| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Interquartile range calculation |
| Difficulty | Standard +0.3 This is a straightforward application of finding quartiles from a piecewise CDF. Students must solve F(x) = 0.25 and F(x) = 0.75 using the given formulas, requiring basic algebraic manipulation of quadratic expressions. While it involves multiple steps and careful attention to which piece of the function to use, it's a standard textbook exercise with no novel insight required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x-2}{6}\); \(\frac{8-x}{12}\); Sketch (two line segments touching x-axis, meeting at point above x-axis) | M1 | Attempt at differentiating both parts of CDF to obtain linear expressions. May be implied by first A1. |
| Correct shape | A1 | Two line segments touch the \(x\)-axis and meet at a point above the \(x\)-axis. |
| Correct shape with 2 and 8 and at least one of 4 or \(\frac{1}{3}\) correctly labelled | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_2^4 \frac{1}{6}x(x-2)\,dx + \int_4^8 \frac{1}{12}x(8-x)\,dx\) | B1 FT | FT their PDF, with correct limits. |
| \(\frac{1}{6}\left[\frac{1}{3}x^3 - x^2\right] + \frac{1}{12}\left[4x^2 - \frac{1}{3}x^3\right]\) | M1 | Attempt at integration with their PDF (not CDF); no limits needed. May be implied by correct answer. |
| \(\frac{14}{3}\) | A1 | Accept 4.67 or any equivalent fraction. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| LQ: \(\frac{1}{12}(x-2)^2 = \frac{1}{4}\), \(x = 2+\sqrt{3}\) | B1 | Allow 3.73. |
| UQ: \(1 - \frac{(8-x)^2}{24} = \frac{3}{4}\), \(x = 8-\sqrt{6}\) | B1 | Allow 5.55. |
| \(\text{IQR} = \text{UQ} - \text{LQ}\), with values found for both quartiles | M1 | |
| \(6 - \sqrt{6} - \sqrt{3}\) | A1 | Exact answer required. |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x-2}{6}$; $\frac{8-x}{12}$; Sketch (two line segments touching x-axis, meeting at point above x-axis) | M1 | Attempt at differentiating both parts of CDF to obtain linear expressions. May be implied by first A1. |
| Correct shape | A1 | Two line segments touch the $x$-axis and meet at a point above the $x$-axis. |
| Correct shape with 2 and 8 and at least one of 4 or $\frac{1}{3}$ correctly labelled | A1 | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_2^4 \frac{1}{6}x(x-2)\,dx + \int_4^8 \frac{1}{12}x(8-x)\,dx$ | B1 FT | FT their PDF, with correct limits. |
| $\frac{1}{6}\left[\frac{1}{3}x^3 - x^2\right] + \frac{1}{12}\left[4x^2 - \frac{1}{3}x^3\right]$ | M1 | Attempt at integration with their PDF (not CDF); no limits needed. May be implied by correct answer. |
| $\frac{14}{3}$ | A1 | Accept 4.67 or any equivalent fraction. |
---
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| LQ: $\frac{1}{12}(x-2)^2 = \frac{1}{4}$, $x = 2+\sqrt{3}$ | B1 | Allow 3.73. |
| UQ: $1 - \frac{(8-x)^2}{24} = \frac{3}{4}$, $x = 8-\sqrt{6}$ | B1 | Allow 5.55. |
| $\text{IQR} = \text{UQ} - \text{LQ}$, with values found for both quartiles | M1 | |
| $6 - \sqrt{6} - \sqrt{3}$ | A1 | Exact answer required. |
---5 The continuous random variable $X$ has cumulative distribution function F given by
$$F ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { ( x - 2 ) ^ { 2 } } { 12 } & 2 \leqslant x < 4 , \\ 1 - \frac { ( 8 - x ) ^ { 2 } } { 24 } & 4 \leqslant x \leqslant 8 , \\ 1 & x > 8 . \end{cases}$$
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(c) Find the exact value of the interquartile range of $X$.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q5 [10]}}