OCR MEI C2 — Question 4 13 marks

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTangents, normals and gradients
TypeTangent meets curve/axis — further geometry
DifficultyModerate -0.3 This is a multi-part question covering standard C2 differentiation and integration techniques. While it has several parts and requires careful working (finding tangent equations, x-intercepts, and areas), all components are routine textbook exercises requiring no novel insight. The most challenging aspect is part (iii) finding where a tangent crosses a cubic again, but this is still a standard algebraic manipulation once the tangent equation is found.
Spec1.02f Solve quadratic equations: including in a function of unknown1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals
4 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-2_604_912_1100_638} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
  2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
  3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-3_643_1034_267_549} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).
  4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  5. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
  6. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.
Question 4:
i
AnswerMarks
\(7 - 2x\)A1
ii
AnswerMarks Guidance
\(x = 2\), gradient \(= 3\)M1 differentiation must be used
\(x = 2\), \(y = 4\)B1
\(y - \text{their } 4 = \text{their grad}(x - 2)\)M1 or use of \(y = \text{their } mx + c\) and subst \((2, \text{their } 4)\), dependent on diffn
subst \(y = 0\) in their linear eqnM1 seen
completion to \(x = 2\) (ans given)A1
iii
AnswerMarks Guidance
\(f(1) = 0\) or factorising to \((x - 1)(6 - x)\) or \((x - 1)(x - 6)\)M1 or using quadratic formula correctly to obtain \(x = 1\)
\(\frac{1}{2}x^2 - \frac{x^3}{3} - 6x\)M1 A1 for two terms correct; ignore \(+c\)
value at \(2 -\) value at \(1\)M1
\(\frac{1}{2}\) o.e.A1 ft attempt at integration only
\(\frac{1}{4} \times \frac{1}{3} \times 4 - \text{their integral}\)M1
\(0.5\) o.e.A1 
**Question 4:**

**i**

$7 - 2x$ | A1

**ii**

$x = 2$, gradient $= 3$ | M1 | differentiation must be used

$x = 2$, $y = 4$ | B1

$y - \text{their } 4 = \text{their grad}(x - 2)$ | M1 | or use of $y = \text{their } mx + c$ and subst $(2, \text{their } 4)$, dependent on diffn

subst $y = 0$ in their linear eqn | M1 | seen

completion to $x = 2$ (ans given) | A1

**iii**

$f(1) = 0$ or factorising to $(x - 1)(6 - x)$ or $(x - 1)(x - 6)$ | M1 | or using quadratic formula correctly to obtain $x = 1$

$\frac{1}{2}x^2 - \frac{x^3}{3} - 6x$ | M1 A1 | for two terms correct; ignore $+c$

value at $2 -$ value at $1$ | M1

$\frac{1}{2}$ o.e. | A1 | ft attempt at integration only

$\frac{1}{4} \times \frac{1}{3} \times 4 - \text{their integral}$ | M1

$0.5$ o.e. | A1
4 Fig. 10 shows a sketch of the graph of $y = 7 x - x ^ { 2 } - 6$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-2_604_912_1100_638}
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{center}
\end{figure}

(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the equation of the tangent to the curve at the point on the curve where $x = 2$.

Show that this tangent crosses the $x$-axis where $x = \frac { 2 } { 3 }$.\\
(ii) Show that the curve crosses the $x$-axis where $x = 1$ and find the $x$-coordinate of the other point of intersection of the curve with the $x$-axis.\\
(iii) Find $\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x$.

Hence find the area of the region bounded by the curve, the tangent and the $x$-axis, shown shaded on Fig. 10.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-3_643_1034_267_549}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{center}
\end{figure}

The equation of the curve shown in Fig. 11 is $y = x ^ { 3 } - 6 x + 2$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Find, in exact form, the range of values of $x$ for which $x ^ { 3 } - 6 x + 2$ is a decreasing function.\\
(iii) Find the equation of the tangent to the curve at the point $( - 1,7 )$.

Find also the coordinates of the point where this tangent crosses the curve again.

\hfill \mbox{\textit{OCR MEI C2  Q4 [13]}}
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