| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find range where function increasing/decreasing |
| Difficulty | Standard +0.3 Part (a) requires finding dy/dx, solving a quadratic inequality for dy/dx < 0 - standard calculus technique. Part (b) requires setting the derivative equal to 9, solving for x, then substituting to find k - straightforward but involves multiple steps. Both parts are routine applications of differentiation with no novel insight required, making this slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| 9(a) | Differentiate to obtain 5+12x−9x2 | B1 |
| Attempt to find two critical values by solving quadratic equation or inequality | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 3 | A1 | SC B1 if no method for solving the quadratic. |
| Answer | Marks | Guidance |
|---|---|---|
| 3 3 | A1FT | SC B1 if no method for solving the quadratic. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 9(b) | Equate first derivative to 9 and simplify to 3 term quadratic | *M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | A1 | SC B1 for solving 5+12x−9x2 = 9 without |
| Answer | Marks |
|---|---|
| Use x-value and corresponding y-value to determine value of k | DM1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | A1 | 28 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(a) ---
9(a) | Differentiate to obtain 5+12x−9x2 | B1
Attempt to find two critical values by solving quadratic equation or inequality | M1
1 5
Obtain values − and
3 3 | A1 | SC B1 if no method for solving the quadratic.
1 5
Conclude x− , x
3 3 | A1FT | SC B1 if no method for solving the quadratic.
4
Question | Answer | Marks | Guidance
--- 9(b) ---
9(b) | Equate first derivative to 9 and simplify to 3 term quadratic | *M1
2
Obtain x=
3 | A1 | SC B1 for solving 5+12x−9x2 = 9 without
simplifying to a 3-term quadratic.
Use x-value and corresponding y-value to determine value of k | DM1
28
Obtain k =
9 | A1 | 28
SC B1 for k = from solving 5+12x−9x2 = 9
9
without simplifying to a 3-term quadratic.
4
Question | Answer | Marks | GuidanceThe equation of a curve is $y = 4 + 5x + 6x^2 - 3x^3$.
\begin{enumerate}[label=(\alph*)]
\item Find the set of values of $x$ for which $y$ decreases as $x$ increases. [4]
\item It is given that $y = 9x + k$ is a tangent to the curve.
Find the value of the constant $k$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q9 [8]}}