SPS SPS SM Statistics (SPS SM Statistics) 2022 February

1. Answer all the questions. $$f ( x ) = 3 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 10$$

1. Use the factor theorem to show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\)
2. Find the values of the constants \(a , b\) and \(c\) such that $$\mathrm { f } ( x ) \equiv ( x - 2 ) \left( a x ^ { 2 } + b x + c \right)$$
3. Using your answer to part (b) show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
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2. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Express as an integral $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x$$
2. Using your answer to part (a) show that $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x = \frac { 98 } { 3 }$$ [BLANK PAGE]

3. The curve \(C\) has equation $$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. 1. Verify that \(C\) has a stationary point at \(x = 1\)
   2. Show that this stationary point is a point of inflection, giving reasons for your answer.
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4. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l l } \mathrm { f } ( x ) = \frac { k x } { 2 x - 1 } & x \in \mathbb { R } & x \neq \frac { 1 } { 2 }
\mathrm {~g} ( x ) = 2 + 3 x - x ^ { 2 } & x \in \mathbb { R } & \end{array}$$ where \(k\) is a non-zero constant.

1. Find in terms of \(k\)
   1. \(\mathrm { fg } ( 4 )\)
   2. the range of f
   3. \(\mathrm { f } ^ { - 1 }\) Given that $$\mathrm { f } ^ { - 1 } ( 2 ) = \frac { 11 } { 3 \mathrm {~g} ( 2 ) }$$
2. find the exact value of \(k\)
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5. $$f ( x ) = \frac { 10 } { \sqrt { 4 - 3 x } }$$

1. Show that the first 4 terms in the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), are $$A + B x + C x ^ { 2 } + \frac { 675 } { 1024 } x ^ { 3 }$$ where \(A , B\) and \(C\) are constants to be found. Give each constant in simplest form. Given that this expansion is valid for \(| x | < k\)
2. state the largest value of \(k\). By substituting \(x = \frac { 1 } { 3 }\) into \(\mathrm { f } ( x )\) and into the answer for part (a),
3. find an approximation for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.
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6. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Express \(3 \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\) where
   • \(\quad R\) and \(\alpha\) are constants
   • \(R > 0\)
   • \(0 < \alpha < \frac { \pi } { 2 }\)
   Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a rabbit hole on a particular day is modelled by the equation $$\theta = 6.5 + 3 \cos \left( \frac { \pi t } { 13 } - 4 \right) + \sin \left( \frac { \pi t } { 13 } - 4 \right) \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
   Using the equation of the model and your answer to part (a)
2. 1. deduce the minimum value of \(\theta\) during this day,
   2. find the time of day when this minimum value occurs, giving your answer to the nearest minute.
3. Find the rate of temperature increase in the rabbit hole at midday.
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7. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula $$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.

1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{59120894-c480-492b-a304-106ddbadacf0-18_613_926_699_699} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
   The student recorded the value of \(T\) for different values of \(l\).
   Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data. The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\) Using this information,
2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(a\).
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8.

1. Use the substitution \(u = 1 + \sin ^ { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \int _ { p } ^ { q } \frac { 4 } { u ( 2 - u ) } \mathrm { d } u$$ where \(p\) and \(q\) are constants to be found.
2. Hence, using algebraic integration, show that $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \ln A$$ where \(A\) is a rational number to be found.
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9. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
The first 3 terms of an arithmetic sequence are $$\ln 3 \quad \ln \left( 3 ^ { k } - 1 \right) \quad \ln \left( 3 ^ { k } + 5 \right)$$ Find the exact value of the constant \(k\).
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10. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\) $$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.
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1. Answer all the questions.

The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where \(c\) and \(k\) are constants.

1. Show that \(16 c + k = 1\)
2. Given that \(\mathrm { P } ( X \geq 3 ) = \frac { 5 } { 8 }\) find the value of \(c\) and the value of \(k\).
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12. The four pie charts illustrate the numbers of employees using different methods of travel in four Local Authorities in 2011.
\includegraphics[max width=\textwidth, alt={}, center]{59120894-c480-492b-a304-106ddbadacf0-30_1234_1160_276_255} \begin{table}[h] \end{table}

1. State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
2. Two of the Local Authorities represent urban areas.
   1. State with a reason which two Local Authorities are likely to be urban.
   2. One urban Local Authority introduced a Park-and-Ride service in 2006. Users of this service drive to the edge of the urban area and then use buses to take them into the centre of the area. A student claims that a comparison of the corresponding pie charts for 2001 (not shown) and 2011 would enable them to identify which Local Authority this was. State with a reason whether you agree with the student.
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13. Sam is playing a computer game. When Sam earns a reward in the game, she randomly receives either a Silver reward or a Gold reward. Each time that Sam earns a reward, the probability of receiving a Gold reward is 0.4 One day Sam plays the computer game and earns 11 rewards.

1. Find the probability that she receives
   1. exactly 2 Gold rewards,
   2. at least 5 Gold rewards. In the next month Sam earns 300 rewards.
      She decides to use a Normal distribution to estimate the probability that she will receive at least 135 Gold rewards.
2. 1. Find the mean and variance of this Normal distribution.
   2. Estimate the probability that Sam will receive at least 135 Gold rewards.
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14. Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.

   Mean	Standard deviation	Median	Lower quartile	Upper quartile
   21.0	4.20	20.5	18.0	22.9

3. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
4. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the \(5 \%\) significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.
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