OCR H240/02 (Pure Mathematics and Statistics) 2018 June

1

1. Express \(2 x ^ { 2 } - 12 x + 23\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Use your result to show that the equation \(2 x ^ { 2 } - 12 x + 23 = 0\) has no real roots.
3. Given that the equation \(2 x ^ { 2 } - 12 x + k = 0\) has repeated roots, find the value of the constant \(k\).

2 The points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1
- 2
5 \end{array} \right)\) and \(\left( \begin{array} { c } - 3
- 1
2 \end{array} \right)\) respectively.

1. Find the exact length of \(A B\).
2. Find the position vector of the midpoint of \(A B\). The points \(P\) and \(Q\) have position vectors \(\left( \begin{array} { l } 1
   2
   0 \end{array} \right)\) and \(\left( \begin{array} { l } 5
   1
   3 \end{array} \right)\) respectively.
3. Show that \(A B P Q\) is a parallelogram.

3 Ayesha, Bob, Chloe and Dave are discussing the relationship between the time, \(t\) hours, they might spend revising for an examination, and the mark, \(m\), they would expect to gain. Each of them draws a graph to model this relationship for himself or herself.
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_437_423_1576_187}
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_439_426_1576_609}
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_439_428_1576_1032}
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_437_419_1576_1462}

1. Assuming Ayesha's model is correct, how long would you recommend that she spends revising?
2. State one feature of Dave's model that is likely to be unrealistic.
3. Suggest a reason for the shape of Bob's graph as compared with Ayesha's graph.
4. What does Chloe's model suggest about her attitude to revision?

4 Prove that \(\sin ^ { 2 } ( \theta + 45 ) ^ { \circ } - \cos ^ { 2 } ( \theta + 45 ) ^ { \circ } \equiv \sin 2 \theta ^ { \circ }\).

5 Charlie claims to have proved the following statement.
"The sum of a square number and a prime number cannot be a square number."

1. Give an example to show that Charlie's statement is not true. Charlie's attempt at a proof is below.
   Assume that the statement is not true.
   ⇒ There exist integers \(n\) and \(m\) and a prime \(p\) such that \(n ^ { 2 } + p = m ^ { 2 }\).
   \(\Rightarrow p = m ^ { 2 } - n ^ { 2 }\)
   \(\Rightarrow p = ( m - n ) ( m + n )\)
   \(\Rightarrow p\) is the product of two integers.
   \(\Rightarrow p\) is not prime, which is a contradiction.
   ⇒ Charlie's statement is true.
2. Explain the error that Charlie has made.
3. Given that 853 is a prime number, find the square number \(S\) such that \(S + 853\) is also a square number.

1. Find the \(x\)-coordinate of the point where the curve crosses the \(x\) axis.
2. The points \(A\) and \(B\) lie on the curve and have \(x\) coordinates 2 and 4. Show that the line \(A B\) is parallel to the \(x\)-axis.
3. Find the coordinates of the turning point on the curve.
4. Determine whether this turning point is a maximum or a minimum.

7 The diagram shows a part \(A B C\) of the curve \(y = 3 - 2 x ^ { 2 }\), together with its reflections in the lines \(y = x\), \(y = - x\) and \(y = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-05_691_673_1957_678}

8

1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
   (a) Find \(\mathrm { P } ( X > 25 )\).
   (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
   (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).

9 Briony suspects that a particular 6-sided dice is biased in favour of 2. She plans to throw the dice 35 times and note the number of times that it shows a 2 . She will then carry out a test at the \(4 \%\) significance level. Find the rejection region for the test.

10 A certain forest contains only trees of a particular species. Dipak wished to take a random sample of 5 trees from the forest. He numbered the trees from 1 to 784. Then, using his calculator, he generated the random digits 14781049 . Using these digits, Dipak formed 5 three-digit numbers. He took the first, second and third digits, followed by the second, third and fourth digits and so on. In this way he obtained the following list of numbers for his sample. $$\begin{array} { l l l l l } 147 & 478 & 781 & 104 & 49 \end{array}$$

1. Explain why Dipak omitted the number 810 from his list.
2. Explain why Dipak's sample is not random. The mean height of all trees of this species is known to be 4.2 m . Dipak wishes to test whether the mean height of trees in the forest is less than 4.2 m . He now uses a correct method to choose a random sample of 50 trees and finds that their mean height is 4.0 m . It is given that the standard deviation of trees in the forest is 0.8 m .
3. Carry out the test at the \(2 \%\) significance level.

11 Christa used Pearson's product-moment correlation coefficient, \(r\), to compare the use of public transport with the use of private vehicles for travel to work in the UK.

1. Using the pre-release data set for all 348 UK Local Authorities, she considered the following four variables.

   Number of employees using public transport	\(x\)
   Number of employees using private vehicles	\(y\)
   Proportion of employees using public transport	\(a\)
   Proportion of employees using private vehicles	\(b\)

   (a) Explain, in context, why you would expect strong, positive correlation between \(x\) and \(y\).
   (b) Explain, in context, what kind of correlation you would expect between \(a\) and \(b\).
2. Christa also considered the data for the 33 London boroughs alone and she generated the following scatter diagram. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{London} \includegraphics[alt={},max width=\textwidth]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-07_467_707_1366_653}
   \end{figure} One London Borough is represented by an outlier in the diagram.
   (a) Suggest what effect this outlier is likely to have on the value of \(r\) for the 32 London Boroughs.
   (b) Suggest what effect this outlier is likely to have on the value of \(r\) for the whole country.
   (c) What can you deduce about the area of the London Borough represented by the outlier? Explain your answer.

12 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.

1. Show that \(a = \frac { 16 } { 31 }\). The discrete probability distribution for \(X\) is given in the table.

   \(x\)	1	2	3	4	5
   \(\mathrm { P } ( X = x )\)	\(\frac { 16 } { 31 }\)	\(\frac { 8 } { 31 }\)	\(\frac { 4 } { 31 }\)	\(\frac { 2 } { 31 }\)	\(\frac { 1 } { 31 }\)

2. Find the probability that \(X\) is odd. Two independent values of \(X\) are chosen, and their sum \(S\) is found.
3. Find the probability that \(S\) is odd.
4. 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. $$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
5. Find \(\mathrm { P } ( Y = 1 )\).
6. 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.

13 In this question you must show detailed reasoning. The probability that Paul's train to work is late on any day is 0.15 , independently of other days.

1. The number of days on which Paul's train to work is late during a 450-day period is denoted by the random variable \(Y\). Find a value of \(a\) such that \(\mathrm { P } ( Y > a ) \approx \frac { 1 } { 6 }\). In the expansion of \(( 0.15 + 0.85 ) ^ { 50 }\), the terms involving \(0.15 ^ { r }\) and \(0.15 ^ { r + 1 }\) are denoted by \(T _ { r }\) and \(T _ { r + 1 }\) respectively.
2. Show that \(\frac { T _ { r } } { T _ { r + 1 } } = \frac { 17 ( r + 1 ) } { 3 ( 50 - r ) }\).
3. The number of days on which Paul's train to work is late during a 50-day period is modelled by the random variable \(X\).
   (a) Find the values of \(r\) for which \(\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )\).
   (b) Hence find the most likely number of days on which the train will be late during a 50-day period.