 7)(2x + 1)$ | M1 A1 [2]
$x > 3\frac{1}{2}$ (or) $< -\frac{1}{2}$ cao
Allow $-\frac{1}{2} > x > 3\frac{1}{2}$2 (i) Express $4 x ^ { 2 } - 12 x$ in the form $( 2 x + a ) ^ { 2 } + b$.\\
(ii) Hence, or otherwise, find the set of values of $x$ satisfying $4 x ^ { 2 } - 12 x > 7$.
\hfill \mbox{\textit{CAIE P1 2014 Q2 [4]}}