| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise with standard algebraic manipulation. Part (a) requires routine factoring and completing the square, while part (b) applies the discriminant condition (one root means the completed square form equals zero). Both parts are mechanical applications of well-practiced techniques with no problem-solving insight required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((4x-3)^2\) or \((4x+(-3))^2\) or \(a=-3\) | B1 | \(k(4x-3)^2\) where \(k \neq 1\) scores B0 but mark final answer, allow recovery. |
| \(+1\) or \(b=1\) | B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [For one root] \(k=1\) or '*their b*' | B1 FT | Either by inspection or solving from \(24^2 - 4 \times 16 \times (10-k) = 0\) WWW |
| [Root or \(x=\)] \(\dfrac{3}{4}\) or \(0.75\) | B1 | SC B2 for correct final answer WWW. |
| Total: 2 |
**Question 1:**
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(4x-3)^2$ or $(4x+(-3))^2$ or $a=-3$ | B1 | $k(4x-3)^2$ where $k \neq 1$ scores B0 but mark final answer, allow recovery. |
| $+1$ or $b=1$ | B1 | |
| **Total: 2** | | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| [For one root] $k=1$ or '*their b*' | B1 FT | Either by inspection or solving from $24^2 - 4 \times 16 \times (10-k) = 0$ WWW |
| [Root or $x=$] $\dfrac{3}{4}$ or $0.75$ | B1 | **SC B2** for correct final answer WWW. |
| **Total: 2** | | |1
\begin{enumerate}[label=(\alph*)]
\item Express $16 x ^ { 2 } - 24 x + 10$ in the form $( 4 x + a ) ^ { 2 } + b$.
\item It is given that the equation $16 x ^ { 2 } - 24 x + 10 = k$, where $k$ is a constant, has exactly one root.
Find the value of this root.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q1 [4]}}