Date | May 2019 | Marks available | 4 | Reference code | 19M.1.AHL.TZ1.H_10 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 1 |
Command term | Question number | H_10 | Adapted from | N/A |
Question
The function p(x) is defined by p(x)=x3−3x2+8x−24 where x∈R.
Find the remainder when p(x) is divided by (x−2).
Find the remainder when p(x) is divided by (x−3).
Prove that p(x) has only one real zero.
Write down the transformation that will transform the graph of y=p(x) onto the graph of y=8x3−12x2+16x−24.
The random variable X follows a Poisson distribution with a mean of λ and 6P(X=3)=3P(X=2)−2P(X=1)+3P(X=0).
Find the value of λ.
Markscheme
p(2)=8−12+16−24 (M1)
Note: Award M1 for a valid attempt at remainder theorem or polynomial division.
= −12 A1
remainder = −12
[2 marks]
p(3)=27−27+24−24 = 0 A1
remainder = 0
[1 mark]
x=3 (is a zero) A1
Note: Can be seen anywhere.
EITHER
factorise to get (x−3)(x2+8) (M1)A1
x2+8≠0 (for x∈R) (or equivalent statement) R1
Note: Award R1 if correct two complex roots are given.
OR
p′(x)=3x2−6x+8 A1
attempting to show p′(x)≠0 M1
eg discriminant = 36 – 96 < 0, completing the square
no turning points R1
THEN
only one real zero (as the curve is continuous) AG
[4 marks]
new graph is y=p(2x) (M1)
stretch parallel to the x-axis (with x=0 invariant), scale factor 0.5 A1
Note: Accept “horizontal” instead of “parallel to the x-axis”.
[2 marks]
6λ3e−λ6=3λ2e−λ2−2λe−λ+3e−λ M1A1
Note: Allow factorials in the denominator for A1.
2λ3−3λ2+4λ−6=0 A1
Note: Accept any correct cubic equation without factorials and e−λ.
EITHER
4(2λ3−3λ2+4λ−6)=8λ3−12λ2+16λ−24=0 (M1)
2λ=3 (A1)
OR
(2λ−3)(λ2+2)=0 (M1)(A1)
THEN
λ = 1.5 A1
[6 marks]
Examiners report
Syllabus sections
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22M.1.SL.TZ2.5a:
Calculate the expected number of people who will pass this polygraph test.
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22M.1.SL.TZ2.5c:
Determine the probability that fewer than 7 people will pass this polygraph test.
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19M.2.AHL.TZ1.H_9a.ii:
only a sandwich.
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19M.2.AHL.TZ2.H_10b:
Copy and complete the probability distribution table for Y.
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18N.2.AHL.TZ0.H_10b.ii:
an estimate for the standard deviation of the number of emails received per working day.
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18M.2.SL.TZ2.S_10d.i:
Find the expected number of bags in this crate that contain at most one small apple.
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18M.2.SL.TZ1.S_9a:
Find the probability that an orange weighs between 289 g and 310 g.
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18N.2.AHL.TZ0.H_10d:
Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.
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EXM.1.SL.TZ0.9:
Six coins are tossed simultaneously 320 times, with the following results.
At the 5% level of significance, test the hypothesis that all the coins are fair.
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EXN.3.AHL.TZ0.1c.ii:
State an assumption that is being made for X to be considered as following a binomial distribution.
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18M.2.AHL.TZ2.H_9a.iii:
Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.
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22M.2.SL.TZ1.5b.ii:
Find the maximum number of tickets that could be sold if the expected number of passengers who arrive to board the flight must be less than or equal to 72.
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22M.2.SL.TZ1.5b.i:
Write down the expected number of passengers who will arrive to board the flight if 72 tickets are sold.
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22M.1.SL.TZ2.5b:
Calculate the probability that exactly 4 people will fail this polygraph test.
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22M.2.SL.TZ2.1a.i:
median reaction time.
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22M.3.AHL.TZ2.2h:
Show that Jonas’s network satisfies the requirement of there being less than a 2% probability of the network failing after a power surge.
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21N.2.AHL.TZ0.3b:
Find the probability that Arianne throws two consecutive darts that land more than 15 cm from O.
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22M.2.SL.TZ1.5a:
The airline sells 74 tickets for this flight. Find the probability that more than 72 passengers arrive to board the flight.
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21N.2.SL.TZ0.5b:
Find the probability that Arianne throws two consecutive darts that land more than 15 cm from O.
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20N.2.SL.TZ0.T_6e:
Find the probability he plays between 13 minutes and 18 minutes in one game and more than 20 minutes in the other game.
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18N.2.AHL.TZ0.H_10a.ii:
Using this distribution model, find the standard deviation of X.
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18N.2.AHL.TZ0.H_10b.i:
an estimate for the mean number of emails received per working day.
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19M.2.AHL.TZ1.H_9c.i:
A customer is selected at random. Find the probability that the customer is male and buys a sandwich.
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19M.2.AHL.TZ2.H_2a:
Calculate the probability that, on a randomly selected day, Timmy makes a profit.
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17M.2.SL.TZ2.S_10b.i:
Write down the probability of drawing three blue marbles.
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18N.2.SL.TZ0.S_9c:
A randomly selected participant has a reaction time greater than 0.65 seconds. Find the probability that the participant is in Group X.
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18M.2.SL.TZ2.S_10d.ii:
Find the probability that at least 48 bags in this crate contain at most one small apple.
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18M.2.AHL.TZ2.H_8b:
It is further given that P(X ≤ 1) = 0.09478 correct to 4 significant figures.
Determine the value of n and the value of p.
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19M.2.AHL.TZ1.H_9a.i:
both a sandwich and a cake.
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19M.2.AHL.TZ1.H_9b.ii:
Find the probability that more than 100 cakes will be sold on a typical day.
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19M.1.AHL.TZ1.H_10a.i:
Find the remainder when p(x) is divided by (x−2).
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19M.2.SL.TZ1.S_10c.i:
Write down the value of x.
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19M.2.SL.TZ1.S_10c.ii:
Hence, find the value of y.
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17N.2.SL.TZ0.S_8c:
Write down the number of low production hives.
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18N.2.AHL.TZ0.H_10a.i:
Using this distribution model, find P(X<60).
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17M.2.AHL.TZ2.H_10c:
Find Var(X).
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17M.2.AHL.TZ2.H_10f:
Find P(Y⩾3).
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18M.2.SL.TZ1.S_9c:
To the nearest gram, find the minimum weight of an orange that the grocer will buy.
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19M.2.AHL.TZ1.H_9b.i:
Find the expected number of cakes sold on a typical day.
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17M.2.SL.TZ1.S_4b.ii:
Hence, find the probability that fewer than k students are left handed.
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EXM.2.AHL.TZ0.28a.iii:
Calculate an appropriate value of χ2 and state your conclusion, using a 1% significance level.
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EXM.2.AHL.TZ0.28a.i:
State suitable hypotheses for testing this belief.
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17M.2.AHL.TZ2.H_10e:
Find E(Y).
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18N.2.AHL.TZ0.H_10e:
Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.
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19M.2.SL.TZ1.S_10a.ii:
Find the probability of rolling two or more red faces.
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19M.2.SL.TZ1.S_10a.i:
Find the probability of rolling exactly one red face.
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18M.2.AHL.TZ2.H_9b:
During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.
Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.
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17N.2.SL.TZ0.S_8e:
Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.
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18N.2.AHL.TZ0.H_10c:
Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.
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16N.2.AHL.TZ0.H_3b:
Given that P(X=2)=0.241667 and P(X=3)=0.112777, use part (a) to find the value of μ.
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17N.2.SL.TZ0.S_8a:
Write down the value of a and of b.
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18M.2.SL.TZ1.S_9b.ii:
Hence, find the value of σ.
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19M.2.AHL.TZ2.H_2b:
The shop is open for 24 days every month.
Calculate the probability that, in a randomly selected month, Timmy makes a profit on between 5 and 10 days (inclusive).
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18M.2.SL.TZ1.S_9d:
Find the probability that the grocer buys more than half the oranges in a box selected at random.
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17M.2.SL.TZ1.S_4b.i:
Hence, find the probability that exactly k students are left handed;
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17N.2.SL.TZ0.S_8d.i:
Find the value of k;
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17M.2.AHL.TZ2.H_10b:
Find E(X).
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18M.2.AHL.TZ2.H_9a.ii:
Find the most likely number of taxis that would arrive during T.
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18M.2.SL.TZ1.S_9b.i:
Find the standardized value for 289 g.
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17N.2.SL.TZ0.S_8b:
Use this regression line to estimate the monthly honey production from a hive that has 270 bees.
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EXM.1.AHL.TZ0.55a.ii:
Hence estimate p, the probability that a randomly chosen egg is brown.
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18M.2.AHL.TZ2.H_9a.i:
Find the probability that exactly 4 taxis arrive during T.
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18N.2.SL.TZ0.S_9d:
Ten of the participants with reaction times greater than 0.65 are selected at random. Find the probability that at least two of them are in Group X.
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EXM.1.AHL.TZ0.55b:
By calculating an appropriate χ2 statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.
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19M.2.AHL.TZ2.H_10a:
Find the probability that on a randomly selected day, Steffi does not visit Will’s house.
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17N.2.SL.TZ0.S_8d.ii:
Find the number of hives that have a high production.
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18M.2.AHL.TZ2.H_8a:
Find the least possible value of n.
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19M.1.AHL.TZ1.H_10a.ii:
Find the remainder when p(x) is divided by (x−3).
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SPM.1.SL.TZ0.13a:
Find the probability that on any given day Mr Burke chooses a female student to answer a question.
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16N.2.SL.TZ0.S_7a:
Write down the probability that the first disc selected is red.
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EXN.3.AHL.TZ0.1c.i:
Write down the value of n and the value of p.
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18M.2.AHL.TZ1.H_6:
The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.
Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.
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17N.2.AHL.TZ0.H_6b:
Find the expected number of weeks in the year in which Lucca eats no bananas.
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17M.2.SL.TZ1.S_4a:
Find k.
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20N.2.SL.TZ0.T_6a:
Sketch a diagram to represent this information.
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20N.2.SL.TZ0.T_6b:
Show that m=15.7.
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20N.2.SL.TZ0.T_6c.i:
Find the probability that Emlyn plays between 13 minutes and 18 minutes in a game.
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20N.2.SL.TZ0.T_6c.ii:
Find the probability that Emlyn plays more than 20 minutes in a game.
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20N.2.SL.TZ0.T_6d:
Find the value of x.
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20N.2.SL.TZ0.T_6f:
Find the expected number of successful shots Emlyn will make on Monday, based on the results from Saturday and Sunday.
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20N.2.SL.TZ0.T_6g:
Emlyn claims the results from Saturday and Sunday show that his expected number of successful shots will be more than Johan’s.
Determine if Emlyn’s claim is correct. Justify your reasoning.
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17M.2.SL.TZ2.S_10d:
Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.
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EXM.2.AHL.TZ0.28b.i:
Find the significance level of this procedure.
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16N.2.AHL.TZ0.H_3a:
Show that P(X=x+1)=μx+1×P(X=x), x∈N.
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19M.1.AHL.TZ1.H_10c:
Write down the transformation that will transform the graph of y=p(x) onto the graph of y=8x3−12x2+16x−24.
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17N.1.AHL.TZ0.H_10b.i:
Determine the mean of X.
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18M.2.SL.TZ2.S_10a.ii:
Show that μ = 106.
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17N.1.AHL.TZ0.H_10a:
Show that the probability that Chloe wins the game is 38.
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19M.2.SL.TZ1.S_10b:
Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is 13.
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SPM.1.AHL.TZ0.17b:
The Head of Year, Mrs Smith, decides to select a student at random from the year group to read the notices in assembly. There are 80 students in total in the year group. Mrs Smith calculates the probability of picking a male student 8 times in the first 20 assemblies is 0.153357 correct to 6 decimal places.
Find the number of male students in the year group.
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EXM.2.AHL.TZ0.28b.ii:
Some time later, the actual value of μ is 503. Find the probability of a Type II error.
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SPM.1.SL.TZ0.13b:
Find the probability he will choose a female student 8 times.
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17M.2.SL.TZ2.S_10a.i:
Find q.
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SPM.1.SL.TZ0.13c:
Find the probability he will choose a male student at most 9 times.
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17N.1.AHL.TZ0.H_10b.ii:
Determine the variance of X.
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19M.2.AHL.TZ2.H_10d:
In any given year of 365 days, the probability that Steffi does not visit Will for at most n days in total is 0.5 (to one decimal place). Find the value of n.
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19M.2.AHL.TZ1.H_9c.ii:
A female customer is selected at random. Find the probability that she buys a sandwich.
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18M.2.SL.TZ2.S_10b:
Find P(M < 95) .
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17M.2.SL.TZ2.S_B10c:
Jill plays the game nine times. Find the probability that she wins exactly two prizes.
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17M.2.SL.TZ2.S_10b.ii:
Explain why the probability of drawing three white marbles is 16.
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19M.1.AHL.TZ1.H_10d:
The random variable X follows a Poisson distribution with a mean of λ and 6P(X=3)=3P(X=2)−2P(X=1)+3P(X=0).
Find the value of λ.
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18M.2.SL.TZ1.S_9e:
The grocer selects two boxes at random.
Find the probability that the grocer buys more than half the oranges in each box.
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18M.2.SL.TZ2.S_10c:
Find the probability that a bag of apples selected at random contains at most one small apple.
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EXM.2.AHL.TZ0.28a.ii:
Calculate the mean of these data and hence estimate the value of p.
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17M.2.SL.TZ2.S_10a.ii:
Find p.
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18M.2.SL.TZ2.S_10a.i:
Write down the value of k.
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17M.2.AHL.TZ2.H_10a:
Show that a=32 and b=112.
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21M.3.AHL.TZ1.2e.iii:
The firm obtains a significant result when comparing section 2 of the written assessment and attribute X. Interpret this result.
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19M.2.AHL.TZ2.H_10e:
Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.
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16N.2.SL.TZ0.S_7b:
Let X be the number of red discs selected. Find the smallest value of m for which Var(X )<0.6.
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17N.2.AHL.TZ0.H_6a:
Find the probability that Lucca eats at least one banana in a particular day.
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19M.2.AHL.TZ2.H_10c:
Hence find the expected number of times per day that Steffi is fed at Will’s house.
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17M.2.SL.TZ2.S_10b.iii:
The bag contains a total of ten marbles of which w are white. Find w.
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17M.2.AHL.TZ2.H_10d:
Find the median of X.
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21M.2.SL.TZ2.4e:
Ten of the cats are chosen at random. Find the probability that exactly one of them weighs over 6.25 kg.
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SPM.1.AHL.TZ0.17a:
Find the probability he will choose a female student 8 times.
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EXM.1.AHL.TZ0.55a.i:
Calculate the mean number of brown eggs in a box.
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21M.2.AHL.TZ2.2d.ii:
Let N be the number of cats weighing over 4.62 kg.
Find the variance of N.
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19M.2.SL.TZ1.S_10d:
Ted will always have another turn if he expects an increase to his winnings.
Find the least value of w for which Ted should end the game instead of having another turn.
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21M.3.AHL.TZ1.2e.ii:
The tests are performed at the 5% significance level.
Assuming that:- there is no correlation between the marks in any of the sections and scores in any of the attributes,
- the outcome of each hypothesis test is independent of the outcome of the other hypothesis tests,
find the probability that at least one of the tests will be significant.
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21M.2.AHL.TZ2.2d.i:
Find the probability that exactly one of them weighs over 4.62 kg.
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21N.2.SL.TZ0.5d.i:
Find the probability that Arianne scores at least 5 points in the competition.
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EXN.2.SL.TZ0.4b:
Find the probability of scoring more than 2 sixes when this die is rolled 5 times.
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21N.2.AHL.TZ0.3d.iv:
Given that Arianne scores at least 5 points, find the probability that Arianne scores less than 8 points.
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21N.2.SL.TZ0.5a.i:
a dart lands less than 13 cm from O.
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21N.2.SL.TZ0.5a.ii:
a dart lands more than 15 cm from O.
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21N.2.SL.TZ0.5c:
Find the probability that Arianne does not score a point on a turn of three darts.
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21N.2.SL.TZ0.5d.ii:
Find the probability that Arianne scores at least 5 points and less than 8 points.
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21N.2.SL.TZ0.5d.iii:
Given that Arianne scores at least 5 points, find the probability that Arianne scores less than 8 points.
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21N.2.AHL.TZ0.3a.i:
a dart lands less than 13 cm from O.
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21N.2.AHL.TZ0.3a.ii:
a dart lands more than 15 cm from O.
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21N.2.AHL.TZ0.3c:
Find the probability that Arianne does not score a point on a turn of three darts.
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21N.2.AHL.TZ0.3d.i:
Find Arianne’s expected score in the competition.
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21N.2.AHL.TZ0.3d.ii:
Find the probability that Arianne scores at least 5 points in the competition.
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21N.2.AHL.TZ0.3d.iii:
Find the probability that Arianne scores at least 5 points and less than 8 points.
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21N.2.AHL.TZ0.7a.i:
Write down null and alternative hypotheses for Loreto’s test.
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21N.2.AHL.TZ0.7a.ii:
Using the data from Loreto’s sample, perform the hypothesis test at a 5% significance level to determine if Loreto should employ extra staff.
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21N.2.AHL.TZ0.7b.i:
Write down null and alternative hypotheses for this test.
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21N.2.AHL.TZ0.7b.ii:
Perform the test, clearly stating the conclusion in context.