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MATHEMATICS 9709

Cambridge International AS and A Level

Name of student Date
Adm. number Year/grade Stream
Subject Probability & Statistics 2 (S2) Variant(s) P72, P73
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 4 5 3 5 9 8 34
Score

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Question 1 Code: 9709/72/O/N/13/1, Topic: -

71

Question 2 Code: 9709/73/O/N/15/1, Topic: -

It is known that the number, $N$, of words contained in the leading article each day in a certain newspaper can be modelled by a normal distribution with mean 352 and variance 29. A researcher takes a random sample of 10 leading articles and finds the sample mean, $\bar{N}$, of $N$.

$\text{(i)}$ State the distribution of $\bar{N}$, giving the values of any parameters. $[2]$

$\text{(ii)}$ Find $\mathrm{P}(\bar{N}>354)$. $[3]$

Question 3 Code: 9709/72/O/N/16/1, Topic: -

71

Question 4 Code: 9709/72/O/N/11/2, Topic: -

Question 5 Code: 9709/73/O/N/12/6, Topic: -

Darts are thrown at random at a circular board. The darts hit the board at distances $X$ centimetres from the centre, where $X$ is a random variable with probability density function given by

$$ \mathrm{f}(x)= \begin{cases}\dfrac{2}{a^{2}} x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise }\end{cases} $$

where $a$ is a positive constant.

$\text{(i)}$ Verify that $\mathrm{f}$ is a probability density function whatever the value of $a$. $[3]$

It is now given that $\mathrm{E}(X)=8$.

$\text{(ii)}$ Find the value of $a$. $[3]$

$\text{(iii)}$ Find the probability that a dart lands more than $6 \mathrm{~cm}$ from the centre of the board. $[3]$

Question 6 Code: 9709/72/O/N/13/6, Topic: -

Worked solutions: P1, P3 & P6 (S1)

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