2015 Fall Semester Review
1a. [1 mark]
The first three terms of an infinite geometric sequence are 32, 16 and 8.
Write down the value of r .
1b. [2 marks]
Find
.
1c. [2 marks]
Find the sum to infinity of this sequence.
2a. [1 mark]
The diagram below shows the probabilities for events A and B , with
Write down the value of p .
2b. [3 marks]
Find
.
2c. [3 marks]
Find
.
3a. [2 marks]
Let
and
.
1
.
Find
.
4a. [3 marks]
Consider the events A and B, where
,
and
The Venn diagram below shows the events A and B, and the probabilities p, q and r.
Write down the value of
(i) p ;
(ii) q ;
(iii) r.
4b. [2 marks]
Find the value of
.
4c. [1 mark]
Hence, or otherwise, show that the events A and B are not independent.
5a. [2 marks]
A standard die is rolled 36 times. The results are shown in the following table.
Write down the standard deviation.
5b. [1 mark]
Write down the median score.
2
.
5c. [3 marks]
Find the interquartile range.
6a. [3 marks]
The velocity v ms of an object after t seconds is given by
, for
On the grid below, sketch the graph of v , clearly indicating the maximum point.
6b. [4 marks]
(i) Write down an expression for d .
(ii) Hence, write down the value of d .
7a. [1 mark]
The n term of an arithmetic sequence is given by
.
Write down the common difference.
7b. [5 marks]
(i) Given that the n term of this sequence is 115, find the value of n .
(ii) For this value of n , find the sum of the sequence.
3
.
8a. [6 marks]
A test has five questions. To pass the test, at least three of the questions must be answered correctly.
The probability that Mark answers a question correctly is
. Let X be the number of questions that
Mark answers correctly.
(i) Find E(X ) .
(ii) Find the probability that Mark passes the test.
8b. [8 marks]
Bill also takes the test. Let Y be the number of questions that Bill answers correctly.
The following table is the probability distribution for Y .
(i) Show that
(ii) Given that
.
, find a and b .
8c. [3 marks]
Find which student is more likely to pass the test.
9a. [2 marks]
Let
. Part of the graph of f is shown below.
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The graph passes through the points (−2, 0), (0, − 4) and (4, 0) .
Write down the value of q and of r.
9b. [1 mark]
Write down the equation of the axis of symmetry.
9c. [3 marks]
Find the value of p.
10a. [1 mark]
Consider the arithmetic sequence 3, 9, 15,
, 1353 .
Write down the common difference.
10b. [3 marks]
Find the number of terms in the sequence.
10c. [2 marks]
Find the sum of the sequence.
11a. [4 marks]
The following frequency distribution of marks has mean 4.5.
Find the value of x.
11b. [2 marks]
Write down the standard deviation.
12a. [2 marks]
Evan likes to play two games of chance, A and B.
For game A, the probability that Evan wins is 0.9. He plays game A seven times.
Find the probability that he wins exactly four games.
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12b. [2 marks]
For game B, the probability that Evan wins is p . He plays game B seven times.
Write down an expression, in terms of p , for the probability that he wins exactly four games.
12c. [3 marks]
Hence, find the values of p such that the probability that he wins exactly four games is 0.15.
13a. [2 marks]
The weights of players in a sports league are normally distributed with a mean of
three significant figures). It is known that
of the players have weights between
The probability that a player weighs less than
, (correct to
and
is 0.05.
Find the probability that a player weighs more than
.
13b. [4 marks]
(i) Write down the standardized value, z, for
.
(ii) Hence, find the standard deviation of weights.
13c. [5 marks]
To take part in a tournament, a player’s weight must be within 1.5 standard deviations of the mean.
(i) Find the set of all possible weights of players that take part in the tournament.
(ii) A player is selected at random. Find the probability that the player takes part in the tournament.
13d. [4 marks]
Of the players in the league,
are women. Of the women,
take part in the tournament.
Given that a player selected at random takes part in the tournament, find the probability that the
selected player is a woman.
14a. [2 marks]
Let
Find
and
.
.
14b. [1 mark]
6
.
Write down
.
14c. [2 marks]
Find
.
15a. [5 marks]
In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music. The
Venn diagram below shows the events art and music. The values p , q , r and s represent numbers of
students.
(i) Write down the value of s .
(ii) Find the value of q .
(iii) Write down the value of p and of r .
15b. [4 marks]
(i) A student is selected at random. Given that the student takes music, write down the probability the
student takes art.
(ii) Hence, show that taking music and taking art are not independent events.
15c. [4 marks]
Two students are selected at random, one after the other. Find the probability that the first student
takes only music and the second student takes only art.
16a. [2 marks]
In an arithmetic sequence,
and
.
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Find d .
16b. [2 marks]
Find
.
16c. [2 marks]
Find
.
17a. [3 marks]
A scientist has 100 female fish and 100 male fish. She measures their lengths to the nearest cm. These
are shown in the following box and whisker diagrams.
Find the range of the lengths of all 200 fish.
17b. [2 marks]
Four cumulative frequency graphs are shown below.
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Which graph is the best representation of the lengths of the female fish?
18a. [2 marks]
The following diagram shows part of the graph of a quadratic function f .
The x-intercepts are at
Write down
and
, and the y-intercept is at
in the form
.
9
.
18b. [4 marks]
Find another expression for
in the form
.
18c. [2 marks]
Show that
can also be written in the form
.
18d. [7 marks]
A particle moves along a straight line so that its velocity,
, for
, at time t seconds is given by
.
(i) Find the value of t when the speed of the particle is greatest.
(ii) Find the acceleration of the particle when its speed is zero.
19a. [3 marks]
In an arithmetic sequence
,
and
.
Find the value of the common difference.
19b. [2 marks]
Find the value of n .
20a. [3 marks]
A random variable X is distributed normally with a mean of 20 and variance 9.
Find
.
20b. [5 marks]
Let
.
(i) Represent this information on the following diagram.
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(ii) Find the value of k .
21a. [2 marks]
A box holds 240 eggs. The probability that an egg is brown is 0.05.
Find the expected number of brown eggs in the box.
21b. [2 marks]
Find the probability that there are 15 brown eggs in the box.
21c. [3 marks]
Find the probability that there are at least 10 brown eggs in the box.
22a. [3 marks]
A company uses two machines, A and B, to make boxes. Machine A makes
of the boxes.
of the boxes made by machine A pass inspection.
of the boxes made by machine B pass inspection.
A box is selected at random.
Find the probability that it passes inspection.
22b. [4 marks]
The company would like the probability that a box passes inspection to be 0.87.
Find the percentage of boxes that should be made by machine B to achieve this.
23a. [2 marks]
Let the random variable X be normally distributed with mean 25, as shown in the following diagram.
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The shaded region between 25 and 27 represents
Find
of the distribution.
.
23b. [5 marks]
Find the standard deviation of X .
24a. [3 marks]
Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are labelled 1, 2, 3, 4.
The score for each die is the number which lands face down.
List the pairs of scores that give a sum of 6.
24b. [3 marks]
The probability distribution for the sum of the scores on the two dice is shown below.
Find the value of p , of q , and of r .
24c. [6 marks]
Fred plays a game. He throws two fair 4-sided dice four times. He wins a prize if the sum is 5 on three
or more throws.
Find the probability that Fred wins a prize.
Printed for El Dorado High School
© International Baccalaureate Organization 2015
International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®
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