OCR H240/02 (Pure Mathematics and Statistics) 2021 November

1 Differentiate the following with respect to \(x\).

1. \(\mathrm { e } ^ { - 4 x }\)
2. \(\frac { x ^ { 2 } } { x + 1 }\)

2 The diagram shows part of the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-04_437_620_909_274} Explain why one of the roots of the equation \(\mathrm { f } ( x ) = 0\) cannot be found by the sign change method.

3 The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310 .
Determine the first term and the common difference.

4 The size, \(P\), of a population of a certain species of insect at time \(t\) months is modelled by the following formula.
\(P = 5000 - 1000 \cos ( 30 t ) ^ { \circ }\)

1. Write down the maximum size of the population.
2. Write down the difference between the largest and smallest values of \(P\).
3. Without giving any numerical values, describe briefly the behaviour of the population over time.
4. Find the time taken for the population to return to its initial size for the first time.
5. Determine the time on the second occasion when \(P = 4500\). A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and \(P\) converges to 5000 .
6. Suggest a change to the model that will take account of this observation.

5 In this question you must show detailed reasoning. Points \(A , B\) and \(C\) have coordinates \(( 0,6 ) , ( 7,5 )\) and \(( 6 , - 2 )\) respectively.

1. Find an equation of the perpendicular bisector of \(A B\).
2. Hence, or otherwise, find an equation of the circle that passes through points \(A , B\) and \(C\).

6 Alex is investigating the area, \(A\), under the graph of \(y = x ^ { 2 }\) between \(x = 1\) and \(x = 1.5\). They draw the graph, together with rectangles of width \(\delta x = 0.1\), and varying heights \(y\).
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-06_531_714_356_251}

1. Use the rectangles in the diagram to show that lower and upper bounds for the area \(A\) are 0.73 and 0.855 respectively.
2. Alex finds lower and upper bounds for the area \(A\), using widths \(\delta x\) of decreasing size. The results are shown in the table. Where relevant, values are given correct to 3 significant figures.

   Width \(\delta x\)	0.1	0.05	0.025	0.0125
   Lower bound for area \(A\)	0.73	0.761	0.776	0.784
   Upper bound for area \(A\)	0.855	0.823	0.807	0.799

   Use Alex's results to estimate the value of \(A\) correct to \(\mathbf { 2 }\) significant figures. Give a brief justification for your estimate.
3. Write down an expression, in terms of \(y\) and \(\delta x\), for the exact value of the area \(A\).

7 Differentiate \(\cos x\) with respect to \(x\), from first principles.

8 The number \(K\) is defined by \(K = n ^ { 3 } + 1\), where \(n\) is an integer greater than 2 .

1. Given that \(n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)\), find the constants \(b\) and \(c\).
2. Prove that \(K\) has at least two distinct factors other than 1 and \(K\).

9 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) relative to an origin \(O\) in 3-dimensional space. Rectangles \(O A D C\) and \(B E F G\) are the base and top surface of a cuboid.
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-07_522_812_952_280}

• The point \(M\) is the midpoint of \(B C\).
• The point \(X\) lies on \(A M\) such that \(A X = 2 X M\).
  1. Find \(\overrightarrow { O X }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), simplifying your answer.
  2. Hence show that the lines \(O F\) and \(A M\) intersect.

10 A researcher plans to carry out a statistical investigation to test whether there is linear correlation between the time ( \(T\) weeks) from conception to birth, and the birth weight ( \(W\) grams) of new-born babies.

1. Explain why a 1-tail test is appropriate in this context. The researcher records the values of \(T\) and \(W\) for a random sample of 11 babies. They calculate Pearson's product-moment correlation coefficient for the sample and find that the value is 0.722 .
2. Use the table below to carry out the test at the \(1 \%\) significance level. \section*{Critical values of Pearson's product-moment correlation coefficient.}

   \multirow{2}{*}{}	1-tail test	5\%	2.5\%	1\%	0.5\%
   	2-tail test	10\%	5\%	2.5\%	1\%
   \multirow{4}{*}{\(n\)}	10	0.5494	0.6319	0.7155	0.7646
   	11	0.5214	0.6021	0.6851	0.7348
   	12	0.4973	0.5760	0.6581	0.7079
   	13	0.4762	0.5529	0.6339	0.6835

11 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.

12 Anika and Beth are playing a game which consists of several points.

• The probability that Anika will win any point is 0.7 .
• The probability that Beth will win any point is 0.3 .
• The outcome of each point is independent of the outcome of every other point.

The first player to win two points wins the game.

1. Write down the probability that the game consists of more than three points.
2. Complete the probability tree diagram in the Printed Answer Booklet showing all the possibilities for the game.
3. Determine the probability that Beth wins the game.
4. Determine the probability that the game consists of exactly three points.
5. Given that Beth wins the game, determine the probability that the game consists of exactly three points.

13 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]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-10_1131_1077_347_242}

1. State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
2. Explain why pie charts are more suitable for answering part (a) than bar charts showing the same data.
3. 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.

14 The probability distribution of a random variable \(X\) is modelled as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 ,
0 & \text { otherwise, } \end{cases}\)
where \(k\) is a constant.

1. Show that \(k = \frac { 12 } { 25 }\).
2. Show in a table the values of \(X\) and their probabilities.
3. The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\). Find \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)\). In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7 .
4. Determine the probability that a total of exactly 7 is first reached on the 5th observation. \section*{OCR} Oxford Cambridge and RSA