| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Direct expected frequency calculation |
| Difficulty | Moderate -0.8 This is a straightforward normal distribution question requiring standard z-score lookups and basic probability calculations. Part (a) is routine (find P(X>160) then multiply by 12), part (b) is standard inverse normal application, and part (c) requires minimal statistical reasoning. All techniques are textbook exercises with no problem-solving insight needed. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Using the model and horizontal motion: \(s = ut\) | M1 | Using the model and correct strategy |
| \(12 = T \times 45\cos10°\) | A1 | Correct equation in \(T\) only |
| \(T = 0.2707...\) | A1 | 0.271 or better |
| Using the model and vertical motion: \(s = ut + \frac{1}{2}at^2\) | M1 | Using the model and correct strategy |
| \(s = 45T\sin10° + 4.9T^2\) | A1 | Correct equation |
| Correct strategy: sub for \(T\) and find \(s\) | M1 | Sub for \(T\) and solve for \(s\) |
| \(d = 3.5 - 2.4752 - 1\) | M1 | Correct method to find \(d\) using their \(s\) |
| \(= 2.5\) (cm) (2 SF) | A1 | 2.5 is the only correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Using the model and vertical motion: \(v = u + at\) | M1 | Using the model and correct strategy |
| \(v = 45\sin10° + 9.8T\) | A1 | Correct equation |
| Speed \(= ((45\cos10°)^2 + v^2)^{0.5}\) | M1 | Must have found a \(v\) and usual rules apply. Square root is needed |
| \(46\ (\text{m s}^{-1})\) (2 SF) | A1 | 46 (2 SF) is only correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Model does not take account of air resistance | B1 | Other appropriate answer e.g. spin of the ball, wind effect |
| Model does not take account of the size of the tennis ball | B1 | Other appropriate answer e.g. spin of the ball, wind effect |
# Question 5:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using the model and horizontal motion: $s = ut$ | M1 | Using the model and correct strategy |
| $12 = T \times 45\cos10°$ | A1 | Correct equation in $T$ only |
| $T = 0.2707...$ | A1 | 0.271 or better |
| Using the model and vertical motion: $s = ut + \frac{1}{2}at^2$ | M1 | Using the model and correct strategy |
| $s = 45T\sin10° + 4.9T^2$ | A1 | Correct equation |
| Correct strategy: sub for $T$ and find $s$ | M1 | Sub for $T$ and solve for $s$ |
| $d = 3.5 - 2.4752 - 1$ | M1 | Correct method to find $d$ using their $s$ |
| $= 2.5$ (cm) (2 SF) | A1 | 2.5 is the only correct answer |
**(8 marks)**
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using the model and vertical motion: $v = u + at$ | M1 | Using the model and correct strategy |
| $v = 45\sin10° + 9.8T$ | A1 | Correct equation |
| Speed $= ((45\cos10°)^2 + v^2)^{0.5}$ | M1 | Must have found a $v$ and usual rules apply. Square root is needed |
| $46\ (\text{m s}^{-1})$ (2 SF) | A1 | 46 (2 SF) is only correct answer |
**(4 marks)**
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Model does not take account of air resistance | B1 | Other appropriate answer e.g. spin of the ball, wind effect |
| Model does not take account of the size of the tennis ball | B1 | Other appropriate answer e.g. spin of the ball, wind effect |
**(2 marks)**
**Total: 14 marks**5. The lifetimes of batteries sold by company $X$ are normally distributed, with mean 150 hours and standard deviation 25 hours.
A box contains 12 batteries from company $X$.
\begin{enumerate}[label=(\alph*)]
\item Find the expected number of these batteries that have a lifetime of more than 160 hours.
The lifetimes of batteries sold by company $Y$ are normally distributed, with mean 160 hours and $80 \%$ of these batteries have a lifetime of less than 180 hours.
\item Find the standard deviation of the lifetimes of batteries from company $Y$.
Both companies sell their batteries for the same price.
\item State which company you would recommend. Give reasons for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 3 Q5 [8]}}