Question 7 - A-Level Maths

CAIE Further Paper 1 2022 November — Question 7 16 marks

Exam Board	CAIE
Module	Further Paper 1 (Further Paper 1)
Year	2022
Session	November
Marks	16
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Curve Sketching
Type	Solve \|f(x)\| > k using sketch
Difficulty	Standard +0.8 This is a substantial Further Maths question requiring multiple techniques: finding asymptotes (vertical and oblique), differentiation with quotient rule, stationary point analysis, curve sketching with asymptotic behavior, then modulus transformation and solving an inequality. The modulus inequality requires careful case analysis and exact algebraic solutions. More demanding than standard A-level but typical for Further Maths.
Spec	1.02k Simplify rational expressions: factorising, cancelling, algebraic division 1.02l Modulus function: notation, relations, equations and inequalities 1.02n Sketch curves: simple equations including polynomials 1.02s Modulus graphs: sketch graph of \|ax+b\|1.07m Tangents and normals: gradient and equations 1.07n Stationary points: find maxima, minima using derivatives 1.07o Increasing/decreasing: functions using sign of dy/dx

7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).

Find the equations of the asymptotes of \(C\).
Find the coordinates of the stationary points on \(C\).
Sketch \(C\).
Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.

Show mark scheme Show mark scheme source

Question 7(a):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(x = \frac{2}{5}\)	B1	Vertical asymptote.
\(y = \dfrac{(5x-2)(x+\frac{2}{5})+\frac{4}{5}}{5x-2}\) leading to \(y = x + \frac{2}{5}\)	M1 A1	Oblique asymptote.
Total: 3

Question 7(b):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\dfrac{dy}{dx} = \dfrac{(5x-2)(10x) - (5x^2)(5)}{(5x-2)^2}\)	M1	Finds \(\dfrac{dy}{dx}\)
\(5x^2 - 4x = 0\)	M1	Sets equal to 0 and forms quadratic equation.
\((0,0)\), \(\left(\frac{4}{5}, \frac{8}{5}\right)\)	A1 A1
Total: 4

Question 7(c):

Answer	Marks	Guidance
Answer	Marks	Guidance
Axes and asymptotes shown	B1	Axes and asymptotes.
Correct upper branch with asymptotic behaviour	B1	Correct upper branch and asymptotic behaviour.
Correct lower branch	B1	Correct lower branch.
Total: 3

Question 7(d):

Answer	Marks	Guidance
Answer	Marks	Guidance
Sketch of \(\left	\dfrac{5x^2}{5x-2}\right	\)
Correct shape as \(x\) tends to infinity	B1	Correct shape as \(x\) tends to infinity.
\(\dfrac{5x^2}{5x-2} = 2\) or \(\dfrac{5x^2}{5x-2} = -2\); \(5x^2 - 10x + 4 = 0\) or \(5x^2 + 10x - 4 = 0\)	M2	Finds critical points, award M1 for each case.
\(x = -1-\frac{3}{5}\sqrt{5}\), \(x = -1+\frac{3}{5}\sqrt{5}\) or \(x = 1-\frac{1}{5}\sqrt{5}\), \(x = 1+\frac{1}{5}\sqrt{5}\)	A1	Must be exact.
\(-1-\frac{3}{5}\sqrt{5} < x < -1+\frac{3}{5}\sqrt{5}\), \(1-\frac{1}{5}\sqrt{5} < x < 1+\frac{1}{5}\sqrt{5}\)	A1 FT	Follow through on use of decimals.
Total: 6

## Question 7(a):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = \frac{2}{5}$ | B1 | Vertical asymptote. |
| $y = \dfrac{(5x-2)(x+\frac{2}{5})+\frac{4}{5}}{5x-2}$ leading to $y = x + \frac{2}{5}$ | M1 A1 | Oblique asymptote. |
| **Total: 3** | | |

## Question 7(b):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dy}{dx} = \dfrac{(5x-2)(10x) - (5x^2)(5)}{(5x-2)^2}$ | M1 | Finds $\dfrac{dy}{dx}$ |
| $5x^2 - 4x = 0$ | M1 | Sets equal to 0 and forms quadratic equation. |
| $(0,0)$, $\left(\frac{4}{5}, \frac{8}{5}\right)$ | A1 A1 | |
| **Total: 4** | | |

## Question 7(c):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Axes and asymptotes shown | B1 | Axes and asymptotes. |
| Correct upper branch with asymptotic behaviour | B1 | Correct upper branch and asymptotic behaviour. |
| Correct lower branch | B1 | Correct lower branch. |
| **Total: 3** | | |

## Question 7(d):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch of $\left|\dfrac{5x^2}{5x-2}\right|$ | B1 FT | FT from sketch in part (c). |
| Correct shape as $x$ tends to infinity | B1 | Correct shape as $x$ tends to infinity. |
| $\dfrac{5x^2}{5x-2} = 2$ or $\dfrac{5x^2}{5x-2} = -2$; $5x^2 - 10x + 4 = 0$ or $5x^2 + 10x - 4 = 0$ | M2 | Finds critical points, award M1 for each case. |
| $x = -1-\frac{3}{5}\sqrt{5}$, $x = -1+\frac{3}{5}\sqrt{5}$ or $x = 1-\frac{1}{5}\sqrt{5}$, $x = 1+\frac{1}{5}\sqrt{5}$ | A1 | Must be exact. |
| $-1-\frac{3}{5}\sqrt{5} < x < -1+\frac{3}{5}\sqrt{5}$, $1-\frac{1}{5}\sqrt{5} < x < 1+\frac{1}{5}\sqrt{5}$ | A1 FT | Follow through on use of decimals. |
| **Total: 6** | | |

Show LaTeX source

7 The curve $C$ has equation $y = \frac { 5 x ^ { 2 } } { 5 x - 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the asymptotes of $C$.
\item Find the coordinates of the stationary points on $C$.
\item Sketch $C$.
\item Sketch the curve with equation $y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|$ and find in exact form the set of values of $x$ for which $\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2$.\\

If you use the following page to complete the answer to any question, the question number must be clearly shown.
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q7 [16]}}

This paper (7 questions)

View full paper

Q1 8 Q2 6 Q3 9 Q4 11 Q5 11 Q6 14 Q7 16