SPS SPS SM Pure 2023 February — Question 8 9 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2023
SessionFebruary
Marks9
TopicProduct & Quotient Rules
TypeFind stationary points and nature
DifficultyStandard +0.3 This is a straightforward multi-part question requiring quotient rule differentiation, solving dy/dx=0 for stationary points, and applying the trapezium rule. All techniques are standard A-level procedures with no novel insight required. The quotient rule application is slightly more involved than basic examples due to the ln(x²+1) term, but the stationary point equation simplifies nicely, and the trapezium rule is routine. Slightly easier than average due to the guided structure and standard techniques.
Spec1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.09f Trapezium rule: numerical integration
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f97853c-812f-4b7b-9d40-2de7a85886c0-18_563_853_274_566} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve \(C\) with equation $$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Using your answer to (a), find the exact coordinates of the stationary point on the curve \(C\) for which \(x > 0\). Write each coordinate in its simplest form.
    (4) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 3\)
  3. Complete the table below with the value of \(y\) corresponding to \(x = 1\)
    \(x\)0123
    \(y\)0\(\frac { 3 } { 5 } \ln 5\)\(\frac { 3 } { 10 } \ln 10\)
    (1)
  4. Use the trapezium rule with all the \(y\) values in the completed table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
    (2)
8.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8f97853c-812f-4b7b-9d40-2de7a85886c0-18_563_853_274_566}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

Figure 3 shows part of the curve $C$ with equation

$$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item Using your answer to (a), find the exact coordinates of the stationary point on the curve $C$ for which $x > 0$. Write each coordinate in its simplest form.\\
(4)

The finite region $R$, shown shaded in Figure 3, is bounded by the curve $C$, the $x$-axis and the line $x = 3$
\item Complete the table below with the value of $y$ corresponding to $x = 1$

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$y$ & 0 &  & $\frac { 3 } { 5 } \ln 5$ & $\frac { 3 } { 10 } \ln 10$ \\
\hline
\end{tabular}
\end{center}

(1)
\item Use the trapezium rule with all the $y$ values in the completed table to find an approximate value for the area of $R$, giving your answer to 4 significant figures.\\
(2)
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q8 [9]}}
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Q1 7 Q2 4 Q3 4 Q4 8 Q5 6 Q6 4 Q7 9 Q8 9 Q9 8 Q10 8 Q11 7 Q12 7 Q13 9 Q14 10