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OCR Further Additional Pure AS 2024 June Q5
14 marks Challenging +1.2
5 The set \(S\) consists of all \(2 \times 2\) matrices having determinant 1 or - 1 . For instance, the matrices \(\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\) are elements of \(S\). It is given that \(\times _ { \mathbf { M } }\) is the operation of matrix multiplication.
  1. State the identity element of \(S\) under \(\times _ { \mathbf { M } }\). The group \(G\) is generated by \(\mathbf { P }\), under \(\times _ { \mathbf { M } }\).
  2. Determine the order of \(G\). The group \(H\) is generated by \(\mathbf { Q }\) and \(\mathbf { R }\), also under \(\times _ { \mathbf { M } }\).
    1. By finding each element of \(H\), determine the order of \(H\).
    2. List all the proper subgroups of \(H\).
  4. State whether each of the following statements is true or false. Give a reason for each of your answers.
OCR Further Additional Pure AS 2024 June Q6
9 marks Challenging +1.8
6 For positive integers \(n\), let \(f ( n ) = 1 + 2 ^ { n } + 4 ^ { n }\).
    1. Given that \(n\) is a multiple of 3 , but not of 9 , use the division algorithm to write down the two possible forms that \(n\) can take.
    2. Show that when \(n\) is a multiple of 3 , but not of 9 , \(f ( n )\) is a multiple of 73 .
  2. Determine the value of \(\mathrm { f } ( n )\), modulo 73 , in the case when \(n\) is a multiple of 9 .
OCR Further Additional Pure AS 2024 June Q7
12 marks Standard +0.3
7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.
  1. Let \(n\) be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
  2. Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
  3. Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
  4. In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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OCR Further Additional Pure AS 2021 November Q1
5 marks Moderate -0.3
1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3 \\ 0 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0 \\ 4 \\ 0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) respectively, relative to the origin \(O\).
    1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
    2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
  2. Use a vector product method to calculate the area of triangle \(A B C\).
OCR Further Additional Pure AS 2021 November Q2
4 marks Moderate -0.8
2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).
    1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
    2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
  2. Write down the equation of any one contour of \(S\) which does not include the origin.
OCR Further Additional Pure AS 2021 November Q3
6 marks Standard +0.8
3 For positive integers \(n\), the sequence of Fibonacci numbers, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), starts with the terms \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , \ldots\) and is given by the recurrence relation \(\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )\).
  1. Show that \(\mathrm { F } _ { 3 \mathrm { k } + 3 } = 2 \mathrm {~F} _ { 3 \mathrm { k } + 1 } + \mathrm { F } _ { 3 \mathrm { k } }\), where \(k\) is a positive integer.
  2. Prove by induction that \(\mathrm { F } _ { 3 n }\) is even for all positive integers \(n\).
OCR Further Additional Pure AS 2021 November Q4
6 marks Standard +0.3
4
  1. Let \(a = 1071\) and \(b = 67\).
    1. Find the unique integers \(q\) and \(r\) such that \(\mathrm { a } = \mathrm { bq } + \mathrm { r }\), where \(q > 0\) and \(0 \leqslant r < b\).
    2. Hence express the answer to (a)(i) in the form of a linear congruence modulo \(b\).
  2. Use the fact that \(358 \times 715 - 239 \times 1071 = 1\) to prove that 715 and 1071 are co-prime.
OCR Further Additional Pure AS 2021 November Q5
11 marks Challenging +1.8
5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).
  1. In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
  2. For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)
OCR Further Additional Pure AS 2021 November Q6
11 marks Challenging +1.8
6 The set \(S\) consists of the following four complex numbers. \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\) For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).
    1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
    2. Verify that ( \(S , \bigcirc\) ) is a group.
    3. State the order of each element of \(( S , \bigcirc )\).
  2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
    1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
    2. List all possible generators of \(( S , \bigcirc )\).
OCR Further Additional Pure AS 2021 November Q7
10 marks Challenging +1.2
7
  1. Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
  2. Use the standard test for divisibility by 11 to prove the following statements.
    1. \(10 ^ { 33 } + 1\) is divisible by 11
    2. \(10 ^ { 33 } + 1\) is divisible by 121
OCR Further Additional Pure AS 2021 November Q8
7 marks Challenging +1.8
8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}
OCR MEI Further Statistics Major Specimen Q1
7 marks Standard +0.3
1 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.
  1. Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs. Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable \(X\) represents the number of packs which Sam opens.
  2. Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of \(X\).
    \(r\)3456
    \(\mathrm { P } ( X = r )\)\(\frac { 2 } { 9 }\)\(\frac { 14 } { 81 }\)
    \section*{(iii) In this question you must show detailed reasoning.} Find
OCR MEI Further Statistics Major Specimen Q2
12 marks Standard +0.3
2 The continuous random variable \(X\) takes values in the interval \(- 1 \leq x \leq 1\) and has probability density function $$f ( x ) = \left\{ \begin{array} { l r } a & - 1 \leq x < 0 \\ a + x ^ { 2 } & 0 \leq x \leq 1 \end{array} \right.$$ where \(a\) is a constant.
  1. (A) Sketch the probability density function.
    (B) Show that \(a = \frac { 1 } { 3 }\).
  2. Find
    (A) \(\mathrm { P } \left( X < \frac { 1 } { 2 } \right)\),
    (B) the mean of \(X\).
  3. Show that the median of \(X\) satisfies the equation \(2 m ^ { 3 } + 2 m - 1 = 0\).
OCR MEI Further Statistics Major Specimen Q3
11 marks Standard +0.3
3 A researcher is investigating factors that might affect how many hours per day different species of mammals spend asleep. First she investigates human beings. She collects data on body mass index, \(x\), and hours of sleep, \(y\), for a random sample of people. A scatter diagram of the data is shown in Fig. 3.1 together with the regression line of \(y\) on \(x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6ee3a4a-3e76-4422-9a78-17b64b458f83-04_885_1584_598_274} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Calculate the residual for the data point which has the residual with the greatest magnitude.
  2. Use the equation of the regression line to estimate the mean number of hours spent asleep by a person with body mass index
    (A) 26,
    (B) 16,
    commenting briefly on each of your predictions. The researcher then collects additional data for a large number of species of mammals and analyses different factors for effect size. Definitions of the variables measured for a typical animal of the species, the correlations between these variables, and guidelines often used when considering effect size are given in Fig. 3.2.
    VariableDefinition
    Body massMass of animal in kg
    Brain massMass of brain in g
    Hours of sleep/dayNumber of hours per day spent asleep
    Life spanHow many years the animal lives
    DangerA measure of how dangerous the animal's situation is when asleep, taking into account predators and how protected the animal's den is: higher value indicates greater danger.
    Correlations (pmcc)Body MassBrain MassHours of sleep/dayLife spanDanger
    Body Mass1.00
    Brain Mass0.931.00
    Hours of sleep/day-0.31-0.361.00
    Life span0.300.51-0.411.00
    Danger0.130.15-0.590.061.00
    \begin{table}[h]
    Product moment
    correlation coefficient
    		Effect size
    0.1Small
    0.3Medium
    0.5Large
    \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{table}
  3. State two conclusions the researcher might draw from these tables, relevant to her investigation into how many hours mammals spend asleep. One of the researcher's students notices the high correlation between body mass and brain mass and produces a scatter diagram for these two variables, shown in Fig. 3.3 below. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e6ee3a4a-3e76-4422-9a78-17b64b458f83-05_675_698_1802_735} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{figure}
  4. Comment on the suitability of a linear model for these two variables.
OCR MEI Further Statistics Major Specimen Q4
10 marks Moderate -0.3
4 A fair six-sided dice is rolled repeatedly. Find the probability of the following events.
  1. A five occurs for the first time on the fourth roll.
  2. A five occurs at least once in the first four rolls.
  3. A five occurs for the second time on the third roll.
  4. At least two fives occur in the first three rolls. The dice is rolled repeatedly until a five occurs for the second time.
  5. Find the expected number of rolls required for two fives to occur. Justify your answer.
OCR MEI Further Statistics Major Specimen Q5
7 marks Standard +0.3
5 A particular brand of pasta is sold in bags of two different sizes. The mass of pasta in the large bags is advertised as being 1500 g ; in fact it is Normally distributed with mean 1515 g and standard deviation 4.7 g . The mass of pasta in the small bags is advertised as being 500 g ; in fact it is Normally distributed with mean 508 g and standard deviation 3.3 g .
  1. Find the probability that the total mass of pasta in 5 randomly selected small bags is less than 2550 g .
  2. Find the probability that the mass of pasta in a randomly selected large bag is greater than three times the mass of pasta in a randomly selected small bag.
OCR MEI Further Statistics Major Specimen Q6
3 marks Standard +0.3
6 Fig. 6 shows the wages earned in the last 12 months by each of a random sample of American males aged between 16 and 65 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6ee3a4a-3e76-4422-9a78-17b64b458f83-07_771_1278_340_392} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} A researcher wishes to test whether the sample provides evidence of a tendency for higher wages to be earned by older men in the age range 16 to 65 in America.
  1. The researcher needs to decide whether to use a test based on Pearson's product moment correlation coefficient or Spearman's rank correlation coefficient. Use the information in Fig. 6 to decide which test is more appropriate.
  2. Should it be a one-tail or a two-tail test? Justify your answer.
OCR MEI Further Statistics Major Specimen Q7
11 marks Standard +0.3
7 A newspaper reports that the average price of unleaded petrol in the UK is 110.2 p per litre. The price, in pence, of a litre of unleaded petrol at a random sample of 15 petrol stations in Yorkshire is shown below together with some output from software used to analyse the data.
116.9114.9110.9113.9114.9
117.9112.999.9114.9103.9
123.9105.7108.9102.9112.7
\begin{table}[h]
\(| l |\)Statistics
n15
Mean111.6733
\(\sigma\)6.1877
s6.4048
\(\Sigma \mathrm { x }\)1675.1
\(\Sigma \mathrm { x } ^ { 2 }\)187638.31
Min99.9
Q 1105.7
Median112.9
Q 3114.9
Max123.9
\captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{table}
\(n\)15
Kolmogorov-Smirnov
test
\(p > 0.15\)
Null hypothesis
The data can be modelled
by a Normal distribution
Alternative hypothesis
The data cannot be
modelled by a Normal
distribution
  1. Select a suitable hypothesis test to investigate whether there is any evidence that the average price of unleaded petrol in Yorkshire is different from 110.2 p. Justify your choice of test.
  2. Conduct the hypothesis test at the \(5 \%\) level of significance.
OCR MEI Further Statistics Major Specimen Q8
12 marks Standard +0.3
8 Natural background radiation consists of various particles, including neutrons. A detector is used to count the number of neutrons per second at a particular location.
  1. State the conditions required for a Poisson distribution to be a suitable model for the number of neutrons detected per second. The number of neutrons detected per second due to background radiation only is modelled by a Poisson distribution with mean 1.1.
  2. Find the probability that the detector detects
    (A) no neutrons in a randomly chosen second,
    (B) at least 60 neutrons in a randomly chosen period of 1 minute. A neutron source is switched on. It emits neutrons which should all be contained in a protective casing. The detector is used to check whether any neutrons have not been contained; these are known as stray neutrons. If the detector detects more than 8 neutrons in a period of 1 second, an alarm will be triggered in case this high reading is due to stray neutrons.
  3. Suppose that there are no stray neutrons and so the neutrons detected are all due to the background radiation. Find the expected number of times the alarm is triggered in 1000 randomly chosen periods of 1 second.
  4. Suppose instead that stray neutrons are being produced at a rate of 3.4 per second in addition to the natural background radiation. Find the probability that at least one alarm will be triggered in 10 randomly chosen periods of 1 second. You should assume that all stray neutrons produced are detected.
OCR MEI Further Statistics Major Specimen Q9
13 marks Standard +0.3
9 A random sample of adults in the UK were asked to state their primary source of news: television (T), internet (I), newspapers (N) or radio (R). The responses were classified by age group, and an analysis was carried out to see if there is any association between age group and primary source of news. Fig. 9 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]
ABCDEF
1SourceAge group
2of news18-3233-4748-6465+
3T63617180275
4I33332212100
5N98112048
6R499527
7109111113117450
8
9Expected frequencies
1066.6167.8369.0671.50
1124.2224.6726.00
1211.6311.8412.0512.48
136.546.666.787.02
14
15Contributions to the test statistic
160.200.690.051.01
173.182.827.54
180.590.094.53
190.990.820.730.58
20test statistic25.45
\captionsetup{labelformat=empty} \caption{Fig. 9}
\end{table}
  1. (A) State the sample size.
    (B) Give the name of the appropriate hypothesis test.
    (C) State the null and alternative hypotheses.
  2. Showing your calculations, find the missing values in cells
OCR MEI Further Statistics Major Specimen Q10
10 marks Standard +0.3
10 The label on a particular size of milk carton states that it contains 1.5 litres of milk. In an investigation at the packaging plant the contents, \(x\) litres, of each of 60 randomly selected cartons are measured. The data are summarised as follows. $$\Sigma x = 89.758 \quad \Sigma x ^ { 2 } = 134.280$$
  1. Estimate the variance of the underlying population.
  2. Find a 95\% confidence interval for the mean of the underlying population.
  3. What does the confidence interval which you have calculated suggest about the statement on the carton? Each day for 300 days a random sample of 60 cartons is selected and for each sample a \(95 \%\) confidence interval is constructed.
  4. Explain why the confidence intervals will not be identical.
  5. What is the expected number of confidence intervals to contain the population mean?
OCR MEI Further Statistics Major Specimen Q11
24 marks Standard +0.3
11 Two girls, Lili and Hui, play a game with a fair six-sided dice. The dice is thrown 10 times. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) represent the scores on the \(1 ^ { \text {st } } , 2 ^ { \text {nd } } , \ldots , 10 ^ { \text {th } }\) throws of the dice. \(L\) denotes Lili's score and \(L = 10 X _ { 1 }\). \(H\) denotes Hui's score and \(H = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).
  1. Calculate
    The spreadsheet below shows a simulation of 25 plays of the game. The cell E3, highlighted, shows the score when the dice is thrown the fourth time in the first game. \begin{table}[h]
    ABCDEFGHIJKLMN
    1Throw of diceLili'sHui's
    212345678910scorescore
    3Game 135211311143022
    4Game 263244353356038
    5Game 364265215236036
    6Game 415166314621035
    7Game 544316441624035
    8Game 621512515232027
    9Game 711344563421033
    10Game 811363445231032
    11Game 922243215562032
    12Game 1035335343113031
    13Game 1153655421155037
    14Game 1264324133536034
    15Game 1323212222212019
    16Game 1441331266134030
    17Game 1551263463645040
    18Game 1636115313333029
    19Game 1752524522345034
    20Game 1836355231123031
    21Game 1966315634166041
    22Game 2026456524332040
    23Game 2153545336615041
    24Game 2263556356116041
    25Game 2354556421365041
    26Game 2435232432333030
    27Game 2552424522525033
    28
    29mean37.6033.68
    30sd17.395.77
    \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{table}
  2. Use the simulation to estimate \(\mathrm { P } ( L > 40 )\) and \(\mathrm { P } ( H > 40 )\).
  3. (A) Calculate the exact value of \(\mathrm { P } ( L > 40 )\).
    (B) Comment on how the exact value compares with your estimate of \(\mathrm { P } ( L > 40 )\) in part (v). Hui wonders whether it is appropriate to use the Central Limit Theorem to approximate the distribution of \(X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).
  4. (A) State what type of diagram Hui could draw, based on the output from the spreadsheet, to investigate this.
    (B) Explain how she should interpret the diagram.
  5. (A) Calculate an approximate value of \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 } > 40 \right)\) using the Central Limit Theorem.
    (B) Comment on how this value compares with your estimate of \(\mathrm { P } ( H > 40 )\) in part (v). \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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WJEC Unit 4 Specimen Q1
6 marks Moderate -0.3
It is known that \(4 \%\) of a population suffer from a certain disease. When a diagnostic test is applied to a person with the disease, it gives a positive response with probability 0.98 . When the test is applied to a person who does not have the disease, it gives a positive response with probability 0.01 .
  1. Using a tree diagram, or otherwise, show that the probability of a person who does not have the disease giving a negative response is 0.9504 . The test is applied to a randomly selected member of the population.
  2. Find the probability that a positive response is obtained.
  3. Given that a positive response is obtained, find the probability that the person has the disease.
WJEC Unit 4 Specimen Q2
9 marks Challenging +1.2
2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.
  1. Determine the probability that Jeff wins the game
    i) with his first shot,
    ii) with his second shot.
  2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
  3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.
WJEC Unit 4 Specimen Q3
7 marks Standard +0.8
3. A string of length 60 cm is cut a random point.
  1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
  2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).