| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find parameter from normal approximation |
| Difficulty | Standard +0.8 This question requires multiple sophisticated steps: (a) involves recognizing a binomial within binomial structure (bags with >20 blue flowers, then sampling 5 bags), and (b) requires working backwards from a normal approximation probability to find n, involving inverse normal calculations and solving a quadratic inequality. The reverse-engineering aspect and nested probability structure elevate this above standard normal approximation questions. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Resolve perpendicular to the plane | M1 | Using appropriate strategy to set up first equation, usual rules applying |
| \(R + 25\sin 30° = 3g\cos 20°\) | A1 | \(g\) does not need to be substituted |
| Equation of motion up the plane | M1 | Using appropriate strategy to set up second equation, usual rules applying |
| \(25\cos 30° - 3g\sin 20° - F = 3a\) | A1 | Neither \(g\) nor \(F\) need to be substituted (-1 each error) |
| \(F = 0.3R\) | B1 | \(F = 0.3R\) seen |
| Correct strategy: sub for \(F\) and solve for \(a\) | M1 | Correct overall strategy to substitute for \(F\) and solve for \(a\) |
| \(a = 2.4\) or \(2.35\ (\text{m s}^{-2})\) | A1 | Only possible answers, since \(g = 9.8\) used |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| e.g. Include air resistance | B1 | e.g. include air resistance, allow for the weight of the rope |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(R = 3g\cos 20°\) so \(F_\text{max} = 0.9g\cos 20°\) | B1 | Correct overall strategy (first equation could be implied) |
| Consider \(3g\sin 20° - 0.9g\cos 20°\) | M1 | Must be difference or comparison of the two values |
| Since \(> 0\), box moves down plane. | A1* | Given answer |
## Question 3(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Resolve perpendicular to the plane | M1 | Using appropriate strategy to set up first equation, usual rules applying |
| $R + 25\sin 30° = 3g\cos 20°$ | A1 | $g$ does not need to be substituted |
| Equation of motion up the plane | M1 | Using appropriate strategy to set up second equation, usual rules applying |
| $25\cos 30° - 3g\sin 20° - F = 3a$ | A1 | Neither $g$ nor $F$ need to be substituted (-1 each error) |
| $F = 0.3R$ | B1 | $F = 0.3R$ seen |
| Correct strategy: sub for $F$ and solve for $a$ | M1 | Correct overall strategy to substitute for $F$ and solve for $a$ |
| $a = 2.4$ or $2.35\ (\text{m s}^{-2})$ | A1 | Only possible answers, since $g = 9.8$ used |
**Total: 7 marks**
## Question 3(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| e.g. Include air resistance | B1 | e.g. include air resistance, allow for the weight of the rope |
**Total: 1 mark**
## Question 3(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $R = 3g\cos 20°$ so $F_\text{max} = 0.9g\cos 20°$ | B1 | Correct overall strategy (first equation could be implied) |
| Consider $3g\sin 20° - 0.9g\cos 20°$ | M1 | Must be difference or comparison of the two values |
| Since $> 0$, box moves down plane. | A1* | Given answer |
**Total: 3 marks**
---3. For a particular type of bulb, $36 \%$ grow into plants with blue flowers and the remainder grow into plants with white flowers. Bulbs are sold in mixed bags of 40
Russell selects a random sample of 5 bags of bulbs.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers.\\
(4)
Maggie takes a random sample of $n$ bulbs.\\
Using a normal approximation, the probability that more than 244 of these $n$ bulbs will grow into blue flowers is 0.0521 to 4 decimal places.
\item Find the value of $n$.\\
(6)\\
(Total 10 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 3 Q3 [10]}}