Questions H240/02 - A-Level Maths

OCR H240/02 2018 June Q1

1

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

OCR H240/02 2018 June Q2

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.

Find the exact length of $A B$.
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.
Show that $A B P Q$ is a parallelogram.

OCR H240/02 2018 June Q3

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}

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

OCR H240/02 2018 June Q4

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

OCR H240/02 2018 June Q5

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

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.
Explain the error that Charlie has made.
Given that 853 is a prime number, find the square number $S$ such that $S + 853$ is also a square number.

OCR H240/02 2018 June Q7

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}

OCR H240/02 2018 June Q8

8

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 $\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5$.
The variable $Y$ has the distribution $\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)$. Find $\mathrm { P } ( Y > 1.5 \mu )$.

OCR H240/02 2018 June Q9

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.

OCR H240/02 2018 June Q10

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. $$\begin{array} { l l l l l } 147 & 478 & 781 & 104 & 49 \end{array}$$

Explain why Dipak omitted the number 810 from his list.
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 .
Carry out the test at the $2 \%$ significance level.

OCR H240/02 2018 June Q11

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.

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 $b$.
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.

OCR H240/02 2018 June Q12

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.

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.

OCR H240/02 2019 June Q1

1

Differentiate the following.
1. $\frac { x ^ { 2 } } { 2 x + 1 }$
2. $\tan \left( x ^ { 2 } - 3 x \right)$
Use the substitution $u = \sqrt { x } - 1$ to integrate $\frac { 1 } { \sqrt { x } - 1 }$.
Integrate $\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }$.

OCR H240/02 2019 June Q2

2

Find the coefficient of $x ^ { 5 }$ in the expansion of $( 3 - 2 x ) ^ { 8 }$.
1. Expand $( 1 + 3 x ) ^ { 0.5 }$ as far as the term in $x ^ { 3 }$.
2. State the range of values of $x$ for which your expansion is valid. A student suggests the following check to determine whether the expansion obtained in part (b)(i) may be correct.
  "Use the expansion to find an estimate for $\sqrt { 103 }$, correct to five decimal places, and compare this with the value of $\sqrt { 103 }$ given by your calculator."
3. Showing your working, carry out this check on your expansion from part (b)(i).

OCR H240/02 2019 June Q3

3

A circle is defined by the parametric equations $x = 3 + 2 \cos \theta , y = - 4 + 2 \sin \theta$.
1. Find a cartesian equation of the circle.
2. Write down the centre and radius of the circle.
In this question you must show detailed reasoning. The curve $S$ is defined by the parametric equations $x = 4 \cos t , y = 2 \sin t$. The line $L$ is a tangent to $S$ at the point given by $t = \frac { 1 } { 6 } \pi$. Find where the line $L$ cuts the $x$-axis.

OCR H240/02 2019 June Q4

4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population $P$ after $t$ years. The diagrams show these models as they apply to the first 20 years.
\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242}
\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}

Without calculation, describe briefly how the rate of growth of $P$ will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is $P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }$.
The equation of the curve for model B is $P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)$.
Describe the behaviour of $P$ that is predicted for $t > 20$
1. using model A,
2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
State what is suggested about the long-term food supply by
1. model A,
2. model B.

OCR H240/02 2019 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-06_371_339_255_251} For a cone with base radius $r$, height $h$ and slant height $l$, the following formulae are given.
Curved surface area, $S = \pi r l$
Volume, $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$
A container is to be designed in the shape of an inverted cone with no lid. The base radius is $r \mathrm {~m}$ and the volume is $V \mathrm {~m} ^ { 3 }$. The area of the material to be used for the cone is $4 \pi \mathrm {~m} ^ { 2 }$.

Show that $V = \frac { 1 } { 3 } \pi \sqrt { 16 r ^ { 2 } - r ^ { 6 } }$.
In this question you must show detailed reasoning. It is given that $V$ has a maximum value for a certain value of $r$.
Find the maximum value of $V$, giving your answer correct to 3 significant figures.

OCR H240/02 2019 June Q6

6 Shona makes the following claim.
" $n$ is an even positive integer greater than $2 \Rightarrow 2 ^ { n } - 1$ is not prime"
Prove that Shona's claim is true.

OCR H240/02 2019 June Q7

7 In this question you must show detailed reasoning.
Use the substitution $u = 6 x ^ { 2 } + x$ to solve the equation $36 x ^ { 4 } + 12 x ^ { 3 } + 7 x ^ { 2 } + x - 2 = 0$.

OCR H240/02 2019 June Q8

8 The stem-and-leaf diagram shows the heights, in centimetres, of 17 plants, measured correct to the nearest centimetre.

5	5	7	9	9
6	3	4	5	5	5	9	9
7	4	5	7	9	9
8
9	9

Key: 5 | 6 means 56

Find the median and inter-quartile range of these heights.
Calculate the mean and standard deviation of these heights.
State one advantage of using the median rather than the mean as a measure of average for these heights.

OCR H240/02 2019 June Q9

9

The masses, in grams, of plums of a certain kind have the distribution $\mathrm { N } ( 55,18 )$.
1. Find the probability that a plum chosen at random has a mass between 50.0 and 60.0 grams.
2. The heaviest $5 \%$ of plums are classified as extra large. Find the minimum mass of extra large plums.
3. The plums are packed in bags, each containing 10 randomly selected plums. Find the probability that a bag chosen at random has a total mass of less than 530 g .
The masses, in grams, of apples of a certain kind have the distribution $\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)$. It is given that half of the apples have masses between 62 g and 72 g . Determine $\sigma$.

OCR H240/02 2019 June Q10

10 The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable $X$, where $X$ has the distribution $\mathrm { N } ( \mu , 0.0000409 )$. In the past the value of $\mu$ has been 0.0340 . This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the 5\% significance level, whether the mean level of pollutant has changed.

OCR H240/02 2019 June Q11

11 A trainer was asked to give a lecture on population profiles in different Local Authorities (LAs) in the UK. Using data from the 2011 census, he created the following scatter diagram for 17 selected LAs. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{17 Selected Local Authorities} \includegraphics[alt={},max width=\textwidth]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-08_560_897_466_246}

\end{figure} He selected the 17 LAs using the following method. The proportions of people aged 18 to 24 and aged 65+ in any Local Authority are denoted by $P _ { \text {young } }$ and $P _ { \text {senior } }$ respectively. The trainer used a spreadsheet to calculate the value of $k = \frac { P _ { \text {young } } } { P _ { \text {senior } } }$ for each of the 348 LAs in the UK. He then used specific ranges of values of $k$ to select the 17 LAs.

Estimate the ranges of values of $k$ that he used to select these 17 LAs.
Using the 17 LAs the trainer carried out a hypothesis test with the following hypotheses.
$\mathrm { H } _ { 0 }$ : There is no linear correlation in the population between $P _ { \text {young } }$ and $P _ { \text {senior } }$.
$\mathrm { H } _ { 1 }$ : There is negative linear correlation in the population between $P _ { \text {young } }$ and $P _ { \text {senior } }$.
He found that the value of Pearson's product-moment correlation coefficient for the 17 LAs is - 0.797 , correct to 3 significant figures.
1. Use the table on page 9 to show that this value is significant at the $1 \%$ level. The trainer concluded that there is evidence of negative linear correlation between $P _ { \text {young } }$ and $P _ { \text {senior } }$ in the population.
2. Use the diagram to comment on the reliability of this conclusion.
Describe one outstanding feature of the population in the areas represented by the points in the bottom right hand corner of the diagram.

The trainer's audience included representatives from several universities. Suggest a reason why the diagram might be of particular interest to these people. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Critical values of Pearson's product-moment correlation coefficient}

\multirow{2}{*}{1-tail test 2-tail test}	5\%	2.5\%	1\%	0.5\%
	10\%	5\%	2\%	1\%
$n$
1	-	-	-	-
2	-	-	-	-
3	0.9877	0.9969	0.9995	0.9999
4	0.9000	0.9500	0.9800	0.9900
5	0.8054	0.8783	0.9343	0.9587
6	0.7293	0.8114	0.8822	0.9172
7	0.6694	0.7545	0.8329	0.8745
8	0.6215	0.7067	0.7887	0.8343
9	0.5822	0.6664	0.7498	0.7977
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
14	0.4575	0.5324	0.6120	0.6614
15	0.4409	0.5140	0.5923	0.6411
16	0.4259	0.4973	0.5742	0.6226
17	0.4124	0.4821	0.5577	0.6055
18	0.4000	0.4683	0.5425	0.5897
19	0.3887	0.4555	0.5285	0.5751
20	0.3783	0.4438	0.5155	0.5614
21	0.3687	0.4329	0.5034	0.5487
22	0.3598	0.4227	0.4921	0.5368
23	0.3515	0.4132	0.4815	0.5256
24	0.3438	0.4044	0.4716	0.5151
25	0.3365	0.3961	0.4622	0.5052
26	0.3297	0.3882	0.4534	0.4958
27	0.3233	0.3809	0.4451	0.4869
28	0.3172	0.3739	0.4372	0.4785
29	0.3115	0.3673	0.4297	0.4705
30	0.3061	0.3610	0.4226	0.4629

\end{table} Turn over for questions 12 and 13

OCR H240/02 2019 June Q12

12 A random variable $X$ has probability distribution defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 1,2,3,4,5 ,
0 & \text { otherwise, } \end{cases}$$ where $k$ is a constant.

Show that $\mathrm { P } ( X = 3 ) = 0.2$.
Show in a table the values of $X$ and their probabilities.
Two independent values of $X$ are chosen, and their total $T$ is found.
1. Find $\mathrm { P } ( T = 7 )$.
2. Given that $T = 7$, determine the probability that one of the values of $X$ is 2 .

OCR H240/02 2019 June Q13

13 It is known that $26 \%$ of adults in the UK use a certain app. A researcher selects a random sample of 5000 adults in the UK. The random variable $X$ is defined as the number of adults in the sample who use the app. Given that $\mathrm { P } ( X < n ) < 0.025$, calculate the largest possible value of $n$.

OCR H240/02 2021 November Q1

1 Differentiate the following with respect to $x$.

$\mathrm { e } ^ { - 4 x }$
$\frac { x ^ { 2 } } { x + 1 }$

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

Questions H240/02 (94 questions)

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