$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	Total
Marks	8	6	7	21
Score

Question 1 Code: 9709/61/M/J/18/4, Topic: The normal distribution

$\text{(a)}$ The distance that car tyres of a certain make can travel before they need to be replaced has a normal distribution. A survey of a large number of these tyres found that the probability of this distance being more than $36800 \mathrm{~km}$ is $0.0082$ and the probability of this distance being more than $31000 \mathrm{~km}$ is $0.6915.$ Find the mean and standard deviation of the distribution. $[5]$

$\text{(b)}$ The random variable $X$ has the distribution $\mathrm{N}\left(\mu, \sigma^{2}\right)$, where $3 \sigma=4 \mu$ and $\mu \neq 0$. Find $\mathrm{P}(X<3 \mu)$. $[3]$

Question 2 Code: 9709/62/O/N/18/4, Topic: Permutations and combinations

$\text{(i)}$ Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together. $[3]$

$\text{(ii)}$ Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy. $[3]$

Question 3 Code: 9709/63/O/N/18/4, Topic: Permutations and combinations

Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.

$\text{(i)}$ Find the probability that the group of 7 includes one particular boy. $[3]$

$\text{(ii)}$ Find the probability that the group of 7 includes at least 2 girls. $[4]$