| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Quadratic inequalities |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise followed by solving a simple quadratic inequality. Part (i) is routine algebraic manipulation, and part (ii) requires only direct substitution and basic inequality solving. Both parts are standard textbook exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| 1(i) | ( x−2 )2 [ +4 ] | |
| | B1 DB1 | 2nd B1 dependent on 2 inside bracket |
| Answer | Marks | Guidance |
|---|---|---|
| 1(ii) | ( x−2 )2 <5 → − 5<x−2 and/or x−2< 5 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2− 5<x<2+ 5 | A1A1 | A1 for each inequality – allow two separate statements but there must |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 1:
--- 1(i) ---
1(i) | ( x−2 )2 [ +4 ]
| B1 DB1 | 2nd B1 dependent on 2 inside bracket
2
--- 1(ii) ---
1(ii) | ( x−2 )2 <5 → − 5<x−2 and/or x−2< 5 | M1 | Allow e.g. x−2<± 5 , x ‒ 2=± 5 and decimal equivalents for √5
For M1, ft from their(i). Also allow √13 instead of √5 for clear slip
2− 5<x<2+ 5 | A1A1 | A1 for each inequality – allow two separate statements but there must
be 2 inequalities for x. Non-hence methods, if completely correct,
score SC 1/3. Condone (cid:45)
[3]
Question | Answer | Marks | GuidanceThe function f is defined by $\text{f}(x) = x^2 - 4x + 8$ for $x \in \mathbb{R}$.
\begin{enumerate}[label=(\roman*)]
\item Express $x^2 - 4x + 8$ in the form $(x - a)^2 + b$. [2]
\item Hence find the set of values of $x$ for which $\text{f}(x) < 9$, giving your answer in exact form. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q1 [5]}}