| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 Part (a) is a routine completing the square exercise with a coefficient, requiring only algebraic manipulation. Part (b) involves differentiating a polynomial and analyzing the sign of the derivative, which is straightforward once factored. Both parts are standard textbook exercises requiring recall of techniques rather than problem-solving insight. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\{5(y-3)^2\} \{+5\}\) | B1 B1 | Accept \(a = -3, b = 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([f'(x) =] 5x^4 - 30x^2 + 50\) | B1 | |
| \(5(x^2-3)^2 + 5\) or \(b^2 < 4ac\) and at least one value of \(f'(x) > 0\) | M1 | |
| \(> 0\) and increasing | A1 | WWW |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\{5(y-3)^2\} \{+5\}$ | B1 B1 | Accept $a = -3, b = 5$ |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[f'(x) =] 5x^4 - 30x^2 + 50$ | B1 | |
| $5(x^2-3)^2 + 5$ or $b^2 < 4ac$ and at least one value of $f'(x) > 0$ | M1 | |
| $> 0$ and increasing | A1 | WWW |
---3
\begin{enumerate}[label=(\alph*)]
\item Express $5 y ^ { 2 } - 30 y + 50$ in the form $5 ( y + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.
\item The function f is defined by $\mathrm { f } ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 50 x$ for $x \in \mathbb { R }$.
Determine whether $f$ is an increasing function, a decreasing function or neither.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q3 [5]}}