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ACET: Practice Exam – Questions Page 1
The Actuarial Education Company © IFE: 2014 Examinations
Practice Exam – Questions
Question E1
What is 0.040783 rounded to 2 significant figures?
A 0.04
B 0.041
C 0.0408
D 0.04078 [1]
FAC 2 1
Question E2
What is na a x
log x log y
Ê ˆ - Á ˜ Ë ¯
A ( 1) a an log x log y- +
B ( )a x ynx a --
C
n
alog x
Ê ˆ
-Á ˜ Ë ¯
D1n
a
xlog
-Ê ˆ Á ˜ Ë ¯
[1]
FAC 4 1
Question E3
Solve 11 3 2 15 x- £ - <
A 7 6 x- £ < B 7 x ≥ and 6 x < - C 7 6 x≥ > - D 7 x £ - and 6 x > [1]
FAC 4 4
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Page 2 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E4
Which of these is the graph of n y x= where n is an even number:
A
x
y
B
x
y
C
x
y
D
x
y
[1]
FAC 3 1
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ACET: Practice Exam – Questions Page 3
The Actuarial Education Company © IFE: 2014 Examinations
Question E5
Interest rates at the start of the year are set at 2.5%. At the end of the year they are
2.75%. What is the absolute change in the interest rate over the year?
A 10%
B 250 basis points
C 9.09%
D 25 basis points [1]
FAC 5 2
Question E6
What is the result if (2 3 )i+ is multiplied by its complex conjugate?
A 5 12i- + B 5- C 13
D 5 12i- - [1]FAC 5 7
Question E7
Differentiate 5 y x= with respect to x .
A 25 --
B ½52
x
C 3 252
x
D5
2 x [1]
FAC 6 3
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Page 4 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E8
Find ( ) f x¢ where2
3( )
x f x
e= .
A2
3
2 xe
B2
3
2 xe-
C2
6 xe
-
D2
6 xe
[1]
FAC 6 4
Question E9
What is
14
0
3 xe dxÚ ?
A 19.17
B 40.20
C 160.79
D 643.18 [1]
FAC 7 2
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ACET: Practice Exam – Questions Page 5
The Actuarial Education Company © IFE: 2014 Examinations
Question E10
What is the integral of3
1 1
x x- ?
A4
41 c
x- +
B4
1ln
4c
x- +
C2
21 c
x- +
D2
1ln
2c
x+ + [1]
FAC 7 1
Question E11
If
1
2
2
Ê ˆ Á ˜ = -Á ˜ Ë ¯
a and
2
3
5
-Ê ˆ Á ˜ =Á ˜ Ë ¯
b then 2-b a is:
A 4.24B 5.83
C 8.12
D 12.37 [1]
FAC 8 1
Question E12
If3 4
2 5
- -Ê ˆ = Á ˜ Ë ¯ B then | |B is given by:
A 23- B 7- C 7
D 23 [1]
FAC 8 2
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Page 6 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E13
The abbreviation etc means:
A for exampleB compared with
C that is to say
D and so on [1]
FAC Gloss
Question E14
The histogram below shows the age distribution of the last 100 policyholders to take out
an insurance policy with a life insurance company.
0.6
5.6
4.2
1.6
0.7 0.48 f r e q u e n c y
d e n s i t y
age
0 15 20 25 35 55 80
1
2
3
4
5
6
Determine the number of policyholders in the 35 to 55 age group.
A 7
B 14
C 56
D 70 [1]SP 1 3
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ACET: Practice Exam – Questions Page 7
The Actuarial Education Company © IFE: 2014 Examinations
Question E15
A company employs 32 skilled labourers and 19 unskilled labourers. The mean salary
of the skilled labourers is £24,638, and the mean salary of the unskilled labourers is
£13,942.
Calculate the mean salary of all 51 labourers.
A £17,926.78
B £19,290.00
C £20,653.22
D £22,428.22 [1]
SP 2 2
Question E16
The numbers of claims last year on a group of home insurance policies was recorded
and resulted in the following frequency distribution:
Number of claims, 0 1 2 3
Number of policies, f 70 30 15 5
Calculate the sample median number of claims per policy.
A 0
B 0.625
C 0.864
D 1.5 [1]
SP 2 3
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Page 8 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E17
For independent events A and B , you are given:
( ) 0.5 ( ) 0.3 P A P B
Determine ( or ) P A B .
A 0.65
B 0.72
C 0.80
D 0.95 [1]
SP 4 2&3
Question E18
A contest has 3 prizes and 8 competitors. Each competitor can receive at most one
prize. How many ways are there of allocating the prizes to the competitors?
A 56
B 168
C 336
D 1512 [1]SP 6 2
Question E19
The number of claims, X , arise on a policy according to the following probability
distribution:
x 0 1 2 3
P X x 0.2 a b 0.4
If the mean number of claims is 1.72, then the value b a- is:
A 0.16- B 0
C 0.12
D 0.28 [1]
SP 7 4
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ACET: Practice Exam – Questions Page 9
The Actuarial Education Company © IFE: 2014 Examinations
Question E20
A discrete random variable X has the following probability function:
1 2 3
( ) P X x= 0.45 0.35 0.2
Calculate2
E X
Ê ˆ Á ˜ Ë ¯
.
A 1.143
B 1.383
C 1.75
D 2 [1]SP 7 5
Question E21
If 5.2 E X and 4.49 sd X , the second moment about zero is:
A 9.69
B 25.36
C 31.53
D 47.20 [1]
SP 7 8
Question E22
The probability density function of a random variable Y is given by:
2364( ) 0 4 f y y y
Calculate ( 2) P Y .
A 0.125
B 0.1875
C 0.8125
D 0.875 [1]
SP 9 3
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Page 10 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E23
A continuous random variable X has a mean of 50 and a standard deviation of 8.
Calculate
4 7
var 10
X -Ê ˆ Á ˜ Ë ¯ .
A 1.28
B 3.2
C 10.24
D 25.6 [1]
SP 9 6
Question E24
The random variable X is uniformly distributed on the interval (1,4) . The CDF of X
is given by:
A1
3
B1
4
C
1
3
x -
D4
x [1]
SP 10 1
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ACET: Practice Exam – Questions Page 11
The Actuarial Education Company © IFE: 2014 Examinations
Question E25
Which of the following scattergraphs is most likely to be the relationship between the
number of cold drinks sold by a cafe and the temperature?
A
B
C
D
[1]
SP 12 2
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Page 12 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E26
Simplify the expression(3 1)! (2 2)
(3 1)(2 )!
n n
n n
- G +G +
, where n is an integer.
A(2 2)(2 1)
(3 1)(3 )
n n
n n
+ ++
B(2 2)(2 1)
3
n n
n
+ +
C(2 1)
(3 1)(3 )
n
n n
++
D(2 1)
3
n
n
+ [2]
FAC 3 3
Question E27
Calculate21 (5 4.712)
2.306 0.106310 2.82596
Ï ¸-Ô Ô+ ¥Ì ˝Ô ÔÓ ˛
.
A 0.088B 0.270
C 0.338
D 0.741 [2]
FAC 2 2
Question E28
Solve the quadratic equation 27 4 0 x x- - = giving your solutions to 1 DP.
A 5.3 and 1.3 x = - B 0.2 and 0.8 x = - C 10.6 and 2.6 x = - D The equation has no real solutions [2]
FAC 4 2
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ACET: Practice Exam – Questions Page 13
The Actuarial Education Company © IFE: 2014 Examinations
Question E29
Use the result( 1) ( 1)( 2)2 32! 3!
(1 ) 1 p p p p p p x px x x
- - -+ = + + + + to expand the
expression ( ) 1.21 x -+ as far as the term in 3 x and hence evaluate it when 0.4 x = - .
A 0.641088
B 1.358912
C 1.781312
D 1.845944 [2]
FAC 4 9
Question E30
Solve for x and : 2 22 33 x y+ = 3 x y+ = -
A 21 x = - and 18 y = B 4 x = - and 1 y = C 5 x = and 2 y = - or 1 x = - and 4 y = D 5 x = - and 2 y = or 1= and 4 y = - [2]
FAC 4 3
Question E31
Calculate the sum of the first 10 terms in the sequence 5, 6, 7.2, 8.64, ...
A 103.995
B 129.793
C 155.752
D 160.752 [2]FAC 4 7
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Page 14 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E32
A population increases by 5% every year. Calculate the minimum whole number of
years until the population has doubled in size.
A 9
B 12
C 15
D 18 [2]
FAC 5 1
Question E33
Given 3 i= + is a complex root of the cubic 3 24 2 20 0 x x x- - + = , find the othertwo roots.
A 3 or 2 x i= - - B ( ) ( )3 or 2i i= - - + C 3 or 3 x i= - D 3 or 2 x i= - - - [2]
FAC 5 7
Question E34
Finddy
diwhere
41 (1 )i y
i
-- += .
A4 5
2
(1 ) 4 (1 ) 1i i i
i
- -+ - + -
B
4 3
2
(1 ) 4 (1 ) 1i i i
i
- -+ - + -
C4 5
2
(1 ) 4 (1 ) 1i i i
i
- -+ + + -
D4 3
2
(1 ) 4 (1 ) 1i i i
i
- -+ + + - [2]
FAC 6 4
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ACET: Practice Exam – Questions Page 15
The Actuarial Education Company © IFE: 2014 Examinations
Question E35
Find the second derivative of ( 1)( )t e M t e m -= evaluated at 0t = , where m is a constant.
A (0) M m =¢¢
B 2(0) M m =¢¢
C 2(0) M m m = +¢¢
D 2(0) e m =¢¢ [2]FAC 6 5
Question E36
What is ( ) ( )( )2
2 3axy b xy c xy
x y
∂+ +
∂ ∂?
A ( ) ( )2
2 3a b yx c yx+ +
B ( ) ( )2
4 9a b yx c yx+ +
C
2 3
2 3
yx yxaxy b c
Ê ˆ Ê ˆ + +Á ˜ Á ˜ Ë ¯ Ë ¯
D
2 3 4
2 3 4
yx yx yxa b c
Ê ˆ Ê ˆ Ê ˆ + +Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯ [2]
FAC 6 7
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Page 16 ACET: Practice Exam – Questions
© IFE: 2014 Examinations The Actuarial Education Company
Question E37
What is 2 48 (3 2) x dx+Ú ?
A2 58 (3 2)
5
x xc
++
B2 58(3 2)
5
xc
++
C2 54(3 2)
15
xc
++
D3 58 ( 2)
5
x xc
++ [2]
FAC 7 3
Question E38
What is
5 2
1 1
(3 4 )
x y
y dy dx
= =
+Ú Ú ?
A24
B 60
C 66
D 84 [2]
FAC 7 5
Question E39
Using the trapezium rule and 6 ordinates, what is the value of
1
3
0
xe dxÚ ?
A 4.330
B 6.552
C 13.103
D 65.516 [2]
FAC 7 6
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ACET: Practice Exam – Questions Page 17
The Actuarial Education Company © IFE: 2014 Examinations
Question E40
If1 2
3 4
Ê ˆ = Á ˜
-Ë ¯
A then T A A is given by:
A5 5
5 20
Ê ˆ Á ˜ -Ë ¯
B2 16
9 2
-Ê ˆ Á ˜ - -Ë ¯
C 1 0
0 1
Ê ˆ Á ˜ Ë ¯
D 10 1010 20-Ê ˆ Á ˜ -Ë ¯
[2]
FAC 8 2
Question E41
The eigenvalues of1 2
2 2
Ê ˆ Á ˜ -Ë ¯
are:
A 2 and 3- B 1 and 6- C 4.37 and 1.37- D do not exist [2]
FAC 8 2
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Page 18 ACET: Practice Exam – Questions
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Question E42
The stem and leaf diagram below shows the values of 20 claim amounts from a certain
portfolio of policies. The stem unit is $1,000 and the leaf unit is $100.
1 2458
2 03899
3 35677
4 1448
5 03
Determine the upper quartile of this sample.
A $4,100
B $4,175
C $4,325
D $4,400 [2]
SP 1 3 & 3 2
Question E43
Calculate the standard deviation of the following data set:
35 40 41 45 45 51 53
A 5.83
B 6.29
C 33.92
D 39.57 [2]
SP 3 3
Question E44
In a certain population, 55% of the lives are male, 30% of males are over 60 and 35% of
females are over 60. A life is randomly selected from this population. What is the
probability that this life is female, given that the life is under 60?
A 0.4317
B 0.4884
C 0.5683
D 0.6775 [2]
SP 5 2
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ACET: Practice Exam – Questions Page 19
The Actuarial Education Company © IFE: 2014 Examinations
Question E45
Amit has a bunch of 12 bananas. The probability of any banana in the bunch being
unripe is 0.1. Calculate the probability that there are 3 unripe bananas in the bunch.
A 0.001
B 0.000387
C 0.0852
D 0.22 [2]
SP 6 4
Question E46
The random variable, X , has probability function:
( 0) 0.4 ( 1) 0.6 P X P X
Calculate the third-order moment of X about 0.8.
A 0.200- B 0.2096
C 0.28
D 0.6 [2]
SP 7 8
Question E47
Given that ~ (1.2) X Poi , calculate ( )2 | 0 P X X = > .
You are given that the probability function of a Poisson distribution with mean is:
( )!
x
P X x e x
A 0.217
B 0.310
C 0.690
D 0.783 [2]
SP 8 4
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Question E48
A continuous random variable X has PDF:
2 0.1( ) 0.005 x X f x x e-= for 0 x >
Calculate the mode of X .
A 0.2
B 2
C 20
D X has no mode. [2]
SP 9 4
Question E49
The random variable X has an exponential distribution with PDF:
0.2( ) 0.2 0 x f x e x
The median of this distribution is:
A 0.2
B 3.466
C 4.581
D 5 [2]
SP 10 2
Question E50
If ~ 120, 25 X N calculate 113 8 P X
.
You are given that ( 0.04) 0.51595 P Z , ( 0.2) 0.57926 P Z ,
( 0.6) 0.72575 P Z and ( 3) 0.99865 P Z .
A 0.2098
B 0.2417
C 0.41939
D 0.5779 1 [2]
SP 11 4
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ACET: Practice Exam – Questions Page 21
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Question E51
You are given that
( ) ( ) ( ) ( )4 3 2
6.6 2 2 2
11 1 1 PV
ii i i
-= + + +
++ + +. By trial and error and
interpolation, calculate the value of i to 2 significant figures such that 0 PV = .
Evaluate the formula(1 ) 1n
n
i s
i
+ -= , using the value of i obtained, where 4n = .
A 4.246n
s =
B 4.297n
s =
C 4.310n
s =
D 4.375n s = [5]FAC 5 5
Question E52
Find and distinguish between the turning points of the function 3 5( ) 15 f x x x= - .
A minima at 1.225 x = ± and point of inflexion at 0 x = B minimum at 3 x = - , point of inflexion at 0 x = and maximum at 3 x = C maximum at 3 x = - , point of inflexion at 0 x = and minimum at 3 x = D maxima at 1.225 x = ± , point of inflexion at 0 x = [5]
FAC 6 6
Question E53
What is
22 0.5
1
2 x x e dx-Ú ?
A 8.437
B 4.218
C 2.109
D 0.875- [5]FAC 7 3
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Page 22 ACET: Practice Exam – Questions
Question E54
The random variable, X , has the following PF:
x 0 1 2 3
P X x 0.5 0.35 0.1 0.05
The coefficient of skewness,3
3
[( ) ] E X
, is:
A 0.507
B 0.938
C 1.113
D 1.861 [5]
SP 7 7
Question E55
The probability density function of a random variable W is given by:
3( ) (1 ) 0 1 f w kw w x
The variance of W is:
A 0.005
B 0.0317
C 0.476
D 0.667 [5]
SP 9 6