Sample QP - 2014

ACET: Practice Exam – Questions

Page 1

Practice Exam – Questions Question E1 What is 0.040783 rounded to 2 significant figures? A B C D

0.04 0.041 0.0408 0.04078

[1] FAC 2 1

Question E2 Ê xˆ What is log a x n - log a Á ˜ Ë y¯ A

(n - 1)log a x + log a y

B

nx a - a ( x - y )

C

Ê xˆ log a Á x n - ˜ y¯ Ë

D

Ê x n -1 ˆ log a Á ˜ Ë y ¯

[1] FAC 4 1

Question E3 Solve -11 £ 3 - 2 x < 15 A B C D

-7 £ x < 6 x ≥ 7 and x < -6 7 ≥ x > -6 x £ -7 and x > 6

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[1] FAC 4 4

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Page 2

ACET: Practice Exam – Questions

Question E4 Which of these is the graph of y = x n where n is an even number: A y

x

B y

x

C y

x

D y

x

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[1] FAC 3 1

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ACET: Practice Exam – Questions

Page 3

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 + 3i ) is multiplied by its complex conjugate? A B C D

-5 + 12i -5

13 -5 - 12i

[1] FAC 5 7

Question E7 Differentiate y = 5 x with respect to x .

A

-5x -2

B

5 ½ x 2 5 32 x 2 5 2 x

C D

[1] FAC 6 3

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Page 4

ACET: Practice Exam – Questions

Question E8 Find f ¢( x) where f ( x) =

A B C D

3 2e 2 x 3 - 2x 2e 6 - 2x e 6 e2 x

3 . e2 x

[1] FAC 6 4

Question E9 1

What is Ú 3e 4 x dx ? 0

A B C D

19.17 40.20 160.79 643.18

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[1] FAC 7 2

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ACET: Practice Exam – Questions

Page 5

Question E10 What is the integral of

A B C D

1 1 ? x x3

4 +c x4 1 ln x - 4 + c 4x 2 1- 2 + c x 1 ln x + 2 + c 2x 1-

[1] FAC 7 1

Question E11 Ê1ˆ Ê -2ˆ If a = Á -2˜ and b = Á 3 ˜ then b - 2a is: Á ˜ Á ˜ Ë 2¯ Ë 5¯

A B C D

4.24 5.83 8.12 12.37

[1] FAC 8 1

Question E12 Ê -3 -4ˆ If B = Á then | B | is given by: Ë 2 5 ˜¯ A B C D

-23 -7 7 23

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[1] FAC 8 2

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Page 6

ACET: Practice Exam – Questions

Question E13 The abbreviation etc means: A B C D

for example compared with that is to say 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.

5.6

frequency density

6 5

4.2

4 3 1.6

2

0.7

0.6

1

0.48

age 0

15 20 25

35

55

80

Determine the number of policyholders in the 35 to 55 age group. A B C D

7 14 56 70

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[1] SP 1 3

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ACET: Practice Exam – Questions

Page 7

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 B C D

£17,926.78 £19,290.00 £20,653.22 £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, x

0

1

2

3

Number of policies, f

70

30

15

5

Calculate the sample median number of claims per policy. A B C D

0 0.625 0.864 1.5

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[1] SP 2 3

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Page 8

ACET: Practice Exam – Questions

Question E17 For independent events A and B , you are given: P ( A)  0.5

P ( B )  0.3

Determine P ( A or B ) . A B C D

0.65 0.72 0.80 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 B C D

56 168 336 1512

[1] SP 6 2

Question E19 The number of claims, X , arise on a policy according to the following probability distribution: x P  X  x

0

1

2

3

0.2

a

b

0.4

If the mean number of claims is 1.72, then the value b - a is: A B C D

-0.16 0 0.12 0.28

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[1] SP 7 4

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Page 9

Question E20 A discrete random variable X has the following probability function: 1 0.45

x

P( X = x)

2 0.35

3 0.2

Ê 2ˆ Calculate E Á ˜ . ËX¯

A B C D

1.143 1.383 1.75 2

[1] SP 7 5

Question E21 If E  X   5.2 and sd  X   4.49 , the second moment about zero is: A B C D

9.69 25.36 31.53 47.20

[1] SP 7 8

Question E22 The probability density function of a random variable Y is given by: f ( y) 

3 64

y2

0 y4

Calculate P (Y  2) . A B C D

0.125 0.1875 0.8125 0.875

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[1] SP 9 3

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Page 10

ACET: Practice Exam – Questions

Question E23 A continuous random variable X has a mean of 50 and a standard deviation of 8. Ê 4X - 7ˆ . Calculate var Á Ë 10 ˜¯ A B C D

1.28 3.2 10.24 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: A B C D

1 3 1 4 x -1 3 x 4

[1] SP 10 1

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Page 11

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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ACET: Practice Exam – Questions

Question E26 Simplify the expression

A B C D

(3n - 1)!G (2n + 2) , where n is an integer. G (3n + 1)(2n)!

(2n + 2)(2n + 1) (3n + 1)(3n) (2n + 2)(2n + 1) 3n (2n + 1) (3n + 1)(3n) (2n + 1) 3n

[2] FAC 3 3

Question E27 ÔÏ 1 (5 - 4.712) 2 Ô¸ Calculate 2.306 Ì + ˝ ¥ 0.1063 . 2.82596 ˛Ô ÓÔ10

A B C D

0.088 0.270 0.338 0.741

[2] FAC 2 2

Question E28 Solve the quadratic equation 7 - 4 x - x 2 = 0 giving your solutions to 1 DP. A B C D

x = -5.3 and 1.3 x = -0.2 and 0.8 x = -10.6 and 2.6 The equation has no real solutions

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[2] FAC 4 2

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ACET: Practice Exam – Questions

Page 13

Question E29 Use

the

result

expression (1 + x )

A B C D

(1 + x) p = 1 + px +

-1.2

p ( p -1) 2 x 2!

+

p ( p -1)( p - 2) 3 x 3!

+  to

expand

the

as far as the term in x3 and hence evaluate it when x = -0.4 .

0.641088 1.358912 1.781312 1.845944

[2] FAC 4 9

Question E30 Solve for x and y :

A B C D

x 2 + 2 y 2 = 33 x + y =-3

x = -21 and y = 18 x = -4 and y = 1 x = 5 and y = -2 or x = -1 and y = 4 x = -5 and y = 2 or x = 1 and y = -4

[2] FAC 4 3

Question E31 Calculate the sum of the first 10 terms in the sequence 5, 6, 7.2, 8.64, ... A B C D

103.995 129.793 155.752 160.752

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[2] FAC 4 7

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ACET: Practice Exam – Questions

Question E32 A population increases by 5% every year. Calculate the minimum whole number of years until the population has doubled in size. A B C D

9 12 15 18

[2] FAC 5 1

Question E33 Given x = 3 + i is a complex root of the cubic x3 - 4 x 2 - 2 x + 20 = 0 , find the other two roots. A B

x = 3 - i or - 2 x = ( -3 - i ) or (2 + i )

C D

x = 3 - i or 3 x = -3 - i or - 2

[2] FAC 5 7

Question E34 dy 1 - (1 + i ) -4 . Find where y = i di

A

(1 + i ) -4 - 4i (1 + i ) -5 - 1 i2

B

(1 + i ) -4 - 4i (1 + i ) -3 - 1 i2

C

(1 + i ) -4 + 4i (1 + i ) -5 - 1 i2

D

(1 + i ) -4 + 4i (1 + i ) -3 - 1 i2

[2] FAC 6 4

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Page 15

Question E35 Find the second derivative of M (t ) = e m (e A

M ¢¢(0) = m

B

M ¢¢(0) = m 2

C

M ¢¢(0) = m + m 2

D

M ¢¢(0) = m 2e m

t

-1)

evaluated at t = 0 , where m is a constant.

[2] FAC 6 5

Question E36 What is

(

)

∂2 2 3 axy + b ( xy ) + c ( xy ) ? ∂x∂y

A

a + 2b ( yx ) + 3c ( yx )

B

a + 4b ( yx ) + 9c ( yx )

C

Ê yx ˆ Ê yx ˆ axy + b Á ˜ + c Á ˜ Ë 2¯ Ë 3¯

D

Ê yx ˆ Ê yx ˆ Ê yx ˆ aÁ ˜ + bÁ ˜ + cÁ ˜ Ë 2¯ Ë 3¯ Ë 4¯

2

2

2 2 3

3

4

[2] FAC 6 7

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ACET: Practice Exam – Questions

Question E37 What is Ú 8 x(3x 2 + 2) 4dx ? A

8 x(3x 2 + 2)5 +c 5

B

8(3x 2 + 2)5 +c 5

C

4(3x 2 + 2)5 +c 15

D

8 x( x3 + 2)5 +c 5

[2] FAC 7 3

Question E38 5

What is

2

Ú Ú

(3 x + 4 y ) dy dx ?

x =1 y =1

A

24

B

60

C

66

D

84

[2] FAC 7 5

Question E39 1

Using the trapezium rule and 6 ordinates, what is the value of Ú e3 x dx ? 0

A B C D

4.330 6.552 13.103 65.516

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[2] FAC 7 6

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Page 17

Question E40 Ê 1 2ˆ If A = Á then AT A is given by: ˜ Ë -3 4¯

A

Ê 5 5ˆ ÁË -5 20˜¯

B

Ê -2 16 ˆ ÁË -9 -2˜¯

C

Ê 1 0ˆ ÁË 0 1˜¯

D

Ê 10 -10ˆ ÁË -10 20 ˜¯

[2] FAC 8 2

Question E41 Ê1 2 ˆ are: The eigenvalues of Á Ë 2 -2˜¯ A B C D

2 and -3 1 and -6 4.37 and -1.37 do not exist

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[2] FAC 8 2

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Page 18

ACET: Practice Exam – Questions

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 2 3 4 5

2458 03899 35677 1448 03

Determine the upper quartile of this sample. A B C D

$4,100 $4,175 $4,325 $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 B C D

5.83 6.29 33.92 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 B C D

0.4317 0.4884 0.5683 0.6775

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[2] SP 5 2

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Page 19

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 B C D

0.001 0.000387 0.0852 0.22

[2] SP 6 4

Question E46 The random variable, X , has probability function: P ( X  0)  0.4

P( X  1)  0.6

Calculate the third-order moment of X about 0.8. A B C D

-0.200 0.2096 0.28 0.6

[2] SP 7 8

Question E47 Given that X ~ Poi (1.2) , calculate P ( X = 2 | X > 0 ) . You are given that the probability function of a Poisson distribution with mean m is: P ( X  x) 

A B C D

x x!

e 

0.217 0.310 0.690 0.783

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[2] SP 8 4

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ACET: Practice Exam – Questions

Question E48 A continuous random variable X has PDF:

f X ( x) = 0.005 x 2e -0.1x for x > 0 Calculate the mode of X . A B C D

0.2 2 20 X has no mode.

[2] SP 9 4

Question E49 The random variable X has an exponential distribution with PDF: f ( x)  0.2e0.2 x

x0

The median of this distribution is: A B C D

0.2 3.466 4.581 5

[2] SP 10 2

Question E50 If X ~ N 120, 25 calculate P  X  113  8  .

P( Z  0.04)  0.51595 , You are given that P ( Z  0.6)  0.72575 and P ( Z  3)  0.99865 . A B C D

0.2098 0.2417 0.41939 0.5779 1

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P ( Z  0.2)  0.57926 ,

[2] SP 11 4

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Page 21

Question E51 You are given that PV =

-6.6

(1 + i )

4

+

2

(1 + i )

3

+

2

(1 + i )

2

+

2 . By trial and error and (1 + i )

interpolation, calculate the value of i to 2 significant figures such that PV = 0 . Evaluate the formula sn = A

sn = 4.246

B

sn = 4.297

C

sn = 4.310

D

sn = 4.375

(1 + i ) n - 1 , using the value of i obtained, where n = 4 . i

[5] FAC 5 5

Question E52 Find and distinguish between the turning points of the function f ( x) = 15 x3 - x5 . A B C D

minima at x = ±1.225 and point of inflexion at x = 0 minimum at x = -3 , point of inflexion at x = 0 and maximum at x = 3 maximum at x = -3 , point of inflexion at x = 0 and minimum at x = 3 maxima at x = ±1.225 , point of inflexion at x = 0 [5] FAC 6 6

Question E53 2

What is Ú 2 x 2e -0.5 x dx ? 1

A B C D

8.437 4.218 2.109 -0.875

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[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, A B C D

E[( X   )3 ]

3

, is:

0.507 0.938 1.113 1.861

[5] SP 7 7

Question E55 The probability density function of a random variable W is given by: f ( w)  kw3 (1  w)

0  x 1

The variance of W is: A B C D

0.005 0.0317 0.476 0.667

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[5] SP 9 6

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