REVISION LECTURE

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REVISION LECTURE
MATHEMATICAL METHODS UNITS 3 AND 4
Exam Preparation
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Examinations
EXAMINATION 1 - Facts, Skills Short-answer
questions EXAMINATION 2 - Multiple Choice and
Analysis Task
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Examination Advice
General Advice

• Answer questions to the required degree of
  accuracy.
• If a question asks for an exact answer then a
  decimal approximation is not acceptable.
• When an exact answer is required, appropriate
  working must be shown.

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Examination Advice
General Advice

• When an instruction to use calculus is stated for
  a question, an appropriate derivative or
  antiderivative must be shown.
• Label graphs carefully coordinates for
  intercepts and stationary points equations for
  asymptotes.
• Pay attention to detail when sketching graphs.

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Examination Advice
General Advice

• Marks will not be awarded to questions worth more
  than one mark if appropriate working is not shown.

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Examination Advice
Notes Pages

• Well-prepared and organised into topic areas.
• Prepare notes throughout the year.
• Include process steps rather than just specific
  examples of questions.

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Examination Advice
Notes Pages

• Some worked examples can certainly be of benefit.
• Include key steps for using your graphic
  calculator for specific purposes.
• Be sure that you know the syntax to use with your
  calculator

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Examination Advice
Strategy - Examination 1

• Use the reading time to plan an approach for the
  paper.
• Make sure that you answer each question. There is
  no penalty for incorrect answers.
• It may be sensible to obtain the working marks.

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Examination Advice
Strategy - Examination 1

• Questions generally require only one or two steps
  however, you should still expect to do some
  calculations.

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Examination Advice
Strategy - Examination 2

• Use the reading time to carefully plan an
  approach for the paper.
• Momentum can be built early in the exam by
  completing the questions for which you feel the
  most confident.
• Read each question carefully and look for key
  words and constraints.

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Examination Advice
Strategy - Examination 2

• If you find you are spending too much time on a
  question, leave it and move on to the next.
• When a question says to show that a certain
  result is true, you can use this information to
  progress through to the next stage of the
  question.

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Revision Quiz
1 2 3
4 5 6
7 8 9
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Question 1
The derivative of
is equal to
a)
b)
c)
d)
e)
A
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Question 2
The range of the function with graph as shown is
B
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Question 3
Angie notes that 2 out of 10 peaches on her peach
tree are spoilt by birds pecking at them. If she
randomly picks 30 peaches the probability that
exactly 10 of them are spoilt is equal to
D
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Question 4
The total area of the shaded region shown is
given by
a)
b)
c)
d)
D
e)
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Question 5
What does V.C.A.A. stand for?
a) Vice-Chancellors Assessment Authority
b) Victorian Curriculum and Assessment Authority
c) Victorian Combined Academic Authority
d) Victorian Certificate of Academic Aptitude
e) None of the above
B
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Question 6
Which one of the following sets of statements is
true?
A
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Bonus Prize!!
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Question 8
B
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Question 9
a)
b)
E
c)
d)
e)
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EXAMINATION 1 - FACTS, SKILLS AND APPLICATIONS
TASK

• Part A
• 27 multiple-choice questions (27 marks)
• Part B
• short-answer questions (23 marks)
• Time limit
• 15 minutes reading time
• 90 minutes writing time

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EXAMINATION 2 - ANALYSIS TASK

• Extended response questions
• 4 questions (55 marks)
• Time limit
• 15 minutes reading time
• 90 minutes writing time

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Question 1
ANSWER B
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Question 4
fully
a) Expand
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b)
is exactly divisible by
Find the value of a.
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Question 5
a)
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b)
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Question 6
ANSWER B
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Question 7
ANSWER D
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Functions and Their Graphs
Vertical line test - to determine whether a
relation is a function
A represents the DOMAIN
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Interval Notation
Square brackets included Round brackets ( )
excluded
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Question 9
The range of the function with graph as shown is
ANSWER D
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Maximal (or implied) Domain
The largest possible domain for which the
function is defined
A function is undefined when a) The denominator
is equal to zero b) The square root of a negative
number is present.
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Consider the function
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Question 10
This question requires EVERY option to be checked
carefully.
a) b) c) d) e)
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ANSWER E
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Question 11
The graph shown could be that of the function f
whose rule is
ANSWER A
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Using Transformations
When identifying the type of transformation that
has been applied to a function it is essential to
state each of the following
NATURE - Translation, Dilation,
Reflection MAGNITUDE (or size) DIRECTION
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1. Translations a) Parallel to the x-axis
horizontal translation. b) Parallel to the
y-axis vertical translation.
To avoid mistakes, let the bracket containing x
equal zero and then solve for x. If the solution
for x is positive move the graph x units to the
RIGHT. If the solution for x is negative move
the graph x units to the LEFT.
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• 2. Dilations
• a) Parallel to the y-axis the dilation factor
  is the number outside the brackets. This can also
  be described as a dilation from the x-axis.
• Parallel to the x-axis the dilation factor is
  the reciprocal of the coefficient of x. This can
  also be described as a dilation from the y-axis.

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3. Reflections
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Reflection about the x-axis
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Reflection about the y-axis
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Reflection about both axes
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Question 13
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Reflection about the x-axis
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ANSWER A
Translation of 1 unit parallel to the y-axis
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EXTRA QUESTION
ANSWER E
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Transform f(x) to g(x)
Question 15
Dilation by a factor of 0.5 from the
y-axis Dilation by a factor of 2 from the x-axis
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Graphs of Rational Functions
Question 16
The equations of the horizontal and vertical
asymptotes of the graph with equation
ANSWER E
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Inverse Functions
Key features
Domain and range are interchanged Reflection
about the line y x The original function must
be one-to-one
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To find the equation of an inverse function
Step 1 Complete a Function, Domain, Range (FDR)
table. Step 2 Interchange x and y in the given
equation. Step 3 Transpose this equation to make
y the subject. Step 4 Express the answer clearly
stating the rule and the domain.
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Question 17
ANSWER A
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Question 18
Graph of the inverse function
ANSWER C
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Question 20
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Label asymptotes
Approach asymptotes
Label coordinates
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Question 21
The equation relating x and y is most likely
ANSWER E
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Question 22
ANSWER B
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Solving indicial equations
Step 1 Use appropriate index laws to reduce both
sides of the equation to one term. Step
2 Manipulate the equation so that either the
bases or the powers are the same. Step 3 Equate
the bases or powers. If this is not possible then
take logarithms of both sides to either base 10
or base e.
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Question 23
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Solving logarithmic equations
Step 1 Use the logarithmic laws to reduce the
given equation to two terms one on each side of
the equality sign. Step 2 Convert the
logarithmic equation to indicial form. Step
3 Manipulate the given equation so that either
the bases or the powers are the same.
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Step 4 Equate the bases or powers. If this is
not possible then take logarithms of both sides
to either base 10 or base e. Step 5 Check to
make sure that the solution obtained does not
cause the initial function to be undefined.
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Question 26
ANSWER A
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Question 27
ANSWER D
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Circular (Trigonometric) Functions
Amplitude a
Horizontal translation c units in the negative
x-direction
Vertical translation d units in the positive
y-direction
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Question 29
ANSWER C
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Question 30
Amplitude 2
ANSWER B
Range
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Question 31
ANSWER C
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Question 32
ANSWER C
Dilation of factor 2 from the x-axis
Reflection in the x-axis
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Solving Trigonometric Equations

• Put the expression in the form sin(ax) B
• Check the domain modify as necessary.
• Use the CAST diagram to mark the relevant
  quadrants.
• Solve the angle as a first quadrant angle.
• Use symmetry properties to find all solutions in
  the required domain.
• Simplify to get x by itself.

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Question 33
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Question 34
ANSWER E
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Question 35
ANSWER E
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Question 36
Analysis Question
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b)
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DIFFERENTIAL CALCULUS
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Further Rules of Differentiation
Square Root Functions
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Further Rules of Differentiation
Trigonometric Functions
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Further Rules of Differentiation
Logarithmic Functions
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Examples
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Further Rules of Differentiation
Exponential Functions
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Examples
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Question 37
ANSWER D
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Question 39
ANSWER A
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Graphs of Derived Functions
Question 40
ANSWER A
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Question 42
ANSWER C
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Question 43
ANSWER B
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Approximations
Question 44
ANSWER B
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Question 46
Analysis Question
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c) Use CALCULUS to find the EXACT values of the
COORDINATES of the turning point.
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d) i)
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ii)
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Antidifferentiation and Integral Calculus
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Examples
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Question 47
ANSWER E
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Rules of Antidifferentiation
Trigonometric Functions
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Rules of Antidifferentiation
Exponential Functions
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Rules of Antidifferentiation
Logarithmic Functions
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Examples
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Definite Integrals
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Example
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Properties of Definite Integrals
Question 49
ANSWER D
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EXTRA QUESTION
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Integration by recognition
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Question 50
ANSWER B
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Question 52
On the interval (a, b) the gradient of g(x) is
positive.
ANSWER B
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Calculating Area

• Sketch a graph of the function, labelling all
  x-intercepts.
• Shade in the region required.
• Divide the area into parts above the x-axis and
  parts below the x-axis.
• Find the integral of each of the separate
  sections, using the x-intercepts as the terminals
  of integration.
• Subtract the negative areas from the positive
  areas to obtain the total area.

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Question 53
The total area of the shaded region is given by
ANSWER C
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Question 54
The total area bounded by the curve and the
x-axis is given by
ANSWER D
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Question 55
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b) Hence, find the exact area of the shaded region
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Area between curves
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Method

• Sketch the curves, locating the points of
  intersection.
• Shade in the required region.
• If the terminals of integration are not given
  use the points of intersection.
• Check to make sure that the upper curve remains
  as the upper curve throughout the required
  region. If this is not the case then the area
  must be divided into separate sections.
• Evaluate the area.

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The area of the shaded region is given by
BOS 1997 CAT 2 Q. 18
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Question 56
Find the exact area of the shaded region
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Numerical techniques for finding area
Question 57
ANSWER A
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Question 58
Analysis Question
a)
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b) i)
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b) ii)
A repeated root at x -1 indicates that the
normal is a tangent to the curve at this point.
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c) i)
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c) i)
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c) ii)
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Discrete Random Variables
A discrete random variable takes only distinct or
discrete values and nothing in between.Discrete
variables are treated using either discrete,
binomial or hypergeometric distributions. A
continuous random variable can take any value
within a given domain. These values are usually
obtained through measurement of a
quantity.Continuous variables are treated using
normal distributions.
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Expected value and expectation theorems
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Variance and Standard Deviation
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Question 60
Melissa constructs a spinner that will fall onto
one of the numbers 1 to 5 with the following
probabilities.
The mean and standard deviation of the number
that the spinner falls onto are, correct to two
decimal places,
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x
1 0.3 0.3 0.3
2 0.2 0.4 0.8
3 0.1 0.3 0.9
4 0.1 0.4 1.6
5 0.3 1.5 7.5
2.9 11.1
ANSWER E
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The Binomial Distribution
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Question 61
In a two-week period of ten school days, the
probability that the traffic lights have been
green on exactly nine occasions is
ANSWER A
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Question 63
ANSWER A
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The Hypergeometric Distribution
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Question 64
A team of four is selected from six women and
four men. What is the probability that the team
consists of exactly one woman and three men.
ANSWER A
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Question 65
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Calculator program
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The Normal Distribution
The mean, mode and median are the same. The total
area under the curve is one unit.
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Same m Different s
Same s Different m
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Question 67
Which one of the following sets of statements is
true?
ANSWER A
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Method
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Using the cumulative normal distribution table
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Question 68
The mass of fruit jubes, in a packet labelled as
containing 200 grams, has been found to be
normally distributed with a mean of 205 grams and
a standard deviation of 4 grams. The percentage
of packets that contain less than 200 grams is,
correct to one decimal place,
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ANSWER C
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Question 71
The eggs laid by a particular breed of chicken
have a mass which is normally distributed with a
mean of 61 g and a standard deviation of 2.5 g.
The probability, correct to four decimal places,
that a single egg has a mass between 60 g and 65
g is
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ANSWER C
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Applications of the normal distribution
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Question 72
Black Mountain coffee is sold in packets labeled
as being of 250 grams weight. The packing process
produces packets whose weight is normally
distributed with a standard deviation of 3 grams.
In order to guarantee that only 1 of packets are
under the labeled weight, the actual mean weight
(in grams) would be required to be closest to
a) 243 b) 247 c) 250 d) 254 e) 257
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ANSWER E
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Question 74
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Analysis Question
Question 75
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c)
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d)
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Conditional probability
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g)
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THE FINAL RESULT