| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent parallel to given line |
| Difficulty | Moderate -0.3 Part (i) requires finding where dy/dx equals -3, solving a quadratic, then writing the tangent equation—standard differentiation and coordinate geometry. Part (ii) asks for the smallest k where f is increasing, requiring setting f'(x) ≥ 0 and solving—a routine application of calculus concepts. Both parts are straightforward applications of A-level techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks |
|---|---|
| 8(i) | dy |
| Answer | Marks |
|---|---|
| dx | B1 |
| Answer | Marks |
|---|---|
| dx | B1 |
| Answer | Marks |
|---|---|
| dx | M1 |
| Obtain the given answer | A1 |
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(ii) | Equate denominator to zero and solve for y | M1* |
| Obtain y = 0 and x = a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| to find y | M1(dep*) | |
| Obtain x = – a | A1 | |
| Obtain y = 2a | A1 | |
| Total: | 5 | |
| Question | Answer | Marks |
Question 8:
--- 8(i) ---
8(i) | dy
State or imply 3y2 as derivative of y3
dx | B1
dy
State or imply 3y2 +6xy as derivative of 3xy2
dx | B1
dy
Equate derivative of LHS to zero and solve for
dx | M1
Obtain the given answer | A1
Total: | 4
--- 8(ii) ---
8(ii) | Equate denominator to zero and solve for y | M1*
Obtain y = 0 and x = a | A1
Obtain y = αx and substitute in curve equation to find x or
to find y | M1(dep*)
Obtain x = – a | A1
Obtain y = 2a | A1
Total: | 5
Question | Answer | Marks\begin{enumerate}[label=(\roman*)]
\item The tangent to the curve $y = x^3 - 9x^2 + 24x - 12$ at a point $A$ is parallel to the line $y = 2 - 3x$. Find the equation of the tangent at $A$. [6]
\item The function f is defined by $\mathrm{f}(x) = x^3 - 9x^2 + 24x - 12$ for $x > k$, where $k$ is a constant. Find the smallest value of $k$ for f to be an increasing function. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q8 [8]}}