| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Standard +0.3 This is a standard multi-part combinations question covering typical S1 topics: conditional selection with constraints, permutations with restrictions, and basic probability. Part (a) requires careful case-work with the twins constraint, parts (b)(i)-(ii) involve standard digit arrangement problems with mild restrictions, and part (c) is a straightforward probability calculation using the multiplication principle. While requiring multiple techniques, all are textbook applications without novel insight, making it slightly easier than average for A-level. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \([P = \pm F + 0.6g\sin25°]\) | M1 | For resolving forces in the direction of \(P\) |
| \(P_{\max} = F + 0.6g\sin25°\) or '\(P = F + 0.6g\sin25°\) when the particle is about to slide upwards' | A1 | |
| \(P_{\min} = -F + 0.6g\sin25°\) or '\(P = -F + 0.6g\sin25°\) when the particle is about to slide downwards' | A1 | |
| \(R = 0.6g\cos25°\) | B1 | |
| \([F = \mu R]\) | M1 | For using \(F = \mu R\) |
| \([P_{\max} = 0.36 \times 0.6g\cos25° + 0.6g\sin25°, P_{\min} = -0.36 \times 0.6g\cos25° + 0.6g\sin25°]\) | DM1 | For substituting for \(F\) to obtain values of \(P_{\max}\) and \(P_{\min}\) |
| \(P_{\max} = 4.49, P_{\min} = 0.578\) (accept 0.58) | A1 | Dependent on first M mark |
| M1 | For identifying range of value for equilibrium. AEF; Accept 0.58 instead of 0.578 and accept < instead of \(\leq\) | |
| Set of values is \(\{P; 0.578 \leq P \leq 4.49\}\) | A1 | 9 marks total |
$[P = \pm F + 0.6g\sin25°]$ | M1 | For resolving forces in the direction of $P$
$P_{\max} = F + 0.6g\sin25°$ or '$P = F + 0.6g\sin25°$ when the particle is about to slide upwards' | A1 |
$P_{\min} = -F + 0.6g\sin25°$ or '$P = -F + 0.6g\sin25°$ when the particle is about to slide downwards' | A1 |
$R = 0.6g\cos25°$ | B1 |
$[F = \mu R]$ | M1 | For using $F = \mu R$
$[P_{\max} = 0.36 \times 0.6g\cos25° + 0.6g\sin25°, P_{\min} = -0.36 \times 0.6g\cos25° + 0.6g\sin25°]$ | DM1 | For substituting for $F$ to obtain values of $P_{\max}$ and $P_{\min}$
$P_{\max} = 4.49, P_{\min} = 0.578$ (accept 0.58) | A1 | Dependent on first M mark
| M1 | For identifying range of value for equilibrium. AEF; Accept 0.58 instead of 0.578 and accept < instead of $\leq$
Set of values is $\{P; 0.578 \leq P \leq 4.49\}$ | A1 | 9 marks total
---6
\begin{enumerate}[label=(\alph*)]
\item A chess team of 2 girls and 2 boys is to be chosen from the 7 girls and 6 boys in the chess club. Find the number of ways this can be done if 2 of the girls are twins and are either both in the team or both not in the team.
\item \begin{enumerate}[label=(\roman*)]
\item The digits of the number 1244687 can be rearranged to give many different 7-digit numbers. How many of these 7 -digit numbers are even?
\item How many different numbers between 20000 and 30000 can be formed using 5 different digits from the digits $1,2,4,6,7,8$ ?
\end{enumerate}\item Helen has some black tiles, some white tiles and some grey tiles. She places a single row of 8 tiles above her washbasin. Each tile she places is equally likely to be black, white or grey. Find the probability that there are no tiles of the same colour next to each other.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2012 Q6 [12]}}