south pacific form seven certificate spfsc statistics q&a.pdfnr x y 3 2 6 0 unistructural 1 0 nr...

South Pacific Form Seven Certificate

MATHEMATICS WITH STATISTICS 2020

INSTRUCTIONS

Write your Student Personal Identification Number (SPIN) in the space provided on the top right-hand corner of this page.

Answer ALL QUESTIONS. Write your answers in the spaces provided in this booklet.

Show all working. Unless otherwise stated, numerical answers correct to three significant figures will be adequate.

If you need more space for answers, ask the Supervisor for extra paper. Write your SPIN on all extra sheets used and clearly number the questions. Attach the extra sheets at the appropriate places in this booklet.

Major Learning Outcomes (Achievement Standards)

Skill Level & Number of Questions Weight/

Time Level 1

Uni-structural

Level 2 Multi-

structural

Level 3 Relational

Level 4 Extended Abstract

Strand 1: Probability Develop knowledge and skills related to probability in order to solve problems and to investigate situations involving elements of chance.

6 2 2 1 20%

60 min

Strand 2: Modelling Using Graphical Methods Model situations using graphical methods in order to solve problems.

6 4 1 0 17%

51 min

Strand 3: Statistical Investigations Carry out statistical investigations and understand statistical processes.

3 2 1 0 10%

30 min

Strand 4: Numerical and Algebraic Methods Use numeric and algebraic methods to solve problems.

2 2 1 1 13%

39 min

TOTAL 17 10 5 2 60%

180 min

Check that this booklet contains pages 2–17 in the correct order and that none of these pages are blank.

HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

QUESTION and ANSWER BOOKLET (2)

Time allowed: Three hours

(An extra 10 minutes is allowed for reading this paper.)

141/1

2

Assessor’s use only

STRAND 1: PROBABILITY

1.1 An experiment consists of throwing two dice at the same time.

Define an outcome of this experiment.

1.2

For question 1.2, circle the letter of the best answer.

Which of the following pair of events is mutually exclusive?

A. A student is tall and is short.

B. A student plays soccer and is tall.

C. A student plays soccer and gets the school dux prize.

D. A student is late to school and travels to school by car.

1.3 A bag contains 2 green balls, 3 yellow balls and 4 red balls.

In how many ways can 3 balls be drawn from the bag if at least 1 yellow

ball is to be included in the draw?

Unistructural

1

0

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Unistructural

1

0

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Relational

3

2

1

0

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3


1.4a State one property of the binomial distribution.

1.4b

For question 1.4b, circle the letter of the best answer.

Which of the following is the parameter of Poisson distribution?

A. Mean, λ

B. Number of trials, n

C. Standard deviation, σ

D. Expected number, Ȇ

1.4c The mean number of bacteria per millilitre (ml) of a liquid is known to be 4.

Assuming that the number of bacteria follows a Poisson distribution, find the probability that in 0.5 ml of liquid there will be at least two bacteria.

Unistructural

1

0

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Unistructural

1

0

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Extended Abstract

4

3

2

1

0

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4


1.5a

For question 1.5a, circle the letter of the best answer.

The expected value of a random variable is the

A. highest value that will occur in the experiment.

B. value that has the highest probability of occurring.

C. mean value over an infinite number of trials.

D. least value that will occur in the experiment.

1.5b Define the term variance.

1.5c

Use the following table to answer questions 1.5c and 1.5d.

A discrete random variable X has the following probability distribution.

X 7 8 9 10

P(X) 0.2 0.3 0.2 0.1

Calculate the expected value of X.

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Unistructural

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Multistructural

2

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5


1.5d Calculate the variance of X.

1.6 The length of life of an instrument produced by a machine is normally distributed with a mean of 12 months and standard deviation of 2 months.

What is the probability that a randomly selected instrument produced by this machine will last less than 9 months?

Multistructural

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Relational

3

2

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6


STRAND 2: MODELLING USING GRAPHICAL METHODS

2.1 A linear function that is derived from a modelling activity is known by

A. the general form y = ax + b

B. the general form y = ax2 + bx + c

C. having the highest power of the variable as 2

D. having the highest power of the variable as 3

2.2 State one property of a continuous function.

2.3a

For question 2.3a, circle the letter of the best answer.

The general equation, 𝑓(𝑥) = 𝑏𝑥 where 𝑏 > 0 represents

A. a power function.

B. a hyperbolic function.

C. an exponential function.

D. a quadratic function.

2.3b Consider the piece-wise function whose graph is shown below.

Write the equation of the function.

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Multistructural

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x

y

1

0

2

1 -1

Unistructural

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7


2.4a Sketch the graph of the function22xy , clearly showing all relevant

intercepts.

2.4b Clearly shade the region whose boundaries are x 0, y 2, and

𝑦 -2𝑥 + 6

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𝑦

x

y

𝑥 3

2

6

0

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8


2.5 Sketch the graph of1

2)(

x

xxf , clearly showing the intercepts and

asymptotes.

𝑦

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𝑥

9


2.6a A function is given by the equation 𝐶(𝑡) = 100𝑒−𝑡

Find the value of 𝑡 when 𝐶(𝑡) = 10

2.6b Use logarithms to solve 102𝑥 = 45

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Multistructural

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10


2.7a Define the term constraints in statistics.

2.7b A shop owner can spend at most $100000 on desktops and laptops in a

month. A desktop costs the shop owner $1000 and a laptop costs $1500.

Each desktop is sold for a profit of $400 while a laptop is sold for a profit

of $700. The shop owner estimates that at least 15 desktops but no more

than 80 are sold each month. He also estimates that the number of

laptops sold is at most half the number of desktops sold.

Let: 𝑥 = number of desktops; and

𝑦 = number of laptops

We are interested in the number of desktops and number of laptops to be

sold to maximise the profit.

State the mathematical function that needs to be maximised.

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Unistructural

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STRAND 3: STATISTICAL INVESTIGATIONS

3.1a

Which of the following features does not apply to a scatter plot?

A. used to determine whether a relationship is linear or not

B. used to represent a large body of data

C. allows the user to identify an outlier

D. works well with a small data set

3.1b There are many methods of sampling that statisticians use. Two common methods are random sampling and stratified sampling.

State one method of sampling.

3.2 Identify the outlier in the data set given below.

28, 26, 29, 31, 82, 33, 37, 20, 30, 35, 22

Unistructural

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Unistructural

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3.3 The relationship between the number of students and their minutes of homework is shown below.

Based on the above representation, describe the strength of the relationship between the number of students and their minutes of homework.

Multistructural

2

1

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Min

ute

s o

f H

om

ew

ork

10 30 20

Number of Students

10

20

30

40

0 40

50

50

13


3.4 A random sample of 200 watermelons of a particular variety on a farm has

a mean weight of 6.5 kg and standard deviation of 1.5 kg.

Calculate the margin of error for a 95% confidence level.

3.5 A random sample of Form Seven students were asked if they prefer

solving math problems using a pen or a pencil. Of the 250 students

surveyed, 100 preferred a pencil and 150 preferred a pen.

Calculate a 99% confidence interval for the proportion of Form Seven

students who prefer to solve their math problems with a pen.

Relational

3

2

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Multistructural

2

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14


STRAND 4: NUMERICAL AND ALGEBRAIC METHODS

4.1 By inspection, how many solutions does the following system have?

6𝑥 − 2𝑦 = 8

3𝑥 − 2𝑦 = 5

DO NOT SOLVE THE SYSTEM

4.2 Give one advantage of using the Newton-Raphson method to approximate

a root of a function.

4.3 A hyperbolic function is given below.

1

4)(

xxf

Describe how the function, 𝑓(𝑥), has no root in the interval [0, 2]

Unistructural

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Multistructural

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4.4 Use the elimination method to solve the following system of linear equations.

5𝑥 − 7𝑦 = 19

5𝑥 − 3𝑦 = 11

Multistructural

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16


4.5 In a local festival, a stall sells bags of popcorn in three sizes: small, medium

and large. The price of each size of popcorn is $3, $5 and $7 respectively.

On a particular night, the stall owner sold a total of 15 bags and the total

amount of money he made was $77. He sold 2 more medium bags than small

bags.

Let: 𝑥 = the number of small bags of popcorn sold;

𝑦 = the number of medium bags of popcorn sold; and

𝑧 = the number of large bags of popcorn sold.

Write down a system of simultaneous equations that represents this

information.

DO NOT ATTEMPT TO SOLVE THE SYSTEM

Relational

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2

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17


THE END

4.6 Let xexxf 52)( 2 ; then xexxf 4)( . With starting point 3.50x ,

use the Newton-Raphson method to compute 1x , 2x , 3x and 4x .

Extended Abstract

4

3

2

1

0

NR

south pacific form seven certificate spfsc statistics q&a.pdfnr x y 3 2 6 0 unistructural 1 0 nr...

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