Probability, NAP-L4-CA08
Probability, NAP-L3-CA09
Number, NAP-L3-CA08
Kelly measures the length of fish she catches for her research.
Which fish has a length closest to 25 cm?
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Complex Numbers, EXT2 N2 2019 HSC 16b
Number, NAP-L4-CA05
At sunrise the temperature is 4°C.
At midday the temperature is 32°C.
Which one of these calculations can be used to work out how many degrees warmer it is at midday than at sunrise?
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Number, NAP-L4-CA03
Candice buys an apple for 75 cents and a banana for 70 cents.
She pays with a $2 coin.
How much change should Candice get?
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Number, NAP-L4-CA04
In 2015, some wilderness parks in Tasmania lost up to
What is
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Number, NAP-L4-CA07
The table shows the fractions of the Australian workforce in some industries.
Which of these industries has the least number of employees in the workforce?
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Automotive |
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Finance |
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Health Care |
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Telecommunications |
Number, NAP-L3-CA07
The table shows the fractions of the Australian workforce in some industries.
Which of these industries has the least number of employees in the workforce?
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Automotive |
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Finance |
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Health Care |
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Telecommunications |
Geometry, NAP-L3-CA06
Blackbeard finds part of a treasure map.
The palm tree is at I5.
The skull is at G4.
Where is the treasure chest?
E1 | F2 | D3 | H3 |
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Combinatorics, EXT1′ S1 2019 HSC 10 MC
An access code consists of 4 digits chosen from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The code will only work if the digits are entered in the correct order.
Some access codes contain exactly two different digits, for example 3377 or 5155.
How many such access codes can be made using exactly two different digits?
- 630
- 900
- 1080
- 2160
Calculus, EXT1 C2 2019 HSC 14c
The diagram shows the two curves
- Explain why
. (1 mark)
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- Show that
. (2 marks)
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- Hence, or otherwise, find
in terms of . (2 marks)
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Proof, EXT2 P2 SM-Bank 5
Use mathematical induction to prove that
is divisible by 4 for integers
Algebra, STD1 A2 2019 HSC 33
The relationship between British pounds
- Write the direct variation equation relating British pounds to Australian dollars in the form
. Leave as a fraction. (1 mark)
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- The relationship between Japanese yen
and Australian dollars on the same day is given by the equation .
Convert 93 100 Japanese yen to British pounds. (2 marks)
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Financial Maths, STD1 F3 2019 HSC 32
Ashley has a credit card with the following conditions:
- There is no interest-free period.
- Interest is charged at the end of each month at 18.25% per annum, compounding daily, from the purchase date (included) to the last day of the month (included).
Ashley's credit card statement for April is shown, with some figures missing.
The minimum payment is calculated as 2% of the closing balance on 30 April.
Calculate the minimum payment. (3 marks)
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Measurement, STD1 M3 2019 HSC 31
Algebra, STD1 A3 2019 HSC 9 MC
The container shown is initially full of water.
Water leaks out of the bottom of the container at a constant rate.
Which graph best shows the depth of water in the container as time varies?
A. | B. | ||
C. | D. |
Statistics, STD1 S1 2019 HSC 7 MC
Functions, EXT1 F1 2019 HSC 10 MC
Calculus, 2ADV C4 2019 HSC 16c
The diagram shows the region
- Show that the tangent to the curve at
meets the -axis at
. (2 marks)
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- Using the result of part (i), or otherwise, show that the area of the region
is
. (2 marks)
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- Find the exact value of
for which the area of is a maximum. (3 marks)
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Financial Maths, 2ADV M1 2019 HSC 16a
A person wins $1 000 000 in a competition and decides to invest this money in an account that earns interest at 6% per annum compounded quarterly. The person decides to withdraw $80 000 from this account at the end of every fourth quarter. Let
- Show that the amount remaining in the account after the withdrawal at the end of the eighth quarter is
. (2 marks)
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- For how many years can the full amount of $80 000 be withdrawn? (3 marks)
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Networks, STD2 N3 2019 HSC 40
A museum is planning an exhibition using five rooms.
The museum manager draws a network to help plan the exhibition. The vertices
- What is the capacity of the cut shown? (1 mark)
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- The museum manager is planning for a maximum of 240 visitors to pass through the exhibition each hour. By using the 'minimum cut-maximum flow' theorem, the manager determines that the plan does not provide sufficient flow capacity.
Draw the minimum cut onto the network below and recommend a change that the manager could make to one or more security checkpoints to increase the flow capacity to 240 visitors per hour. (2 marks)
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Statistics, STD2 S5 2019 HSC 15 MC
The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6. Michael received a score on the examination between the lower quartile and the upper quartile of the scores.
Which shaded region most accurately represents where Michael's score lies?
A. | B. | ||
C. | D. |
Statistics, STD2 S1 2019 HSC 10 MC
Financial Maths, STD2 F1 2019 HSC 9 MC
What is the interest earned, in dollars, if $800 is invested for
Statistics, EXT1 S1 2012 MET2 3
Steve and Jess are two students who have agreed to take part in a psychology experiment. Each has to answer several sets of multiple-choice questions. Each set has the same number of questions,
- Steve decides to guess the answer to every question, so that for each question he chooses A, B, C or D at random.
Let the random variable
be the number of questions that Steve answers correctly in a particular set.- What is the probability that Steve will answer the first three questions of this set correctly? (1 mark)
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- Use the fact that the variance of
is to show that the value of is 25. (1 mark)
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- What is the probability that Steve will answer the first three questions of this set correctly? (1 mark)
- The probability that Jess will answer any question correctly, independently of her answer to any other question, is
. Let the random variable be the number of questions that Jess answers correctly in any set of 25.If
, show that the value of . (2 marks)
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Vectors, EXT1 V1 2018 SPEC2 12 MC
If
A.
B.
C.
D.
Calculus, MET1 2018 VCAA 9
Consider a part of the graph of
- i. Given that
, evaluate when is a positive even integer or 0.
Give your answer in simplest form. (2 marks)
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ii. Given that
, evaluate when is a positive odd integer.
Give your answer in simplest form. (1 mark)
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- Find the equation of the tangent to
at the point . (2 marks)
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- The translation
maps the graph of onto the graph of , where - and
is a real constant. - State the value of
. (1 mark)
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- Let
and .
The line is the tangent to the graph of at the point and the line is the tangent to the graph of at , as shown in the diagram below.
Find the total area of the shaded regions shown in the diagram above. (2 marks)
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Statistics, 2ADV S3 SM-Bank 4
The continuous random variable
Find the value of
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Calculus, SPEC2 2011 VCAA 16 MC
The gradient of the the perpendicular line to a curve at any point
The coordinate of points on the curve satisfy the differential equation
Graphs, SPEC1 2012 VCAA 10
Consider the functions with rules
-
- Find the maximal domain of
(1 mark)
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- Find the maximal domain of
(1 mark)
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- Find the largest set of values of
for which is defined. (1 mark)
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- Find the maximal domain of
- Given that
and that , evaluateGive your answer in the form
(3 marks)
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Mechanics, SPEC2 2015 VCAA 16 MC
Vectors, SPEC2 2017 VCAA 5
On a particular morning, the position vectors of a boat and a jet ski on a lake
- On the diagram above, mark the initial positions of the boat and the jet ski, clearly identifying each of them. Use arrows to show the directions in which they move. (2 marks)
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-
- Find the first time for
when the speeds of the boat and the jet ski are the same. (2 marks)
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- State the coordinates of the boat at this time. (1 mark)
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- Find the first time for
-
- Write down an expression for the distance between the jet ski and the boat at any time
. (1 mark)
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- Find the minimum distance separating the boat and the jet ski. Give your answer in kilometres, correct to two decimal places. (1 mark)
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- Write down an expression for the distance between the jet ski and the boat at any time
- On another morning, the boat’s position vector remained the same but the jet skier considered starting from a different location with a new position vector given by
, where is a real constant. Both vessels are to start at the same time.
Assuming the vessels would collide shortly after starting, find the time of the collision and the value of . (3 marks)
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Complex Numbers, SPEC2 2017 VCAA 4
- Express
in polar form. (1 mark)
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- Show that the roots of
are and . (1 mark)
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- Express the roots of
in terms of . (1 mark)
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- Show that the cartesian form of the relation
is (2 marks)
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- Sketch the line represented by
and plot the roots of on the Argand diagram below. (2 marks)
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- The equation of the line passing through the two roots of
can be expressed as , where .Find
in terms of . (1 mark)
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- Find the area of the major segment bounded by the line passing through the roots of
and the major arc of the circle given by . (2 marks)
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Complex Numbers, SPEC2-NHT 2017 VCAA 2
One root of a quadratic equation with real coefficients is
-
- Write down the other root of the quadratic equation. (1 mark)
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- Hence determine the quadratic equation, writing it in the form
. (2 marks)
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- Write down the other root of the quadratic equation. (1 mark)
- Plot and label the roots of
on the Argand diagram below. (3 marks)
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- Find the equation of the line that is the perpendicular bisector of the line segment joining the origin and the point
. Express your answer in the form . (2 marks)
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- The three roots plotted in part b. lie on a circle.
Find the equation of this circle, expressing it in the form
, where . (3 marks)
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Calculus, SPEC2-NHT 2018 VCAA 5
A horizontal beam is supported at its endpoints, which are 2 m apart. The deflection
- Given that both the inclination,
, and the deflection, , of the beam from the horizontal at are zero, use the differential equation above to show that . (2 marks)
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- Find the angle of inclination of the beam to the horizontal at the origin
. Give your answer as a positive acute angle in degrees, correct to one decimal place. (2 marks)
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- Find the value of
, in metres, where the maximum deflection occurs, and find the maximum deflection, in metres. (3 marks)
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- Find the maximum angle of inclination of the beam to the horizontal in the part of the beam where
. Give your answer as a positive acute angle in degrees, correct to one decimal place. (2 marks)
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Statistics, SPEC2 2018 VCAA 6
The heights of mature water buffaloes in northern Australia are known to be normally distributed with a standard deviation of 15 cm. It is claimed that the mean height of the water buffaloes is 150 cm.
To decide whether the claim about the mean height is true, rangers selected a random sample of 50 mature water buffaloes. The mean height of this sample was found to be 145 cm.
A one-tailed statistical test is to be carried out to see if the sample mean height of 145 cm differs significantly from the claimed population mean of 150 cm.
Let
- State suitable hypotheses
and for the statistical test. (1 mark)
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- Find the standard deviation of
. (1 mark)
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- Write down an expression for the
value of the statistical test and evaluate your answer to four decimal places. (2 marks)
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- State with a reason whether
should be rejected at the 5% level of significance. (1 mark)
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- What is the smallest value of the sample mean height that could be observed for
to be not rejected? Give your answer in centimetres, correct to two decimal places. (1 mark)
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- If the true mean height of all mature water buffaloes in northern Australia is in fact 145 cm, what is the probability that
will be accepted at the 5% level of significance? Give your answer correct to two decimal places. (1 mark)
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- Using the observed sample mean of 145 cm, find a 99% confidence interval for the mean height of all mature water buffaloes in northern Australia. Express the values in your confidence interval in centimetres, correct to one decimal place. (1 mark)
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Calculus, SPEC1 2015 VCAA 9
Consider the curve represented by
- Find the gradient of the curve at any point
(2 marks) - Find the equation of the tangent to the curve at the point
and find the equation of the tangent to the curve at the pointWrite each equation in the form
(2 marks) - Find the acute angle between the tangent to the curve at the point
and the tangent to the curve at the pointGive your answer in the form
, where is a real constant (2 marks)
Calculus, SPEC1 2014 VCAA 7
Consider
- Write down the range of
. (1 mark)
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- Show that
. (1 mark)
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- Hence evaluate the area enclosed by the graph of
, the -axis and the lines and . (3 marks)
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Calculus, MET1 2018 VCAA 8
Let
- Show that
. (1 mark)
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- Find the value of
for which the graphs of and have exactly one point of intersection. (3 marks)
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Let
Let
- Write down a definite integral that gives the value of
. (1 mark)
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- Using your result from part a., or otherwise, find the value of
such that . (3 marks)
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Algebra, MET2 2018 VCAA 20 MC
The differentiable function
The transformation
The transformation
Algebra, MET2 2018 VCAA 18 MC
Consider the functions
If
A.
B.
C.
D.
E.
Financial Maths, STD2 2014 HSC 27a
Alex is buying a used car which has a sale price of $13 380. In addition to the sale price there are the following costs:
- Stamp Duty for this car is calculated at $3 for every $100, or part thereof, of the sale price.
- Calculate the Stamp Duty payable. (1 mark)
- Alex wishes to take out comprehensive insurance for the car for 12 months. The cost of comprehensive insurance is calculated using the following:
- Find the total amount that Alex will need to pay for comprehensive insurance. (3 marks)
- Alex has decided he will take out the comprehensive car insurance rather than the less expensive non-compulsory third-party car insurance.
- What extra cover is provided by the comprehensive car insurance? (1 mark)
Graphs, MET2 2018 VCAA 11 MC
The graph of
The value of
GEOMETRY, FUR2 2018 VCAA 3
Frank owns a tennis court.
A diagram of his tennis court is shown below
Assume that all intersecting lines meet at right angles.
Frank stands at point
- What is the straight-line distance, in metres, between point
and point ?
Round your answer to one decimal place. (1 mark)
- Frank hits a ball when it is at a height of 2.5 m directly above point
.Assume that the ball travels in a straight line to the ground at point
.
What is the straight-line distance, in metres, that the ball travels?
Round your answer to the nearest whole number. (1 mark)
Frank hits two balls from point
For Frank’s first hit, the ball strikes the ground at point
For Frank’s second hit, the ball strikes the ground at point
Point
Point
The angle,
-
- Determine two possible values for angle
.
Round your answers to one decimal place. (1 mark)
- If point
is within the boundary of the court, what is the value of ?
Round your answer to the nearest metre. (1 mark)
- Determine two possible values for angle
NETWORKS, FUR1 2018 VCAA 8 MC
Annie, Buddhi, Chuck and Dorothy work in a factory.
Today each worker will complete one of four tasks, 1, 2, 3 and 4.
The usual completion times for Annie, Chuck and Dorothy are shown in the table below.
Buddhi takes 3 minutes for Task 3.
He takes
Today the factory supervisor allocates the tasks as follows
-
- Task 1 to Dorothy
- Task 2 to Annie
- Task 3 to Buddhi
- Task 4 to Chuck
This allocation will achieve the minimum total completion time if the value of
- 0
- 1
- 2
- 3
- 4
NETWORKS, FUR2 2018 VCAA 4
Parcel deliveries are made between five nearby towns,
The roads connecting these five towns are shown on the graph below. The distances, in kilometres, are also shown.
A road inspector will leave from town
- How many roads will the inspector have to travel on more than once? (1 mark)
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- Determine the minimum distance, in kilometres, that the inspector will travel. (1 mark)
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CORE, FUR2 2018 VCAA 6
Julie has retired from work and has received a superannuation payment of $492 800.
She has two options for investing her money.
Option 1
Julie could invest the $492 800 in a perpetuity. She would then receive $887.04 each fortnight for the rest of her life.
- At what annual percentage rate is interest earned by this perpetuity? (1 mark)
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Option 2
Julie could invest the $492 800 in an annuity, instead of a perpetuity.
The annuity earns interest at the rate of 4.32% per annum, compounding monthly.
The balance of Julie’s annuity at the end of the first year of investment would be $480 242.25
-
- What monthly payment, in dollars, would Julie receive? (1 mark)
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-
How much interest would Julie’s annuity earn in the second year of investment?
-
Round your answer to the nearest cent. (1 mark)
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- What monthly payment, in dollars, would Julie receive? (1 mark)
CORE, FUR2 2018 VCAA 3
Table 3 shows the yearly average traffic congestion levels in two cities, Melbourne and Sydney, during the period 2008 to 2016. Also shown is a time series plot of the same data.
The time series plot for Melbourne is incomplete.
- Use the data in Table 3 to complete the time series plot above for Melbourne. (1 mark)
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- A least squares line is used to model the trend in the time series plot for Sydney. The equation is
- i. Draw this least squares line on the time series plot. (1 mark)
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- ii. Use the equation of the least squares line to determine the average rate of increase in percentage congestion level for the period 2008 to 2016 in Sydney. (1 mark)
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iii. Use the least squares line to predict when the percentage congestion level in Sydney will be 43%. (1 mark)
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The yearly average traffic congestion level data for Melbourne is repeated in Table 4 below.
- When a least squares line is used to model the trend in the data for Melbourne, the intercept of this line is approximately –1514.75556
- Round this value to four significant figures. (1 mark)
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- Use the data in Table 4 to determine the equation of the least squares line that can be used to model the trend in the data for Melbourne. The variable year is the explanatory variable.
- Write the values of the intercept and the slope of this least squares line in the appropriate boxes provided below.
- Round both values to four significant figures. (2 marks)
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congestion level = |
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+ |
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× year |
- Since 2008, the equations of the least squares lines for Sydney and Melbourne have predicted that future traffic congestion levels in Sydney will always exceed future traffic congestion levels in Melbourne.
Explain why, quoting the values of appropriate statistics. (2 marks)
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Trigonometry, 2ADV T1 SM-Bank 1
A tower is built on flat ground.
Three tourists,
The angles of elevation to the top of the tower from
What is the bearing of
Calculus, SPEC2 2017 VCAA 10 MC
A function
and
The coordinates of any points of inflection of
Functions, 2ADV F1 SM-Bank 11
Given
- Find integers
and such that (2 marks)
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- State the domain for which
is defined. (2 marks)
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Calculus, 2ADV C4 2007* HSC 10a
An object is moving on the
- The object is initially at the origin. During which time(s) is the displacement of the object decreasing? (1 mark)
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- If the object travels 7 units in the first 4 seconds, estimate the time at which the object returns to the origin. Justify your answer. (2 marks)
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- Sketch the displacement,
, as a function of time. (2 marks)
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Calculus, 2ADV C4 2013* HSC 15a
The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m apart. The height of each outer pole is 1.5 m, and the height of the middle pole is 1.8 m. The roof hangs between the poles.
The front of the tent has area
- Use the trapezoidal rule to estimate
. (1 mark)
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- Does the Trapezoidal rule give a higher or lower estimate of the actual area? Justify your answer. (1 mark)
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Probability, 2ADV S1 2007 MET1 6
Two events,
- Calculate
when . (1 mark)
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- Calculate
when and are mutually exclusive events. (1 mark)
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Probability, 2ADV S1 2016 MET2 19 MC
Consider the discrete probability distribution with random variable
The smallest and largest possible values of
Measurement, STD2 M6 2011 HSC 24c
A ship sails 6 km from
size of angle
- What is the bearing of
from ? (1 mark)
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- Find the distance
. Give your answer correct to the nearest kilometre. (2 marks)
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- What is the bearing of
from ? Give your answer correct to the nearest degree. (3 marks)
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Measurement, STD2 M1 2017 HSC 29a*
A new 200-metre long dam is to be built.
The plan for the new dam shows evenly spaced cross-sectional areas.
- Using the Trapezoidal rule, show that the volume of the dam is approximately 44 500 m³. (2 marks)
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- It is known that the catchment area for this dam is 2 km².
Assuming no wastage, calculate how much rainfall is needed, to the nearest mm, to fill the dam. (2 marks)
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Probability, NAP-J2-34
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