419
End Of Week Test Question 1.
Derek has put together a fantasy football squad and has
recorded the heights of all the players in the squad rounded to
the nearest centimetre. The data is illustrated in the following
box and whisker diagram.
(a) Write down the median height of the players.
(b) Write down the upper quartile height.
(c) Find the interquartile range of the heights.
The height of these players are normally distributed.
(d) Find the height of the tallest possible player that is not an
outlier. Give your answer to the nearest centimetre.
428
End Of Week Test Question 2.
A celebrity football match is planned to take place in a large
stadium.
The most expensive tickets are in the first row. The ticket price
for each row forms an arithmetic sequence. Prices for the first
four rows are shown in the following table.
Row number:
1
2
3
4
Price per seat £50 £48.50 £47 £45.50
(a) Write down the value of the common difference, d.
(b) Calculate the price of a ticket in the 19th row.
(c) Find the total cost of buying 5 tickets in each of the first 10
rows.
429
End Of Week Test Question 3.
The Student Council in a large school conducted a survey to find
out how many classes assigned homework for students to
complete.
The data is shown in the following table.
Number of classes with homework 0
Number of students
1
2
3
4 5
16 25 28 18 n 3
(a) State whether the data is discrete or continuous.
(b) Given that mean number of classes in which a student was
given homework is 1.9; Find the value of n.
The Student Council selected students at random from each Year
Group to take part in this survey. The number of students
selected from each Year Group was proportional to the number of
students in that Year Group.
(c) Identify the sampling technique used in the survey.
431
End Of Week Test Question 4.
The circumference of a given circle C can be represented by the
−−
−
function C(A) = 2√Aπ , A ≥ 0 where A is the area of the circle.
The graph of the function C is shown for 0 ≤ A ≤ 16.
(a) Use the graph to find the value of C(8) to the nearest whole
number.
(b) The range of C(A) is 0 ≤ C(A) ≤ n. Write down the value of
n.
(c) On the axes above, draw the graph of the inverse function,
C −1 .
(d) In the context of the question, explain the meaning of
C −1 (12).
434
End Of Week Test Question 5.
Doctor Octothorpe investigated the decreasing population of a
colony of ants in a remote province of China. His investigation
took place in 1958.
He found that during the summer season their population, P ,
could be modelled by the exponential equation
P = 560 + 9560(1.3)−t
where
t≥0
where t is the number of days into the season (t = 1 represents
the beginning of 1st June).
(a) Find the population of the ants at the beginning of 31st May
1958.
(b) Find the population of the ants at the beginning of 10th June.
(c) Calculate the date when the population first fell below 1000.
(d) According to this model, find the smallest possible population
of ants.
436
End Of Week Test Question 6.
Ucayali, a multinational technology company that focuses on
e-commerce, has a distribution centre in Camelot. As part of a
study into healthy lifestyles, the number of cups of coffee drunk
by 420 staff members each day were recorded. Results are
shown in the table.
Cups of coffee per day
none 1 or 2 3 or 4 5 or more
Office Staff
55
74
82
20
Warehouse Staff
21
33
49
21
Transport Staff
4
21
23
17
The company conducted a χ2 test for independence at a 5%
level of significance.
(a) State the null hypothesis.
(b) Calculate the p-value for this test.
(c) State, giving a reason, whether the null hypothesis should be
accepted.
438
End Of Week Test Question 7.
Points A (-1 , -1) , B (-3 , 1) , C (1 , -3) , D (3 , 1) and E (-3 , -3)
represent the locations of toilet buildings on a large caravan site.
These points are illustrated on the following coordinate axes.
Horizontal scale: 2 units represents 1 km.
Vertical scale: 2 units represents 1 km.
(a) Calculate the gradient of the line segment AD.
The Site Manager draws three straight lines to form an
incomplete Voronoi diagram.
(b) Find the equation of the line which would complete the
Voronoi cell containing site A. Give your answer in the form
ax + by + d = 0 where a, b, d ∈ Z.
(c) In the context of the question, explain the significance of the
Voronoi cell containing site A.
440
End Of Week Test Question 8.
In an old science fiction book the author described the intensity
of reverse polarity, P measured in treckons, is a function of the
nebula thrust, N measures in whovians. The intensity level is
given by the following formula.
P = 7 log10 (N × 108 ), N ≥ 0
(a) An space shuttle has a nebula thrust of 9.1 × 10−3 whovians.
Calculate the intensity level, P of the shuttle.
(b) A different space shuttle has an intensity level of 112 trekons.
Find its nebula thrust, N .
443
End Of Week Test Question 9.
The 2nd Rutherford American Scouts joined the 37th
Wolverhampton British Scouts for an International Camp. Skipper
Jones is interested to see if the mean height of American Scouts,
μ1 , is the same as the mean height of British Scouts, μ2 . The
information is recorded in the following table.
American Scout
height (cm)
British Scout
height (cm)
147 153 151 142 155 149 154 156 143 152 149 158
142 146 155 145 149 148 152 143 147 150 149 154 150 144 146
At the 10% level of significance, a t-test was used to compare
the means of the two groups. The data is assumed to be
normally distributed and the standard deviations are equal
between the two groups.
(a) State the null hypothesis.
(b) State the alternative hypothesis.
(c) Calculate the p-value for this test.
(d) State, giving a reason, whether Skipper Jones should accept
the null hypothesis.
444
End Of Week Test Question 10.
Mr and Mrs Moule are considering purchasing a new car with a
price of £12000
The dealership offers two options to
finance a loan.
Finance Plan A:
A five year loan at a nominal annual
interest rate of 12 % compounded
quarterly. No deposit is required and
repayments are made each quarter.
(a) Find the repayment made each
quarter.
(b) Find the total amount that will have been paid for the car by
the end of the five years.
(c) Find the total amount interest that will have been paid on the
loan by the end of the five years.
Finance Plan B:
A five year loan at a nominal annual interest rate of r%
compounded monthly. An Initial 12% deposit is required and
monthly repayments of £255.
(d) Find the annual interest rate, r.
(e) State which option Mr and Mrs Moule should choose and
justify your answer.
This new car will depreciates at an annual rate of 20% per year.
(f) Find the value of Mr and Mrs Moule’s car when it is ten years
old.
446
End Of Week Test Question 11.
The following diagram shows part of the graph of:
f(x) =
(9 + 4x)(6 − x)
,x ∈ R
10
The shaded region A is bounded by the x-axis, y-axis and the
graph of f .
(a) Write down an integral for the area of region A.
(b) Find the area of region A.
The three points A (0 , 0) , B (6 , 9) and C (p , 0) define the
vertices of a triangle.
(c) Find the value of p, the x-coordinate of C , such that the area
of the triangle is equal to the area of region A.
451
End Of Week Test Question 12.
A wedge is to be cut from a log in the shape of a cylinder as
shown in the diagram below (not to scale).
The length of the log is 240cm and its radius is 40cm. The cross
section of the wedge to be removed is a sector with an angle of
130o.
What is the volume of the remaining piece of the log after the
wedge has been removed?
455
End Of Week Test Question 13.
A game is played with a biased five-sided spinner. The possible
scores, X and their probabilities are shown in the following table.
Score x
-5 -1
0
2
10
P (X = x) 0.2 p 0.3 0.3 0.1
(a) Find the exact value of p.
Jaedee plays the game once.
(b) Calculate the expected score.
Jaedee plays the game twice and adds the two scores together.
(c) Find the probability Jaedee has a total score of 10.
457
End Of Week Test Question 14.
Bostock and Chandler play one game of Ultimate Noughts and
Crosses online each day they are in quarantine.
The probability that Bostock wins a game is three times the
probability that Chandler wins a game. It is not possible to have
a draw.
(a) Find the probability that on any given day Bostock will win
the game.
The quarantine lasts for 30 days.
(b) Find the probability the Bostock will win 20 times.
(c) Find the probability Chandler will win at most 9 times.
460
End Of Week Test Question 15.
Melissa has a webcam on the top of her computer screen. She
has the zoom level adjusted so that that her head is framed
nicely in the image produced. The diagram shows the setup.
The webcam is at point C directly above point D on the floor. The
area covered by the camera is shown by the shaded region
enclosed by triangle ABC. The distance from A to C is 4 m and
the distance from A to B is 1.3 m. Angle ACB is 16o.
(a) Find angle ABC.
Melissa's younger brother is 95cm tall. He walks into the room
from the door at the opposite end of the room from Melissa's
computer. When he reaches point A his feet first appear in the
video being recorded by the webcam. He continues to walk until
the top of his head is in shot at which points he stops.
(b) How far is Melissa's brother from point A?
463
End Of Week Test Question 16.
The Scrumptious Sweet Company sell a variety pack of colourful,
shaped sweets. The sweets are produced such that 60% are
square and 40% are circular. It is known that 20% of the square
shaped sweets and 40% of the circular sweets are coloured red.
(a) Show this information in a tree diagram.
A sweet is selected at random.
(b) Find the probability that the sweet is red.
(c) Given that the sweet is red, find the probability it is circular.
The Scrumptious Sweet Company also produce variety packs of
Rainbow Gums. Their specifications state that the colours in each
variety pack should be distributed as follows.
Colour
Red Orange Yellow Green Blue
Percentage (%)
15
25
15
25
20
Inspector Lou Spowels opens a pack of 90 sweets and records
the frequency of each colour:
Colour
Observed Frequency
Red Orange Yellow Green Blue
12
21
15
20
22
To investigate if the sample is consistent with the company's
specifications, Mr Spowels conducts a χ2 goodness of fit test.
The test is carried out at a 5% significance level.
(d) Write down the null hypothesis for this test.
(e) Copy and complete the following table giving the frequencies
correct to one decimal place:
Colour
Red Orange Yellow Green Blue
Expected Frequency
(f) Write down the number of degrees of freedom.
(g) Find the p-value for the test.
(h) State the conclusion of the test. Give a reason for your
answer.
466
End Of Week Test Question 17.
The Farang Parkour Team hosted a Free Running event. The
judges, Anan and Jason awarded 7 competitors a score out of 10.
The scores are shown in the following table.
Free Runners
A
B
C
D
E
F
G
Anan's Score (x) 7.8 9.1 8.3 6.9 7.0 8.5 9.3
Jason's Score (y) 7.2 9.0 8.7 7.5 6.9 8.7 8.9
(a) Find the Pearson’s product–moment correlation coefficient, r,
of these scores.
(b) Using the value of r, interpret the relationship between
Anan’s scores and Jason’s score.
(c) Write down the equation of the regression line y on x.
(d) Use your regression equation from part (c) to estimate
Jason’s score to one decimal place when Anan awards a score of
5.
(e) State whether this estimate is reliable. Justify your answer.
(f) The adjudicator for the event would like to find the
Spearman’s rank correlation coefficient of the scores. Copy and
complete the information in the following table.
Free Runners A B C D E F G
Anan's Rank
Jason's Rank
7
1
1
7
(g) Find the value of the Spearman’s rank correlation coefficient,
rs .
(h) Comment on the result obtained for rs .
The adjudicator believes Jason’s score for competitor E is too
high and so decreases the score from 6.9 to 6.5.
(i) Explain why the value of the Spearman’s rank correlation
coefficient rs does not change.
469
End Of Week Test Question 18.
The circumference of a given circle C can be represented by the
−−
−
function C(A) = 2√Aπ , A ≥ 0 , where A is the area of the
circle. The graph of the function C is shown for 0 ≤ A ≤ 10.
(a) Write down the value of C(5).
The range of C(A) is 0 ≤ C(A) ≤ k.
(b) Find the value of k .
(c) On the axes above, draw the graph of the inverse function,
C −1 .
(d) In the context of the question, explain the meaning of
C −1 (10) ≈ 7.96.
472
End Of Week Test Question 19.
In a fantasy story the power value of a dream catcher varies
depending on its length. The power values of various dream
catchers are recorded in the following table:
Length, x cm 0 10 15
Power, p W
0 12 22
This information was used to create Model A, where p is a
function of x , x ≥ 0.
Model A: p(x) = ax2 + bx , where a, b ∈ Z.
When the length is 10 cm, Model A can be represented by the
equation 50a + 5b = 6.
(a) Write down a second equation to represent Model A, when
the length is 15cm.
(b) Find the values of a and b.
(c) Find the coordinates of the vertex of the graph of y = p(x).
(d) Using the values in the table and your answer to part (c),
sketch the graph of y = p(x) for 0 ≤ x ≤ 15 and 0 ≤ p ≤ 22.
Additional data was used to create Model B, a revised model for
the power of a dream catcher.
Model B: p(x) = 0.06x2 + 0.68x
(e) Use Model B to calculate an estimate for the power of a
dream catcher of length 18cm.
The actual power of a dream catcher of length 18cm is 30 W.
(f) Calculate the percentage error in the estimate in part (e).