| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2024 |
| Session | November |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Range restriction with discriminant (quadratic denominator) |
| Difficulty | Standard +0.8 This is a comprehensive Further Maths curve sketching question requiring multiple techniques: asymptote analysis, range restriction proof (involving algebraic manipulation and completing the square or inequality reasoning), differentiation of quotients, and comparative analysis of related functions. Part (b) requires proving bounds algebraically, which is non-routine. The multi-part structure and the need to synthesize several concepts places this above average difficulty, though each individual step uses standard Further Maths techniques. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2+1=0\) has no real roots | B1 | |
| \(y=1\) | B1 | Horizontal asymptote |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(yx^2+y=x^2+3 \Rightarrow (1-y)x^2-y+3=0\) | M1 A1 | Forms quadratic in \(x\) or uses \(y=1+\frac{2}{x^2+1}\) |
| \(-4(1-y)(3-y)\geq 0\) | M1 | Uses that discriminant is \(\geq 0\) or \(0<\frac{2}{x^2+1}\leq 2\) |
\(1| A1 |
Explanation of why \(y\neq 1\), AG |
|
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{(x^2+1)(2x)-(x^2+3)(2x)}{(x^2+1)^2} = 0\) | M1 | Differentiates |
| \((0, 3)\) | A1 | |
| Alternative method: | ||
| When \(y=3\), \(2x^2=0\) | M1 | Using inequality from (b) |
| \((0, 3)\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch showing correct axes and asymptote labelled | B1 | Axes and correct asymptote labelled |
| Correct shape and position | B1 | Correct shape and position |
| \((0, 3)\) coordinates of intersection with axes | B1 | States \((0,3)\) coordinates of intersection with axes, may be seen on diagram |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch of \(\frac{1}{x^2+3}\) | B1FT | FT from sketch in (d) |
| \(\frac{x^2+1}{x^2+3} = \frac{1}{2}\) | M1 | Finds critical points |
| \(x^2 = 1 \Rightarrow x = \pm 1\) | A1 | |
| \(-1 < x < 1\) | A1 | |
| 4 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2+1=0$ has no real roots | B1 | |
| $y=1$ | B1 | Horizontal asymptote |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $yx^2+y=x^2+3 \Rightarrow (1-y)x^2-y+3=0$ | M1 A1 | Forms quadratic in $x$ or uses $y=1+\frac{2}{x^2+1}$ |
| $-4(1-y)(3-y)\geq 0$ | M1 | Uses that discriminant is $\geq 0$ or $0<\frac{2}{x^2+1}\leq 2$ |
| $1<y\leq 3$ | A1 | Explanation of why $y\neq 1$, AG |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{(x^2+1)(2x)-(x^2+3)(2x)}{(x^2+1)^2} = 0$ | M1 | Differentiates |
| $(0, 3)$ | A1 | |
| **Alternative method:** | | |
| When $y=3$, $2x^2=0$ | M1 | Using inequality from (b) |
| $(0, 3)$ | A1 | |
| | **2** | |
## Question 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch showing correct axes and asymptote labelled | B1 | Axes and correct asymptote labelled |
| Correct shape and position | B1 | Correct shape and position |
| $(0, 3)$ coordinates of intersection with axes | B1 | States $(0,3)$ coordinates of intersection with axes, may be seen on diagram |
| | **3** | |
## Question 6(e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch of $\frac{1}{x^2+3}$ | B1FT | FT from sketch in (d) |
| $\frac{x^2+1}{x^2+3} = \frac{1}{2}$ | M1 | Finds critical points |
| $x^2 = 1 \Rightarrow x = \pm 1$ | A1 | |
| $-1 < x < 1$ | A1 | |
| | **4** | |6 The curve $C$ has equation $y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote.
\item Show that $1 < y \leqslant 3$ for all real values of $x$.
\item Find the coordinates of any stationary points on $C$.\\
\includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-12_2718_42_107_2007}\\
\includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-13_2720_40_106_18}
\item Sketch $C$, stating the coordinates of any intersections with the axes and labelling the asymptote.
\item Sketch the curve with equation $y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }$ and find the set of values of $x$ for which $\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q6 [15]}}