| Exam Board | WJEC |
|---|---|
| Module | Unit 4 (Unit 4) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Breaking/cutting problems |
| Difficulty | Standard +0.8 This question requires students to recognize that the longer piece follows a non-standard distribution (uniform on [30,60]), then apply a geometric transformation (perimeter to area) and solve an inequality involving π. The conceptual leap from 'longer piece' to its distribution and the multi-step probability calculation involving circle geometry elevate this above routine uniform distribution questions. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02e Discrete uniform distribution5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\text{Continuous uniform distribution on } [30,60]\) | B1 | AO3 |
| \(\text{Mean} = 45\) | B1 | AO1 |
| \(\text{Variance} = 75\) | B1 | AO1 |
| (b) \(P(\pi R^2 > 100) = P\left(R > \sqrt{\frac{100}{\pi}}\right)\) | M1 | AO3 |
| \(= P\left(L > 2\pi\sqrt{\frac{100}{\pi}}\right)\) | A1 | AO2 |
| \(= P(L > 35.45)\) | A1 | AO1 |
| \(= \frac{60 - 35.45}{30} = 0.818(3) \text{ or } \frac{491}{600}\) | A1 | AO1 |
**(a)** $\text{Continuous uniform distribution on } [30,60]$ | B1 | AO3 | |
| $\text{Mean} = 45$ | B1 | AO1 | |
| $\text{Variance} = 75$ | B1 | AO1 | |
**(b)** $P(\pi R^2 > 100) = P\left(R > \sqrt{\frac{100}{\pi}}\right)$ | M1 | AO3 | |
| $= P\left(L > 2\pi\sqrt{\frac{100}{\pi}}\right)$ | A1 | AO2 | |
| $= P(L > 35.45)$ | A1 | AO1 | |
| $= \frac{60 - 35.45}{30} = 0.818(3) \text{ or } \frac{491}{600}$ | A1 | AO1 | |3. A string of length 60 cm is cut a random point.
\begin{enumerate}[label=(\alph*)]
\item Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
\item The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than $100 \mathrm {~cm} ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 4 Q3 [7]}}