SPS
SPS FM Pure
2024
September
Q5
5. (a) Show that the binomial expansion of
$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$
in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is
$$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$
giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.
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SPS
SPS FM Pure
2024
September
Q8
8. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places.
Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula
$$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$
where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017.
Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute.
Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017.
Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula
$$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$
where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).
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SPS
SPS FM Statistics
2024
September
Q1
1.
The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{a65400d1-fadc-4bc7-ba4b-af2df57e390a-04_551_894_395_169}
A student is chosen at random from the 100 students. Then another student is chosen from the remaining students.
Find the probability that the first student studies History and the second student studies Geography but not Psychology.
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SPS
SPS SM Statistics
2024
September
Q3
3.
The discrete random variable \(X\) takes values \(1,2,3,4\) and 5 , and its probability distribution is defined as follows.
$$\mathrm { P } ( X = x ) = \begin{cases} a & x = 1
\frac { 1 } { 2 } \mathrm { P } ( X = x - 1 ) & x = 2,3,4,5
0 & \text { otherwise } \end{cases}$$
where \(a\) is a constant.
- Show that \(a = \frac { 16 } { 31 }\).
The discrete probability distribution for \(X\) is given in the table.
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 16 } { 31 }\) | \(\frac { 8 } { 31 }\) | \(\frac { 4 } { 31 }\) | \(\frac { 2 } { 31 }\) | \(\frac { 1 } { 31 }\) |
- Find the probability that \(X\) is odd.
Two independent values of \(X\) are chosen, and their sum \(S\) is found.
- Find the probability that \(S\) is odd.
- Find the probability that \(S\) is greater than 8 , given that \(S\) is odd.
Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows.
$$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
- Find \(\mathrm { P } ( Y = 1 )\).
- Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car.
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SPS
SPS SM Statistics
2024
September
Q4
4.
The radar diagrams illustrate some population figures from the 2011 census results.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-10_723_776_360_159}
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-10_725_775_358_1055}
Each radius represents an age group, as follows:
| Radius | 1 | 2 | 3 | 4 | 5 | 6 |
| \(0 - 17\) | \(18 - 29\) | \(30 - 44\) | \(45 - 59\) | \(60 - 74\) | \(75 +\) |
The distance of each dot from the centre represents the number of people in the relevant age group.
- The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities.
- Approximately how many people aged 45 to 59 were there in Liverpool?
- State the main two differences between the age profiles of the two Local Authorities.
- James makes the following claim.
"Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland."
Use the radar diagrams to give a justification for this claim.
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SPS
SPS SM Statistics
2024
September
Q7
7.
The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-16_542_883_459_148}
A student is chosen at random from the 100 students. Then another student is chosen from the remaining students.
Find the probability that the first student studies History and the second student studies Geography but not Psychology.
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