18 \({ } ^ { \text {th } }\) April 2024}
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Name: □
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Instructions
- Answer all the questions.
- Use black or blue ink. Pencil may be used for graphs and diagrams only.
- There are blank pages at the end of the paper for additional working. You must clearly indicate when you have moved onto additional pages on the question itself. Make sure to include the question number.
- You are permitted to use a scientific or graphical calculator in this paper.
- Where appropriate, your answer should be supported with working. Marks might be given for using a correct method, even if your answer is wrong.
- Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question.
- The acceleration due to gravity is denoted by \(g \mathrm {~ms} ^ { - 2 }\). When a numerical value is needed use \(g = 9.8\) unless a different value is specified in the question.
Information
- The total mark for this paper is \(\mathbf { 6 0 }\) marks.
- The marks for each question are shown in brackets.
- You are reminded of the need for clear presentation in your answers.
- You have \(\mathbf { 6 0 }\) minutes for this paper.
\section*{Formulae
A Level Mathematics A (H240)}
Arithmetic series
\(S _ { n } = \frac { 1 } { 2 } n ( a + l ) = \frac { 1 } { 2 } n \{ 2 a + ( n - 1 ) d \}\)
\section*{Geometric series}
\(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\)
\(S _ { \infty } = \frac { a } { 1 - r }\) for \(| r | < 1\)
\section*{Binomial series}
\(( a + b ) ^ { n } = a ^ { n } + { } ^ { n } \mathrm { C } _ { 1 } a ^ { n - 1 } b + { } ^ { n } \mathrm { C } _ { 2 } a ^ { n - 2 } b ^ { 2 } + \ldots + { } ^ { n } \mathrm { C } _ { r } a ^ { n - r } b ^ { r } + \ldots + b ^ { n } \quad ( n \in \mathbb { N } )\)
where \({ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }\)
$$( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \ldots \quad ( | x | < 1 , n \in \mathbb { R } )$$
\section*{Differentiation}
| \(\mathrm { f } ( x )\) | \(\mathrm { f } ^ { \prime } ( x )\) |
| \(\tan k x\) | \(k \sec ^ { 2 } k x\) |
| \(\sec x\) | \(\sec x \tan x\) |
| \(\cot x\) | \(- \operatorname { cosec } ^ { 2 } x\) |
| \(\operatorname { cosec } x\) | \(- \operatorname { cosec } x \cot x\) |
Quotient rule \(y = \frac { u } { v } , \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { v \frac { \mathrm {~d} u } { \mathrm {~d} x } - u \frac { \mathrm {~d} v } { \mathrm {~d} x } } { v ^ { 2 } }\)
\section*{Differentiation from first principles}
\(\mathrm { f } ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\)
\section*{Integration}
\(\int \frac { \mathrm { f } ^ { \prime } ( x ) } { \mathrm { f } ( x ) } \mathrm { d } x = \ln | \mathrm { f } ( x ) | + c\)
\(\int \mathrm { f } ^ { \prime } ( x ) ( \mathrm { f } ( x ) ) ^ { n } \mathrm {~d} x = \frac { 1 } { n + 1 } ( \mathrm { f } ( x ) ) ^ { n + 1 } + c\)
Integration by parts \(\int u \frac { \mathrm {~d} v } { \mathrm {~d} x } \mathrm {~d} x = u v - \int v \frac { \mathrm {~d} u } { \mathrm {~d} x } \mathrm {~d} x\)
\section*{Small angle approximations}
\(\sin \theta \approx \theta , \cos \theta \approx 1 - \frac { 1 } { 2 } \theta ^ { 2 } , \tan \theta \approx \theta\) where \(\theta\) is measured in radians
\section*{Trigonometric identities}
$$\begin{aligned}
& \sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B
& \cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B
& \tan ( A \pm B ) = \frac { \tan A \pm \tan B } { 1 \mp \tan A \tan B } \quad \left( A \pm B \neq \left( k + \frac { 1 } { 2 } \right) \pi \right)
\end{aligned}$$
\section*{Numerical methods}
Trapezium rule: \(\int _ { a } ^ { b } y \mathrm {~d} x \approx \frac { 1 } { 2 } h \left\{ \left( y _ { 0 } + y _ { n } \right) + 2 \left( y _ { 1 } + y _ { 2 } + \ldots + y _ { n - 1 } \right) \right\}\), where \(h = \frac { b - a } { n }\)
The Newton-Raphson iteration for solving \(\mathrm { f } ( x ) = 0 : x _ { n + 1 } = x _ { n } - \frac { \mathrm { f } \left( x _ { n } \right) } { \mathrm { f } ^ { \prime } \left( x _ { n } \right) }\)
\section*{Probability}
\(\mathrm { P } ( A \cup B ) = \mathrm { P } ( A ) + \mathrm { P } ( B ) - \mathrm { P } ( A \cap B )\)
\(\mathrm { P } ( A \cap B ) = \mathrm { P } ( A ) \mathrm { P } ( B \mid A ) = \mathrm { P } ( B ) \mathrm { P } ( A \mid B ) \quad\) or \(\quad \mathrm { P } ( A \mid B ) = \frac { \mathrm { P } ( A \cap B ) } { \mathrm { P } ( B ) }\)
\section*{Standard deviation}
\(\sqrt { \frac { \sum ( x - \bar { x } ) ^ { 2 } } { n } } = \sqrt { \frac { \sum x ^ { 2 } } { n } - \bar { x } ^ { 2 } }\) or \(\sqrt { \frac { \sum f ( x - \bar { x } ) ^ { 2 } } { \sum f } } = \sqrt { \frac { \sum f x ^ { 2 } } { \sum f } - \bar { x } ^ { 2 } }\)
\section*{The binomial distribution}
If \(X \sim \mathrm {~B} ( n , p )\) then \(\mathrm { P } ( X = x ) = \binom { n } { x } p ^ { x } ( 1 - p ) ^ { n - x }\), mean of \(X\) is \(n p\), variance of \(X\) is \(n p ( 1 - p )\)
\section*{Hypothesis test for the mean of a normal distribution}
If \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) then \(\bar { X } \sim \mathrm {~N} \left( \mu , \frac { \sigma ^ { 2 } } { n } \right)\) and \(\frac { \bar { X } - \mu } { \sigma / \sqrt { n } } \sim \mathrm {~N} ( 0,1 )\)
\section*{Percentage points of the normal distribution}
If \(Z\) has a normal distribution with mean 0 and variance 1 then, for each value of \(p\), the table gives the value of \(z\) such that \(\mathrm { P } ( Z \leqslant z ) = p\).
| \(p\) | 0.75 | 0.90 | 0.95 | 0.975 | 0.99 | 0.995 | 0.9975 | 0.999 | 0.9995 |
| \(z\) | 0.674 | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 2.807 | 3.090 | 3.291 |
\section*{Kinematics}
Motion in a straight line
\(v = u + a t\)
\(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\)
\(s = \frac { 1 } { 2 } ( u + v ) t\)
\(v ^ { 2 } = u ^ { 2 } + 2 a s\)
\(s = v t - \frac { 1 } { 2 } a t ^ { 2 }\)
Motion in two dimensions
\(\mathbf { v } = \mathbf { u } + \mathbf { a } t\)
\(\mathbf { s } = \mathbf { u } t + \frac { 1 } { 2 } \mathbf { a } t ^ { 2 }\)
\(\mathbf { s } = \frac { 1 } { 2 } ( \mathbf { u } + \mathbf { v } ) t\)
\(\mathbf { s } = \mathbf { v } t - \frac { 1 } { 2 } \mathbf { a } t ^ { 2 }\)
1.
200 candidates took each of two examination papers. The diagram shows the cumulative frequency graphs for their marks.
\includegraphics[max width=\textwidth, alt={}, center]{5b55a372-2cc8-454e-a10a-cabdc9801421-04_1091_1484_429_285}
- State, with a reason, which of the two papers was the easier one.
- The minimum mark for grade A , the top grade, on Paper 1 was 10 marks lower than the minimum mark for grade A on Paper 2. Given that 25 candidates gained grade A in Paper 1, find the number of candidates who gained grade A in Paper 2.
- The mean and standard deviation of the marks on Paper 1 were 36.5 and 28.2 respectively. Later, a marking error was discovered and it was decided to add 1 mark to each of the 200 marks on Paper 1. State the mean and standard deviation of the new marks on Paper 1.
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2.
In this question you must show detailed reasoning.
A disease that affects trees shows no visible evidence for the first few years after the tree is infected.
A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not \(100 \%\) reliable, and a researcher uses the following model.
- If the tree has the disease, the probability of a positive result is 0.95 .
- If the tree does not have the disease, the probability of a positive result is 0.1 .
(a) It is known that in a certain county, \(A , 35 \%\) of the trees have the disease. A tree in county \(A\) is chosen at random and is tested.
Given that the result is positive, determine the probability that this tree has the disease.
A forestry company wants to determine what proportion of trees in another county, \(B\), have the disease. They choose a large random sample of trees in county \(B\).
Each tree in the sample is tested and it is found that the result is positive for \(43 \%\) of these trees.
(b) By carrying out a calculation, determine an estimate of the proportion of trees in county \(B\) that have the disease.
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3.
A group of 8 people, including Kathy, David and Harpreet, are planning a theatre trip. - Four of the group are chosen at random, without regard to order, to carry the refreshments. Find the probability that these 4 people include Kathy and David but not Harpreet.
- The 8 people sit in a row. Kathy and David sit next to each other and Harpreet sits at the left-hand end of the row. How many different arrangements of the 8 people are possible?
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4.
A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that \(12 \%\) of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.
(a) Find the probability that the fifth person the researcher asks is the first to have watched the programme.
(b) Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
(c) Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
(d) Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.
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5.
The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 105.0 ) = 0.2420\) and \(\mathrm { P } ( H > 110.0 ) = 0.6915\). Find the values of \(\mu\) and \(\sigma\), giving your answers to a suitable degree of accuracy.
6.
For the events \(A\) and \(B\),
$$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \quad \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \quad \text { and } \quad \mathrm { P } ( A \cup B ) = 0.65$$ - Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
- Determine whether or not \(A\) and \(B\) are independent.
7.
An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered. - Find the probability that
(A) exactly 8 of these orders are delivered within 24 hours,
(B) at least 8 of these orders are delivered within 24 hours.
The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours. - Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
- A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18 , find the critical region for this test, showing all of your calculations.
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8.
The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 ,
0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant and \(n\) is a parameter whose value is positive.
It is given that the median of \(X\) is 0.8816 correct to 4 decimal places.
Ten independent observations of \(X\) are obtained.
Find the expected number of observations that are less than 0.8 .
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