Questions — Edexcel (9685 questions)

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Edexcel C4 Q5
12 marks Standard +0.8
5. $$f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  3. State the set of values of \(x\) for which your expansion is valid.
    5. continued
Edexcel C4 Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{beeaedf6-62e8-4649-b023-1b7e2be9957e-10_524_734_146_532} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with parametric equations $$x = t + \sin t , \quad y = \sin t , \quad 0 \leq t \leq \pi .$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis.
  3. Show that the region bounded by the curve and the \(x\)-axis has area 2 .
    6. continued
Edexcel C4 Q7
14 marks Standard +0.3
7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
  2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
  4. Find the position vector of \(C\).
    7. continued
    7. continued
Edexcel S1 Q1
7 marks Moderate -0.8
  1. A histogram is to be drawn to represent the following grouped continuous data:
Group\(0 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 50\)\(50 - 100\)
Frequency\(2 x\)\(3 x\)\(5 x\)\(6 x\)\(2 x\)\(x\)
The ' \(10 - 20\) ' bar has height 6 cm and width 4 cm . Calculate
  1. the height of the ' \(20 - 25\) ' bar,
  2. the total area under the histogram.
Edexcel S1 Q2
7 marks Moderate -0.8
2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find
  1. \(\mathrm { P } ( B )\),
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
  4. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
Edexcel S1 Q3
9 marks Moderate -0.8
3. The random variable \(X\) has the discrete uniform distribution over the set of consecutive integers \(\{ - 7 , - 6 , \ldots , 10 \}\).
Calculate (a) the expectation and variance of \(X\),
(b) \(\mathrm { P } ( X > 7 )\),
(c) the value of \(n\) for which \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).
Edexcel S1 Q4
9 marks Moderate -0.8
4. The marks, \(x\) out of 100 , scored by 30 candidates in an examination were as follows:
5192021232531373941
42444751565760616265
677071737577818298100
Given that \(\sum x = 1600\) and \(\sum x ^ { 2 } = 102400\),
  1. find the median, the mean and the standard deviation of these marks. The marks were scaled to give modified scores, \(y\), using the formula \(y = \frac { 4 x } { 5 } + 20\).
  2. Find the median, the mean and the standard deviation of the modified scores. \section*{STATISTICS 1 (A) TEST PAPER 1 Page 2}
Edexcel S1 Q5
12 marks Moderate -0.8
  1. The table shows the numbers of cars and vans in a company's fleet having registrations with the prefix letters shown.
Registration letter\(K\)\(L\)\(M\)\(N\)\(P\)\(R\)\(S\)\(T\)\(V\)
Number of cars \(( x )\)67911151412107
Number of vans \(( y )\)810141313151498
  1. Plot a scatter graph of this data, with the number of cars on the horizontal axis and the number of vans on the vertical axis.
  2. If there were \(4 J\)-registered cars, estimate the number of \(J\)-registered vans. Given that \(\sum x ^ { 2 } = 1001 , \sum y ^ { 2 } = 1264\) and \(\sum x y = 1106\),
  3. calculate the product-moment correlation coefficient between \(x\) and \(y\). Give a brief interpretation of your answer.
Edexcel S1 Q6
14 marks Moderate -0.3
6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables:
\(x\)012
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 10 }\)
\(y\)01
\(\mathrm { P } ( Y = y )\)\(\frac { 5 } { 8 }\)\(\frac { 3 } { 8 }\)
The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).
  1. Tabulate the probability distribution for \(Z\).
  2. Calculate \(\mathrm { E } ( Z )\).
  3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
  4. Calculate Var (3Z-4).
Edexcel S1 Q7
17 marks Standard +0.3
7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.
  1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
  2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
  3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
  4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.
Edexcel S1 Q1
10 marks Moderate -0.8
  1. (a) Explain briefly what is meant by a discrete random variable.
A family has 3 cats and 4 dogs. Two of the family's animals are to be chosen at random. The random variable \(X\) represents the number of dogs chosen.
(b) Copy and complete the table to show the probability distribution of \(X\) :
\(x\)012
\(\mathrm { P } ( X = x )\)
(c) Calculate
  1. \(\mathrm { E } ( X )\),
  2. \(\operatorname { Var } ( X )\),
  3. \(\operatorname { Var } ( 2 X )\).
Edexcel S1 Q2
11 marks Moderate -0.8
2. The discrete random variable \(X\) can take any value in the set \(\{ 1,2,3,4,5,6,7,8 \}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that \(\mathrm { P } ( X = 1 ) = 0.125\) and that \(\mathrm { E } ( X ) = 4.5\).
Beatrice finds that \(\mathrm { P } ( X = 2 ) = \mathrm { P } ( X = 3 ) = \mathrm { P } ( X = 4 ) = p\).
Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).
  1. Calculate the values of \(p\) and \(q\).
  2. Give the name for the distribution of \(X\).
  3. Calculate the standard deviation of \(X\).
Edexcel S1 Q3
13 marks Standard +0.3
3. The marks obtained by ten students in a Geography test and a History test were as follows:
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Geography \(( x )\)34574921845310776185
History \(( y )\)404955407139476573
  1. Given that \(\sum y = 547\), calculate the mark obtained by student \(E\) in History. Given further that \(\sum x ^ { 2 } = 34087 , \sum y ^ { 2 } = 31575\) and \(\sum x y = 31342\), calculate
  2. the product moment correlation coefficient between \(x\) and \(y\),
  3. an equation of the regression line of \(y\) on \(x\),
  4. an estimate of the History mark of student \(K\), who scored 70 in Geography.
  5. State, with a reason, whether you would expect your answer to part (d) to be reliable. \section*{STATISTICS 1 (A) TEST PAPER 2 Page 2}
Edexcel S1 Q4
13 marks Standard +0.3
  1. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
    1. If \(2 \mu = 3 \sigma\), find \(\mathrm { P } ( X < 2 \mu )\).
    2. If, instead, \(\mathrm { P } ( X < 3 \mu ) = 0 \cdot 86\),
      1. find \(\mu\) in terms of \(\sigma\),
      2. calculate \(\mathrm { P } ( X > 0 )\).
    3. The stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).
    \(A\)\(B\)
    \(8,7,4,1,0\)1\(1,1,2,5,6,8,9\)
    \(9,8,7,6,6,5,2\)2\(0,3,4,6,7,7,9\)
    \(9,7,6,4,2,1,0\)3\(1,4,5,5,8\)
    \(8,6,3,2,2\)4\(0,2,6,6,9,9\)
    \(6,4,0\)5\(2,3,5,7\)
    \(5,3,1\)60,1
    Key : 3| 1 | 2 means $$A = 13 , B = 12$$
  2. For each set of data, calculate estimates of the median and the quartiles.
  3. Calculate the 42nd percentile for \(A\).
  4. On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data.
  5. Describe the skewness of the distribution of \(A\) and of \(B\).
Edexcel S1 Q6
14 marks Standard +0.3
6. The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack A. Let \(A _ { i }\) represent the event that the first digit on this card is \(i\).
  1. Write down the value of \(\mathrm { P } \left( A _ { 2 } \right)\). The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B _ { i }\) represent the event that the first digit on this card is \(i\).
  2. Show that \(\mathrm { P } \left( A _ { 1 } \cap B _ { 1 } \right) = \frac { 1 } { 24 }\).
  3. Show that \(\mathrm { P } \left( A _ { 6 } \mid B _ { 5 } \right) = \frac { 4 } { 41 }\).
  4. Find the value of \(\mathrm { P } \left( A _ { 1 } \cup B _ { 3 } \right)\).
Edexcel S1 Q1
4 marks Moderate -0.8
  1. (a) Explain briefly what is meant by a random variable.
    (b) Write down a quantity which could be modelled as
    1. a discrete random variable,
    2. a continuous random variable.
    3. The discrete random variable \(X\) has the probability function given by the following table:
    \(x\)0123456
    \(\mathrm { P } ( X = x )\)0.090.120.220.16\(p\)\(2 p\)0.2
    (a) Show that \(p = 0.07\) (b) Find the value of \(\mathrm { E } ( X + 2 )\).
    (c) Find the value of \(\operatorname { Var } ( 3 X - 1 )\).
Edexcel S1 Q3
13 marks Standard +0.3
3. Twenty pairs of observations are made of two variables \(x\) and \(y\), which are believed to be related. It is found that $$\sum x = 200 , \quad \sum y = 174 , \quad \sum x ^ { 2 } = 6201 , \quad \sum y ^ { 2 } = 5102 , \quad \sum x y = 5200 .$$ Find
  1. the product-moment correlation coefficient between \(x\) and \(y\),
  2. the equation of the regression line of \(y\) on \(x\). Given that \(p = x + 30\) and \(q = y + 50\),
  3. find the equation of the regression line of \(q\) on \(p\), in the form \(q = m p + c\).
  4. Estimate the value of \(q\) when \(p = 46\), stating any assumptions you make.
Edexcel S1 Q4
14 marks Standard +0.8
4. The heights of the students at a university are assumed to follow a normal distribution. \(1 \%\) of the students are over 200 cm tall and 76\% are between 165 cm and 200 cm tall. Find
  1. the mean and the variance of the distribution,
  2. the percentage of the students who are under 158 cm tall.
  3. Comment briefly on the suitability of a normal distribution to model such a population. \section*{STATISTICS 1 (A) TEST PAPER 3 Page 2}
Edexcel S1 Q5
16 marks Moderate -0.8
  1. In a survey of natural habitats, the numbers of trees in sixty equal areas of land were recorded, as follows:
171292340321153422318
154510521413294369301547
356241319269312718620
22183051493550258102631
332940373844243442381123
  1. Construct a stem-and-leaf diagram to illustrate this data, using the groupings 5-9, 10-14, 15-19, 20-24, etc.
  2. Find the three quartiles for the distribution.
  3. On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers.
Edexcel S1 Q6
17 marks Standard +0.8
6. Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac { 1 } { 3 }\).
  1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r ( r - 1 ) = 420$$
  2. Hence or otherwise find the value of \(r\).
  3. Find the probability that, when three cards are drawn at random from the pack,
    1. at least two are red,
    2. the first one is red given that at least two are red.
Edexcel S1 Q1
6 marks Moderate -0.8
  1. Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.
    1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution?
    2. Making this modelling assumption, find the expectation and the variance of \(X\).
    3. (a) Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class.
    In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was \(15 - 20\) (i.e. \(\mathrm { f } x\), where \(15000 \leq x < 20000\) ), with frequency 300 . The highest frequency in a class was 400 , for the class \(30 - 40\), and the bar representing this class was 8 cm high. The total area under the histogram was \(50 \mathrm {~cm} ^ { 2 }\).
  2. Find the height and the width of the bar representing the modal class.
Edexcel S1 Q3
10 marks Moderate -0.8
3. The variable \(X\) represents the marks out of 150 scored by a group of students in an examination. The following ten values of \(X\) were obtained: $$60,66,76,80,94,106,110,116,124,140 .$$
  1. Write down the median, \(M\), of the ten marks.
  2. Using the coding \(y = \frac { x - M } { 2 }\), and showing all your working clearly, find the mean and the standard deviation of the marks.
  3. Find \(\mathrm { E } ( 3 X - 5 )\).
Edexcel S1 Q4
11 marks Moderate -0.3
4. The discrete random variable \(X\) has probability function \(\mathrm { P } ( X = x ) = k ( x + 4 )\). Given that \(X\) can take any of the values \(- 3 , - 2 , - 1,0,1,2,3,4\),
  1. find the value of the constant \(k\).
  2. Find \(\mathrm { P } ( X < 0 )\).
  3. Show that the cumulative distribution \(\mathrm { F } ( x )\) is given by $$\mathrm { F } ( x ) = \lambda ( x + 4 ) ( x + 5 )$$ where \(\lambda\) is a constant to be found. \section*{STATISTICS 1 (A) TEST PAPER 4 Page 2}
Edexcel S1 Q5
12 marks Moderate -0.8
  1. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A \cap B ) = 0.24 , \mathrm { P } ( A \cup B ) = 0.88\) and \(\mathrm { P } ( B ) = 0.52\).
    1. Find \(\mathrm { P } ( A )\).
    2. Determine, with reasons, whether \(A\) and \(B\) are
      1. mutually exclusive,
      2. independent.
    3. Find \(\mathrm { P } ( B \mid A )\).
    4. Find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).
    5. The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours.
      Use this model to calculate
    6. the probability that a person chosen at random took between 5 and 9 hours to complete the task,
    7. the range, symmetrical about the mean, within which \(80 \%\) of the people's times lie.
      (5 marks)
      It is found that, in fact, \(80 \%\) of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.
    8. Find the standard deviation of the times in the modified model.
    9. The following data was collected for seven cars, showing their engine size, \(x\) litres, and their fuel consumption, \(y \mathrm {~km}\) per litre, on a long journey.
    Car\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
    \(x\)0.951.201.371.762.252.502.875
    \(y\)21.317.215.519.114.711.49.0
    \(\sum x = 12 \cdot 905 , \sum x ^ { 2 } = 26 \cdot 8951 , \sum y = 108 \cdot 2 , \sum y ^ { 2 } = 1781 \cdot 64 , \sum x y = 183 \cdot 176\).
  2. Calculate the equation of the regression line of \(x\) on \(y\), expressing your answer in the form \(x = a y + b\).
  3. Calculate the product moment correlation coefficient between \(y\) and \(x\) and give a brief interpretation of its value.
  4. Use the equation of the regression line to estimate the value of \(x\) when \(y = 12\). State, with a reason, how accurate you would expect this estimate to be.
  5. Comment on the use of the line to find values of \(x\) as \(y\) gets very small.
Edexcel S1 Q1
7 marks Moderate -0.3
  1. \(70 \%\) of the households in a town have a CD player and \(45 \%\) have both a CD player and a personal computer (PC). 18\% have neither a CD player nor a PC.
    1. Illustrate this information using a Venn diagram.
    2. Find the percentage of the households that do not have a PC.
    3. Find the probability that a household chosen at random has a CD player or a PC but not both.
    4. The random variable \(X\) has the normal distribution \(\mathrm { N } \left( 2,1 \cdot 7 ^ { 2 } \right)\).
    5. State the standard deviation of \(X\).
    6. Find \(\mathrm { P } ( X < 0 )\).
    7. Find \(\mathrm { P } ( 0 \cdot 6 < X < 3 \cdot 4 )\).
    8. The discrete random variable \(X\) has probability function
    $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } c x ^ { 2 } & x = - 3 , - 2 , - 1,1,2,3 \\ 0 & \text { otherwise. } \end{array} \right.$$
  2. Show that \(c = \frac { 1 } { 28 }\).
  3. Calculate
    1. \(\mathrm { E } ( X )\),
    2. \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Calculate
    1. \(\operatorname { Var } ( X )\),
    2. \(\operatorname { Var } ( 10 - 2 X )\).
      (3 marks)