Questions S1 (2020 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel S1 2016 June Q6
9 marks Moderate -0.3
6. The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes.
  1. Find the proportion of men that take longer than 300 minutes to run a marathon.
    (3) Nathaniel is preparing to run a marathon. He aims to finish in the first 20\% of male runners.
  2. Using the above model estimate the longest time that Nathaniel can take to run the marathon and achieve his aim.
    (3) The time, \(W\) minutes, taken by women to run a marathon is modelled by a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( W < \mu + 30 ) = 0.82\)
  3. find \(\mathrm { P } ( W < \mu - 30 \mid W < \mu )\)
Edexcel S1 2017 June Q1
14 marks Moderate -0.5
  1. A clothes shop manager records the weekly sales figures, \(\pounds s\), and the average weekly temperature, \(t ^ { \circ } \mathrm { C }\), for 6 weeks during the summer. The sales figures were coded so that \(w = \frac { s } { 1000 }\)
The data are summarised as follows $$\mathrm { S } _ { w w } = 50 \quad \sum w t = 784 \quad \sum t ^ { 2 } = 2435 \quad \sum t = 119 \quad \sum w = 42$$
  1. Find \(\mathrm { S } _ { w t }\) and \(\mathrm { S } _ { t t }\)
  2. Write down the value of \(\mathrm { S } _ { s s }\) and the value of \(\mathrm { S } _ { s t }\)
  3. Find the product moment correlation coefficient between \(s\) and \(t\). The manager of the clothes shop believes that a linear regression model may be appropriate to describe these data.
  4. State, giving a reason, whether or not your value of the correlation coefficient supports the manager's belief.
  5. Find the equation of the regression line of \(w\) on \(t\), giving your answer in the form \(w = a + b t\)
  6. Hence find the equation of the regression line of \(s\) on \(t\), giving your answer in the form \(s = c + d t\), where \(c\) and \(d\) are correct to 3 significant figures.
  7. Using your equation in part (f), interpret the effect of a \(1 ^ { \circ } \mathrm { C }\) increase in average weekly temperature on weekly sales during the summer.
Edexcel S1 2017 June Q2
14 marks Moderate -0.8
2. An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, \(\pounds x\) per square foot. His results are given in the table below.
Cost (£ \(\boldsymbol { x }\) )Frequency (f)Midpoint (£y)
\(20 \leqslant x < 40\)1230
\(40 \leqslant x < 45\)1342.5
\(45 \leqslant x < 50\)2547.5
\(50 \leqslant x < 60\)3255
\(60 \leqslant x < 80\)870
A histogram is drawn for these data and the bar representing \(50 \leqslant x < 60\) is 2 cm wide and 8 cm high.
  1. Calculate the width and height of the bar representing \(20 \leqslant x < 40\)
  2. Use linear interpolation to estimate the median cost.
  3. Estimate the mean cost of office space for these data.
  4. Estimate the standard deviation for these data.
  5. Describe, giving a reason, the skewness. Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean \(\pounds 50\) and standard deviation \(\pounds 10\)
  6. With reference to your answer to part (e), comment on Rika's suggestion.
  7. Use Rika's model to estimate the 80th percentile of the cost of office space in London.
Edexcel S1 2017 June Q3
11 marks Standard +0.3
  1. The Venn diagram shows three events \(A , B\) and \(C\), where \(p , q , r , s\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{319667e7-3f8b-4a33-8fc5-ef72154d1421-10_647_972_306_488}
    (b) Find the value of \(r\).
    (c) Hence write down the value of \(s\) and the value of \(t\).
    (d) State, giving a reason, whether or not the events \(A\) and \(B\) are independent.
    (e) Find \(\mathrm { P } ( B \mid A \cup C )\). \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } ( C ) = 0.25\) and the events \(B\) and \(C\) are independent.
    (a) Find the value of \(p\) and the value of \(q\).
Edexcel S1 2017 June Q4
6 marks Easy -1.2
4. The discrete random variable \(X\) has probability distribution
\(x\)- 1012
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(b\)\(c\)
The cumulative distribution function of \(X\) is given by
\(x\)- 1012
\(\mathrm {~F} ( x )\)\(\frac { 1 } { 3 }\)\(d\)\(\frac { 5 } { 6 }\)\(e\)
  1. Find the values of \(a , b , c , d\) and \(e\).
  2. Write down the value of \(\mathrm { P } \left( X ^ { 2 } = 1 \right)\).
    \section*{} \section*{
    \includegraphics[max width=\textwidth, alt={}]{image-not-found}
    } \(T\)
Edexcel S1 2017 June Q5
12 marks Standard +0.8
5. Yuto works in the quality control department of a large company. The time, \(T\) minutes, it takes Yuto to analyse a sample is normally distributed with mean 18 minutes and standard deviation 5 minutes.
  1. Find the probability that Yuto takes longer than 20 minutes to analyse the next sample. (3) The company has a large store of samples analysed by Yuto with the time taken for each analysis recorded. Serena is investigating the samples that took Yuto longer than 15 minutes to analyse. She selects, at random, one of the samples that took Yuto longer than 15 minutes to analyse.
  2. Find the probability that this sample took Yuto more than 20 minutes to analyse. Serena can identify, in advance, the samples that Yuto can analyse in under 15 minutes and in future she will assign these to someone else.
  3. Estimate the median time taken by Yuto to analyse samples in future.
Edexcel S1 2017 June Q6
18 marks Moderate -0.8
6. The score, \(X\), for a biased spinner is given by the probability distribution
\(x\)036
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 12 }\)\(\frac { 2 } { 3 }\)\(\frac { 1 } { 4 }\)
Find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\) A biased coin has one face labelled 2 and the other face labelled 5 The score, \(Y\), when the coin is spun has $$\mathrm { P } ( Y = 5 ) = p \quad \text { and } \quad \mathrm { E } ( Y ) = 3$$
  3. Form a linear equation in \(p\) and show that \(p = \frac { 1 } { 3 }\)
  4. Write down the probability distribution of \(Y\). Sam plays a game with the spinner and the coin.
    Each is spun once and Sam calculates his score, \(S\), as follows $$\begin{aligned} & \text { if } X = 0 \text { then } S = Y ^ { 2 } \\ & \text { if } X \neq 0 \text { then } S = X Y \end{aligned}$$
  5. Show that \(\mathrm { P } ( S = 30 ) = \frac { 1 } { 12 }\)
  6. Find the probability distribution of \(S\).
  7. Find \(\mathrm { E } ( S )\). Charlotte also plays the game with the spinner and the coin.
    Each is spun once and Charlotte ignores the score on the coin and just uses \(X ^ { 2 }\) as her score. Sam and Charlotte each play the game a large number of times.
  8. State, giving a reason, which of Sam and Charlotte should achieve the higher total score.
    END
Edexcel S1 2018 June Q1
4 marks Moderate -0.8
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)24710
\(\mathrm { P } ( X = x )\)\(a\)\(b\)0.1\(c\)
where \(a , b\) and \(c\) are probabilities.
The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\) and \(\mathrm { F } ( 3 ) = 0.2\) and \(\mathrm { F } ( 6 ) = 0.8\)
  1. Find the value of \(a\), the value of \(b\) and the value of \(c\).
  2. Write down the value of \(\mathrm { F } ( 7 )\).
Edexcel S1 2018 June Q2
12 marks Moderate -0.8
2. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway one Monday.
Delay (minutes)Number of motorists (f)Delay midpoint (x)
3-6384.5
7-8257.5
9-10189.5
11-151213
16-20718
(You may use \(\sum \mathrm { f } x ^ { 2 } = 8096.25\) ) A histogram has been drawn to represent these data. The bar representing a delay of (3-6) minutes has a width of 2 cm and a height of 9.5 cm .
  1. Calculate the width and the height of the bar representing a delay of (11-15) minutes.
  2. Use linear interpolation to estimate the median delay.
  3. Calculate an estimate of the mean delay.
  4. Calculate an estimate of the standard deviation of the delays. One coefficient of skewness is given by \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\)
  5. Evaluate this coefficient for the above data, giving your answer to 2 significant figures. On the following Friday, the coefficient of skewness for the delays on this stretch of motorway was - 0.22
  6. State, giving a reason, how the delays on this stretch of motorway on Friday are different from the delays on Monday.
Edexcel S1 2018 June Q3
5 marks Moderate -0.8
3. The random variable \(Y\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) The \(\mathrm { P } ( Y > 17 ) = 0.4\) Find
  1. \(\mathrm { P } ( \mu < Y < 17 )\)
  2. \(\mathrm { P } ( \mu - \sigma < Y < 17 )\)
Edexcel S1 2018 June Q4
13 marks Standard +0.3
4.A bag contains 64 coloured beads.There are \(r\) red beads,\(y\) yellow beads and 1 green bead and \(r + y + 1 = 64\) Two beads are selected at random,one at a time without replacement.
  1. Find the probability that the green bead is one of the beads selected. The probability that both of the beads are red is \(\frac { 5 } { 84 }\)
  2. Show that \(r\) satisfies the equation \(r ^ { 2 } - r - 240 = 0\)
  3. Hence show that the only possible value of \(r\) is 16
  4. Given that at least one of the beads is red,find the probability that they are both red.
Edexcel S1 2018 June Q5
14 marks Standard +0.3
5. The score when a spinner is spun is given by the discrete random variable \(X\) with the following probability distribution, where \(a\) and \(b\) are probabilities.
\(x\)- 10245
\(\mathrm { P } ( X = x )\)\(b\)\(a\)\(a\)\(a\)\(b\)
  1. Explain why \(\mathrm { E } ( X ) = 2\)
  2. Find a linear equation in \(a\) and \(b\). Given that \(\operatorname { Var } ( X ) = 7.1\)
  3. find a second equation in \(a\) and \(b\) and simplify your answer.
  4. Solve your two equations to find the value of \(a\) and the value of \(b\). The discrete random variable \(Y = 10 - 3 X\)
  5. Find
    1. \(\mathrm { E } ( Y )\)
    2. \(\operatorname { Var } ( Y )\) The spinner is spun once.
  6. Find \(\mathrm { P } ( Y > X )\).
Edexcel S1 2018 June Q6
14 marks Moderate -0.8
6. A group of climbers collected information about the height above sea level, \(h\) metres, and the air temperature, \(t ^ { \circ } \mathrm { C }\), at the same time at 8 different points on the same mountain. The data are summarised by $$\sum h = 6370 \quad \sum t = 61 \quad \sum t h = 31070 \quad \sum t ^ { 2 } = 693$$
  1. Show that \(\mathrm { S } _ { \text {th } } = - 17501.25\) and \(\mathrm { S } _ { \text {tt } } = 227.875\) The product moment correlation coefficient for these data is - 0.985
  2. State, giving a reason, whether or not this value supports the use of a regression equation to predict the air temperature at different heights on this mountain.
  3. Find the equation of the regression line of \(t\) on \(h\), giving your answer in the form \(t = a + b h\). Give the value of your coefficients to 3 significant figures.
  4. Give an interpretation of your value of \(a\). One of the climbers has just stopped for a short break before climbing the next 150 metres.
  5. Estimate the drop in temperature over this 150 metre climb.
Edexcel S1 2018 June Q7
13 marks Standard +0.3
7. Farmer Adam grows potatoes. The weights of potatoes, in grams, grown by Adam are normally distributed with a mean of 140 g and a standard deviation of 40 g . Adam cannot sell potatoes with a weight of less than 92 g .
  1. Find the percentage of potatoes that Adam grows but cannot sell. The upper quartile of the weight of potatoes sold by Adam is \(q _ { 3 }\)
  2. Find the probability that the weight of a randomly selected potato grown by Adam is more than \(q _ { 3 }\)
  3. Find the lower quartile, \(q _ { 1 }\), of the weight of potatoes sold by Adam. Betty selects a random sample of 3 potatoes sold by Adam.
  4. Find the probability that one weighs less than \(q _ { 1 }\), one weighs more than \(q _ { 3 }\) and one has a weight between \(q _ { 1 }\) and \(q _ { 3 }\)
    END
Edexcel S1 Q1
8 marks Challenging +1.2
  1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.
Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)
Edexcel S1 Q2
11 marks Easy -1.3
2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.
Number of daisies\(1 \mid 1\) means 11
11223444(7)
15567899(7)
200133334(8)
25567999(7)
3001244(6)
366788(5)
413(2)
  1. Write down the modal value of these data.
  2. Find the median and the quartiles of these data.
  3. On graph paper and showing your scale clearly, draw a box plot to represent these data.
  4. Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
  5. Comment on how the student might summarise both sets of raw data before drawing box plots.
    (1 mark)
Edexcel S1 Q3
11 marks Standard +0.3
3. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a group frequency table. The mid-point of each group in the table was represented by \(x\) and the corresponding frequency for that group by \(f\). The data were then coded using \(y = \frac { ( x - 755.0 ) } { 2.5 }\) and summarised as follows: $$\Sigma f y = - 467 , \Sigma f y ^ { 2 } = 9179 .$$
  1. Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
    (9 marks)
    An estimate of the interquartile range for these data was 27.7 hours.
  2. Explain, giving a reason, whether you would recommend the manufacturer to use the interquartile range or the standard deviation to represent the spread of lifetimes of the bulbs from this production line.
    (2 marks)
Edexcel S1 Q4
14 marks Standard +0.3
4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .
  1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
    (2 marks)
    The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
  2. Find the probability distribution of \(A\).
  3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
  4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
  5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).
Edexcel S1 Q5
15 marks Moderate -0.8
5. A keep-fit enthusiast swims, runs or cycles each day with probabilities \(0.2,0.3\) and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.
  1. Represent this information on a tree diagram.
  2. Find the probability that on any particular day he uses the sauna.
  3. Given that he uses the sauna one day, find the probability that he had been swimming.
  4. Given that he did not use the sauna one day, find the probability that he had been swimming.
Edexcel S1 Q6
16 marks Moderate -0.8
6. To test the heating of tyre material, tyres are run on a test rig at chosen speeds under given conditions of load, pressure and surrounding temperature. The following table gives values of \(x\), the test rig speed in miles per hour (mph), and the temperature, \(y ^ { \circ } \mathrm { C }\), generated in the shoulder of the tyre for a particular tyre material.
\(x ( \mathrm { mph } )\)1520253035404550
\(y \left( { } ^ { \circ } \mathrm { C } \right)\)53556365788391101
  1. Draw a scatter diagram to represent these data.
  2. Give a reason to support the fitting of a regression line of the form \(y = a + b x\) through these points.
  3. Find the values of \(a\) and \(b\).
    (You may use \(\Sigma x ^ { 2 } = 9500 , \Sigma y ^ { 2 } = 45483 , \Sigma x y = 20615\) )
  4. Give an interpretation for each of \(a\) and \(b\).
  5. Use your line to estimate the temperature at 50 mph and explain why this estimate differs from the value given in the table. A tyre specialist wants to estimate the temperature of this tyre material at 12 mph and 85 mph .
  6. Explain briefly whether or not you would recommend the specialist to use this regression equation to obtain these estimates.
Edexcel S1 2003 November Q1
16 marks Moderate -0.8
  1. A company wants to pay its employees according to their performance at work. The performance score \(x\) and the annual salary, \(y\) in \(\pounds 100\) s, for a random sample of 10 of its employees for last year were recorded. The results are shown in the table below.
\(x\)15402739271520301924
\(y\)216384234399226132175316187196
$$\text { [You may assume } \left. \Sigma x y = 69798 , \Sigma x ^ { 2 } = 7266 \right]$$
  1. Draw a scatter diagram to represent these data.
  2. Calculate exact values of \(S _ { x y }\) and \(S _ { x x }\).
    1. Calculate the equation of the regression line of \(y\) on \(x\), in the form \(y = a + b x\). Give the values of \(a\) and \(b\) to 3 significant figures.
    2. Draw this line on your scatter diagram.
  4. Interpret the gradient of the regression line. The company decides to use this regression model to determine future salaries.
  5. Find the proposed annual salary for an employee who has a performance score of 35 .
Edexcel S1 2003 November Q2
18 marks Standard +0.3
2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.
  1. Find the probability that Linda scores 30 points in a round. The random variable \(X\) is the number of points Linda scores in a round.
  2. Find the probability distribution of \(X\).
  3. Find the mean and the standard deviation of \(X\). A game consists of 2 rounds.
  4. Find the probability that Linda scores more points in round 2 than in round 1.
Edexcel S1 2003 November Q3
9 marks Moderate -0.8
3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .
    1. Find the probability of a jar containing less than the stated weight.
    2. In a box of 30 jars, find the expected number of jars containing less than the stated weight. The mean weight of sauce is changed so that \(1 \%\) of the jars contain less than the stated weight. The standard deviation stays the same.
  2. Find the new mean weight of sauce.
Edexcel S1 2003 November Q4
7 marks Easy -1.2
4. Explain what you understand by
  1. a sample space,
  2. an event. Two events \(A\) and \(B\) are independent, such that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
    Find
  3. \(\mathrm { P } ( A \cap B )\),
  4. \(\mathrm { P } ( A B )\),
  5. \(\mathrm { P } ( A \cup B )\).
Edexcel S1 2003 November Q5
9 marks Moderate -0.8
5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),
  1. show that \(n = 9\). Find
  2. \(\mathrm { P } ( X < 7 )\),
  3. \(\operatorname { Var } ( X )\).