| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch with |x| in function |
| Difficulty | Standard +0.8 Part (a) is a standard rational function sketch requiring asymptote identification and basic curve behavior. Part (b) requires understanding the transformation effect of |x| on the graph (creating y-axis symmetry), then solving a modulus inequality involving a rational expression, which demands careful case analysis and sign considerations—more sophisticated than typical A-level pure maths questions but within Further Maths scope. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph showing correct axes and asymptotes (vertical asymptote at \(x = 0\), horizontal asymptote at \(y = 0\)) | B1 | Axes and asymptotes correct |
| Correct branches drawn (curve in first quadrant decreasing from top-left to bottom-right, approaching both asymptotes; corresponding branch in third quadrant) | B1 | Branches correct |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph symmetrical about \(x = 0\) | B1 FT | FT from sketch in part (a), symmetrical about \(x = 0\) |
| Correct shape at \(x = 0\) (reflection not turning point) | B1 | Correct shape at \(x = 0\) (reflection not turning point) |
| \(\frac{ | x | +1}{ |
| \(-1 < x < -\frac{1}{3}\), \(\frac{1}{3} < x < 1\) | A1 | |
| Total: 4 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph showing correct axes and asymptotes (vertical asymptote at $x = 0$, horizontal asymptote at $y = 0$) | B1 | Axes and asymptotes correct |
| Correct branches drawn (curve in first quadrant decreasing from top-left to bottom-right, approaching both asymptotes; corresponding branch in third quadrant) | B1 | Branches correct |
| | **Total: 2** | |
## Question 1:
**Part 1(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph symmetrical about $x = 0$ | B1 FT | FT from sketch in part (a), symmetrical about $x = 0$ |
| Correct shape at $x = 0$ (reflection not turning point) | B1 | Correct shape at $x = 0$ (reflection not turning point) |
| $\frac{|x|+1}{|x|-1} = -2$ leading to $|x| = \frac{1}{3}$ | M1 | Finds critical point(s) |
| $-1 < x < -\frac{1}{3}$, $\frac{1}{3} < x < 1$ | A1 | |
| **Total: 4** | | |
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\begin{enumerate}[label=(\alph*)]
\item Sketch the curve with equation $\mathrm { y } = \frac { \mathrm { x } + 1 } { \mathrm { x } - 1 }$.
\item Sketch the curve with equation $\mathrm { y } = \frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 }$ and find the set of values of x for which $\frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 } < - 2$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q1 [6]}}