Sample QP - 2014

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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 log x_a n log_a x y Ê ˆ - _{Á ˜} Ë ¯ A (n-1)log x log y_a + _a B nxa -a(x y- ) C log_a xn x y Ê _- ˆ Á ˜ Ë ¯ D log_a xn 1 y -Ê ˆ Á ˜ Ë ¯ [1] FAC 4 1 Question E3 Solve - £ -11 3 2x<15 A - £ <7 x 6 B x≥7 and x< -6 C 7≥ > -x 6 D x£ -7 and x>6 [1] FAC 4 4

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Which of these is the graph of y=xn 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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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 y=5 xwith respect to x. A -5x-2 B 5₂x½ C 5₂x3 2 D 5 2 x [1] FAC 6 3

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Find ( )f x¢ where f x( ) 3₂_x e = . A 3₂ 2e x B 3₂ 2e x -C 6₂_x e -D 6₂_x e [1] FAC 6 4 Question E9 What is 1 4 0 3e x dx

Ú

? A 19.17 B 40.20 C 160.79 D 643.18 [1] FAC 7 2

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What is the integral of 1 1₃ x - x ? A 1 4₄ c x - + B ln 1₄ 4 x c x - + C 1 2₂ c x - + D ln 1₂ 2 x c 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 b-2a is: A 4.24 B 5.83 C 8.12 D 12.37 [1] FAC 8 1 Question E12 If 3 4 2 5 - -Ê ˆ = Á_Ë ˜_¯

B then | |B is given by:

A -23

B -7

C 7

D 23 [1]

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The abbreviation etc means: A for example B 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 frequency density 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]

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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, x 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]

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For independent events A and B, you are given: ( ) 0.5 ( ) 0.3 P A  P B  Determine P A( or )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]

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A discrete random variable X has the following probability function: x 1 2 3 ( ) P X = x 0.45 0.35 0.2 Calculate E 2 X Ê ˆ Á ˜ Ë ¯. A 1.143 B 1.383 C 1.75 D 2 [1] SP 7 5 Question E21

If E X

 

5.2 and sd X

 

4.49, 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:

2 3 64 ( ) 0 4 f y  y  y Calculate (P Y 2). A 0.125 B 0.1875 C 0.8125 D 0.875 [1] SP 9 3

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A continuous random variable X has a mean of 50 and a standard deviation of 8. Calculate var 4 7 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: A 1 3 B 1 4 C 1 3 x -D 4 x [1] SP 10 1

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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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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 Calculate 2.306 1 (5 4.712)2 0.1063 10 2.82596 Ï _- ¸ Ô ₊ Ô_¥ Ì ˝ Ô Ô Ó ˛ . A 0.088 B 0.270 C 0.338 D 0.741 [2] FAC 2 2 Question E28

Solve the quadratic equation 7 4- x x- 2= giving your solutions to 1 DP. 0 A x= -5.3 and 1.3

B x= -0.2 and 0.8

C x= -10.6 and 2.6

D The equation has no real solutions [2] FAC 4 2

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Use the result (1+x)p = +1 px+ p p(_2!-1)x2+ p p( -1)(_3!p-2)x3+  to expand the expression

(

1 x+

)

-1.2as far as the term in x and hence evaluate it when 3 x= -0.4_.

A 0.641088 B 1.358912 C 1.781312 D 1.845944 [2] FAC 4 9 Question E30

Solve for x and y : x2+2y2 =33

3 x y+ = - A x= -21 and y=18 B x= -4 and y = 1 C x=5 and y= - or 2 x= -1 and y= 4 D x= -5 and y= or 2 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 103.995

B 129.793 C 155.752

D 160.752 [2]

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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 x= +3 i is a complex root of the cubic x3-4x2-2x+20 0= , find the other two roots. A x= -3 i or -2 B x= - -

(

3 i

) (

or 2+ i

)

C x= -3 i or 3 D x= - -3 i or -2 [2] FAC 5 7 Question E34 Find dy di where 4 1 (1 i) y i -- + = . A (1 i) 4 4 (1₂i i) 5 1 i - -+ - + -B (1 i) 4 4 (1₂i i) 3 1 i - -+ - + -C (1 i) 4 4 (1₂i i) 5 1 i - -+ + + -D 4 3 2 (1 i) 4 (1i i) 1 i - -+ + + [2] FAC 6 4

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Find the second derivative of M t( )=em(et-1) evaluated at t =0, where m is a constant. A (0)M¢¢ =m B M¢¢(0)=m2 C M¢¢(0)= +m m2 D M¢¢(0)=m2em [2] FAC 6 5 Question E36 What is

(

( )

)

2 2 3 axy b xy c xy x y ∂ ₊ ₊ ∂ ∂ ? A a+2b yx

( )

+3c yx

( )

2 B a+4b yx

( )

+9c yx

( )

2 C 2 3 2 3 yx yx axy b+ Ê ˆ_{Á ˜}_{Ë ¯} +cÊ ˆ_{Á ˜}_{Ë ¯} D 2 3 4 2 3 4 yx yx yx aÊ ˆ_{Á ˜}_{Ë ¯} +bÊ ˆ_{Á ˜}_{Ë ¯} +cÊ ˆ_{Á ˜}_{Ë ¯} [2] FAC 6 7

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What is

Ú

8 (3x x2+2)4dx? A 8 (3 2 2)5 5 x x c + ₊ B 8(3 2 2)5 5 x c + ₊ C 4(3 2 2)5 15 x c + ₊ D 8 ( 3 2)5 5 x x c + _{+ [2]} FAC 7 3 Question E38 What is 5 2 1 1 (3 4 ) x y x y dy dx = = +

Ú Ú

? A ₂₄ 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 x e dx

Ú

? A 4.330 B 6.552 C 13.103 D 65.516 [2] FAC 7 6

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If 1 2 3 4

Ê ˆ

= Á_Ë_- ˜_¯

A then A AT is given by:

A 5 5 5 20 Ê ˆ Á_- ˜ Ë ¯ B 2 16 9 2 -Ê ˆ Á_- _- ˜ Ë ¯ C 1 0 0 1 Ê ˆ Á ˜ Ë ¯ D 10 10 10 20 -Ê ˆ Á_- ˜ Ë ¯ [2] FAC 8 2 Question E41 The eigenvalues of 1 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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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]

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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: (P X 0) 0.4 P X(  1) 0.6

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 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:

( ) ! 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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A continuous random variable X has PDF:

2 0.1

( ) 0.005 x X

f x = x e- for x>0

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 x 0

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 X ~ N



120, 25



calculate P X



113 8



.

You are given that P Z( 0.04) 0.51595 _, P Z( 0.2) 0.57926 , ( 0.6) 0.72575 P Z   and (P Z 3) 0.99865 . A 0.2098 B 0.2417 C 0.41939 D 0.5779 1 [2] SP 11 4

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You are given that

( ) ( ) ( ) ( )

4 3 2 6.6 2 2 2 1 1 1 1 PV i i i i -= + + + +

+ + + . By trial and error and interpolation, calculate the value of i to 2 significant figures such that PV =0. Evaluate the formula s_n (1 i)n 1

i

+

-= , using the value of i obtained, where n=4. A s_n =4.246 B s_n =4.297 C s_n =4.310 D s_n =4.375 [5] FAC 5 5 Question E52

Find and distinguish between the turning points of the function f x( ) 15= x3-x5. A minima at x= ±1.225 and point of inflexion at x=0

B minimum at x= -3, point of inflexion at x=0 and maximum at x=3

C maximum at x= -3, point of inflexion at x=0 and minimum at x=3

D maxima at x= ±1.225, point of inflexion at x=0 [5] FAC 6 6 Question E53 What is 2 2 0.5 1 2x e- x dx

Ú

? A 8.437 B 4.218 C 2.109 D -0.875 [5] FAC 7 3

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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]