| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Breaking/cutting problems |
| Difficulty | Standard +0.8 Part (a) is routine variance calculation for uniform distribution. Part (b) involves standard probability calculations. Part (c) requires significant problem-solving: translating the cutting scenario into a mathematical model, recognizing that the first cut position follows a uniform distribution, expressing rectangle area in terms of this random variable, then solving an inequality. This multi-step contextual application with geometric insight elevates it above typical S2 questions. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{(a+6)^2}{12} = 27\) | M1 | For setting up the correct equation; do not allow verification |
| \(a = \sqrt{27\times 12} - 6 \Rightarrow 12^*\) or \(a^2+12a-288=0 \Rightarrow a=12^*\) | A1* | For un-simplified expression for \(a\) leading to \(a=12\), or for correct \(3TQ=0\) leading to \(a=12\); condone any letter for \(a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{12-b}{18} = \frac{3}{5}\) or \(\frac{b+6}{18}=\frac{2}{5}\) | M1 | For setting up the correct equation |
| \(b = 1.2\) | A1 | Cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(-6 < W < "0.6") = \frac{"0.6"+6}{18}\) | M1 | For correct method; do not ISW |
| \(= \frac{11}{30}\) or \(0.3666...\) | A1ft | Ft their value for \(b\), provided answer is between 0 and 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{x}{2}\) and \(\left(\frac{160-x}{2}\right)\); \(L+W=80\) and \(LW=975\) | M1 | For both expressions seen; allow any letters; may be implied by correct equation for area |
| \(\frac{x}{2}\times\left(\frac{160-x}{2}\right) = 975 \Rightarrow x=30\) or \(130\); \(L(80-L)=975 \Rightarrow L=15\) or \(65\) | M1 | For correct equation for area in terms of any letter; condone an inequality |
\(P("30"| dM1 |
Dep on previous M mark; for fully correct method ft their \(x\) values provided add to 160 or 80; do not ISW |
|
| \(= \frac{5}{8}\) oe | A1 | Cao |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{(a+6)^2}{12} = 27$ | M1 | For setting up the correct equation; do not allow verification |
| $a = \sqrt{27\times 12} - 6 \Rightarrow 12^*$ or $a^2+12a-288=0 \Rightarrow a=12^*$ | A1* | For un-simplified expression for $a$ leading to $a=12$, or for correct $3TQ=0$ leading to $a=12$; condone any letter for $a$ |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{12-b}{18} = \frac{3}{5}$ or $\frac{b+6}{18}=\frac{2}{5}$ | M1 | For setting up the correct equation |
| $b = 1.2$ | A1 | Cao |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(-6 < W < "0.6") = \frac{"0.6"+6}{18}$ | M1 | For correct method; do not ISW |
| $= \frac{11}{30}$ or $0.3666...$ | A1ft | Ft their value for $b$, provided answer is between 0 and 1 |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{x}{2}$ and $\left(\frac{160-x}{2}\right)$; $L+W=80$ and $LW=975$ | M1 | For both expressions seen; allow any letters; may be implied by correct equation for area |
| $\frac{x}{2}\times\left(\frac{160-x}{2}\right) = 975 \Rightarrow x=30$ or $130$; $L(80-L)=975 \Rightarrow L=15$ or $65$ | M1 | For correct equation for area in terms of any letter; condone an inequality |
| $P("30"<x<"130") = \frac{"130"-"30"}{160}\left[=\frac{5}{8}\right]$ oe or $P("15"<x<"65")=\frac{"65"-"15"}{80}\left[=\frac{5}{8}\right]$ oe | dM1 | Dep on previous M mark; for fully correct method ft their $x$ values provided add to 160 or 80; do not ISW |
| $= \frac{5}{8}$ oe | A1 | Cao |\begin{enumerate}
\item The random variable $W$ has a continuous uniform distribution over the interval $[ - 6 , a ]$ where $a$ is a constant.
\end{enumerate}
Given that $\operatorname { Var } ( W ) = 27$\\
(a) show that $a = 12$
Given that $\mathrm { P } ( W > b ) = \frac { 3 } { 5 }$\\
(b) (i) find the value of $b$\\
(ii) find $\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)$
A piece of wood $A B$ has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.\\
(c) Calculate the probability that the area of the rectangle is greater than $975 \mathrm {~cm} ^ { 2 }$
\hfill \mbox{\textit{Edexcel S2 2024 Q5 [10]}}