| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a routine AS-level question testing standard completing the square technique with a negative coefficient, followed by straightforward reading of the maximum value and solving a composite function equation. All parts are textbook exercises requiring only procedural fluency with no problem-solving insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-2(x-3)^2 + 15\) \((a = -3,\ b = 15)\) | B1 B1 | Or seen as \(a = -3\), \(b = 15\). B1 for each value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((f(x) \leqslant)\ 15\) | B1 | FT for \((\leqslant)\) their "\(b\)". Don't accept \((3,15)\) alone |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(gf(x) = 2(-2x^2 + 12x - 3) + 5 = -4x^2 + 24x - 6 + 5\) | B1 | |
| \(gf(x) + 1 = 0 \rightarrow -4x^2 + 24x = 0\) | M1 | |
| \(x = 0\) or \(6\) | A1 | Forms and attempts to solve a quadratic. Both answers given |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-2(x-3)^2 + 15$ $(a = -3,\ b = 15)$ | B1 B1 | Or seen as $a = -3$, $b = 15$. B1 for each value |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(f(x) \leqslant)\ 15$ | B1 | FT for $(\leqslant)$ their "$b$". Don't accept $(3,15)$ alone |
## Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $gf(x) = 2(-2x^2 + 12x - 3) + 5 = -4x^2 + 24x - 6 + 5$ | B1 | |
| $gf(x) + 1 = 0 \rightarrow -4x^2 + 24x = 0$ | M1 | |
| $x = 0$ or $6$ | A1 | Forms and attempts to solve a quadratic. Both answers given |
---5 The function f is defined by $\mathrm { f } ( x ) = - 2 x ^ { 2 } + 12 x - 3$ for $x \in \mathbb { R }$.\\
(i) Express $- 2 x ^ { 2 } + 12 x - 3$ in the form $- 2 ( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.\\
(ii) State the greatest value of $\mathrm { f } ( x )$.\\
The function g is defined by $\mathrm { g } ( x ) = 2 x + 5$ for $x \in \mathbb { R }$.\\
(iii) Find the values of $x$ for which $\operatorname { gf } ( x ) + 1 = 0$.\\
\hfill \mbox{\textit{CAIE P1 2019 Q5 [6]}}