| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | September |
| Marks | 9 |
| Topic | Tangents, normals and gradients |
| Type | Increasing/decreasing intervals |
| Difficulty | Easy -1.2 This is a straightforward multi-part question testing routine differentiation skills: differentiating a polynomial (trivial), substituting into the second derivative test (standard procedure), rearranging an inequality, and solving a quadratic inequality. All parts are textbook exercises requiring only direct application of learned techniques with no problem-solving or insight needed. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.07d Second derivatives: d^2y/dx^2 notation1.07n Stationary points: find maxima, minima using derivatives |
9. The gradient, $\frac { \mathrm { d } y } { \mathrm {~d} x }$, at the point $( x , y )$ on a curve is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.\\[0pt]
[2 marks]
\item The curve passes through the point $P \left( - 1 \frac { 1 } { 2 } , 4 \right)$.
Verify that the curve has a minimum point at $P$.\\[0pt]
[2 marks]
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that at the points on the curve where $y$ is decreasing
$$2 x ^ { 2 } - 9 x - 18 > 0$$
\item Solve the inequality $2 x ^ { 2 } - 9 x - 18 > 0$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q9 [9]}}