What is the Cartesian equation of the line
Complex Numbers, EXT2 N2 2020 HSC 2 MC
Given that
Algebra, STD1 A3 2020 HSC 19
Each year the number of fish in a pond is three times that of the year before.
- The table shows the number of fish in the pond for four years.
Complete the table above showing the number of fish in 2021 and 2022. (2 marks)
- Plot the points from the table in part (a) on the grid. (2 marks)
- Which model is more suitable for this dataset: linear or exponential? Briefly explain your answer. (2 marks)
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Statistics, STD1 S1 2020 HSC 6 MC
When blood pressure is measured, two numbers are recorded: systolic pressure and diastolic pressure. If the measurements recorded are 140 systolic and 90 diastolic, then the blood pressure is written as 140/90 mmHg.
Blood pressure measurements are categorized as shown in the diagram.
Amy has a blood pressure of 97/76 mmHg and Betty has a blood pressure of 125/75 mmHg.
Which row of the table describes the blood pressure for Amy and Betty?
Networks, STD1 N1 2020 HSC 1 MC
Functions, EXT1 F1 2020 HSC 11c
Functions, EXT1 F2 2020 HSC 11a
Let
- Show that
. (1 mark)
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- Hence, factor the polynomial
as , where is a quadratic polynomial. (2 marks)
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Measurement, STD2 M6 2020 HSC 31
Mr Ali, Ms Brown and a group of students were camping at the site located at
- Show that the angle
is 65°. (1 mark)
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- Find the distance
. (2 marks)
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- Find the bearing of Ms Brown's group from Mr Ali's group. Give your answer correct to the nearest degree. (2 marks)
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Vectors, EXT1 V1 2020 HSC 6 MC
The vectors
Which diagram below shows the vector
A. | |
B. | |
C. | |
D. |
Statistics, EXT1 S1 2020 HSC 12b
When a particular biased coin is tossed, the probability of obtaining a head is
This coin is tossed 100 times.
Let
- Find the expected value,
. (1 mark)
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- By finding the variance,
, show that the standard deviation of is approximately 5. (1 mark)
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- By using a normal approximation, find the approximate probability that
is between 55 and 65. (1 mark)
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Proof, EXT1 P1 2020 HSC 12a
Use the principle of mathematical induction to show that for all integers
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Functions, 2ADV F1 2020 HSC 11
There are two tanks on a property, Tank
- Tank
begins to lose water at a constant rate of 20 litres per minute. The volume of water in Tank is modelled by where is the volume in litres and is the time in minutes from when the tank begins to lose water. (1 mark)
On the grid below, draw the graph of this model and label it as Tank.
- Tank
remains empty until when water is added to it at a constant rate of 30 litres per minute.
By drawing a line on the grid (above), or otherwise, find the value of
when the two tanks contain the same volume of water. (2 marks) - Using the graphs drawn, or otherwise, find the value of
(where ) when the total volume of water in the two tanks is 1000 litres. (1 mark)
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Algebra, STD2 A4 2020 HSC 24
There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.
- Tank A begins to lose water at a constant rate of 20 litres per minute.
The volume of water in Tank A is modelled by
where is the volume in litres and is the time in minutes from when the tank begins to lose water.
On the grid below, draw the graph of this model and label it as Tank A. (1 mark)
- Tank B remains empty until
when water is added to it at a constant rate of 30 litres per minute.
By drawing a line on the grid (above), or otherwise, find the value ofwhen the two tanks contain the same volume of water. (2 marks)
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- Using the graphs drawn, or otherwise, find the value of
(where ) when the total volume of water in the two tanks is 1000 litres. (1 mark)
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Calculus, EXT1 C3 2020 HSC 11e
Solve
Vectors, EXT1 V1 2020 HSC 11b
For what values(s) of
Vectors, EXT1 V1 2020 HSC 4 MC
Maria starts at the origin and walks along all of the vector
How far from the origin is she?
Calculus, EXT1 C2 2020 HSC 3 MC
Which of the following is an anti-derivative of
Functions, EXT1 F1 2020 HSC 1 MC
Financial Maths, 2ADV M1 2020 HSC 26
Tina inherits $60 000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withdraws $800.
The amount in the account immediately after the
where
- Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal. (2 marks)
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- Calculate the amount of interest earned in the first three months. (2 marks)
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- Calculate the amount of money in the account immediately after the 94th withdrawal. (3 marks)
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Probability, 2ADV S1 2020 HSC 14
History and Geography are two of the subjects students may decide to study. For a group of 40 students, the following is known.
-
- 7 students study neither History nor Geography
- 20 students study History
- 18 students study Geography
- A student is chosen at random. By a using a Venn diagram, or otherwise, find the probability that the student studies both History and Geography. (2 marks)
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- A students is chosen at random. Given that the student studies Geography, what is the probability that the student does NOT study History? (1 mark)
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- Two different students are chosen at random, one after the other. What is the probability that the first student studies History and the second student does NOT study History? (2 marks)
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Calculus, 2ADV C3 2020 HSC 21
Hot tea is poured into a cup. The temperature of tea can be modelled by
- What is the temperature of the tea 4 minutes after it has been poured? (1 mark)
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- At what rate is the tea cooling 4 minutes after it has been poured? (2 marks)
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- How long after the tea is poured will it take for its temperature to reach 55°C? (3 marks)
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Trigonometry, 2ADV T1 2020 HSC 15
Mr Ali, Ms Brown and a group of students were camping at the site located at
- Show that the angle
is 65°. (1 mark)
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- Find the distance
. (2 marks)
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- Find the bearing of Ms Brown's group from Mr Ali's group. Give your answer correct to the nearest degree. (2 marks)
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Measurement, STD2 M6 2020 HSC 16
Calculus, 2ADV C4 2020 HSC 18
- Differentiate
. (2 marks)
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- Hence, find
. (1 marks)
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Calculus, 2ADV C4 2020 HSC 13
Evaluate
Financial Maths, 2ADV M1 2020 HSC 12
Calculate the sum of the arithmetic series
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Trigonometry, 2ADV T3 2020 HSC 6 MC
Which interval gives the range of the function
Algebra, STD2 A1 2020 HSC 3 MC
The distance between Bricktown and Koala Creek is 75 km. A person travels from Bricktown to Koala Creek at an average speed of 50 km/h.
How long does it take the person to complete the journey?
- 40 minutes
- 1 hour 25 minutes
- 1 hour 30 minutes
- 1 hour 50 minutes
Functions, 2ADV F2 2020 HSC 2 MC
Mechanics, EXT2 M1 EQ-Bank 4
A torpedo with a mass of 80 kilograms has a propeller system that delivers a force of
- Explain why
where is a positive constant. (1 mark)
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- If the torpedo increases its velocity from
to , show that the distance it travels in this time, , is given by
(3 marks)
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Mechanics, EXT2 M1 EQ-Bank 1
A canon ball of mass 9 kilograms is dropped from the top of a castle at a height of
The canon ball experiences a resistance force due to air resistance equivalent to
- Show the equation of motion is given by
(1 mark)
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- Show, by integrating using partial fractions, that
(5 marks)
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- If the canon hits the ground after 4 seconds, calculate the height of the castle, to the nearest metre. (3 marks)
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Complex Numbers, EXT2 N1 SM-Bank 4
- Express
in the form , given (2 marks)
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- Hence, using the quadratic formula, solve
(1 mark)
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Complex Numbers, EXT2 N1 EQ-Bank 3
Find the values of
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Complex Numbers, EXT2 N1 EQ-Bank 2
Find the values of
Algebra, STD2 A2 SM-Bank 11 MC
Algebra, STD2 A1 SM-Bank 15
Fabio drove 300 km in
His average speed for the first 210 km was 70 km per hour.
How long did he take to travel the last 90 km? (2 marks)
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Calculus, EXT1 C3 EQ-Bank 1
- Sketch the region bounded by the curve
and the lines and . (1 mark)
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- Calculate the area of this region. (3 marks)
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Functions, 2ADV F2 EQ-Bank 2 MC
Which diagram best shows the graph
A. | B. | ||
C. | D. |
Functions, 2ADV F2 EQ-Bank 16
- Describe each transformation and state the equation of the graph after each transformation. (2 marks)
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- Sketch the graph. (1 mark)
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Calculus, EXT1 C1 EQ-Bank 1
A particle is moving along the
Find the acceleration of the particle when
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Statistics, 2ADV S3 EQ-Bank 2
Let
The random variable
The diagram shows the graph of
- Complete the table of values for the given function, correct to four decimal places. (1 mark)
- Use the trapezoidal rule and 5 function values in the table in part i. to estimate
(2 marks)
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- The weights of Rhodesian ridgebacks are normally distributed with a mean of 48 kilograms and a standard deviation of 6 kilograms.
Using the result from part ii., calculate the probability of a randomly selected Rhodesian ridgeback weighing less than 36 kilograms. (2 marks)
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Trigonometry, 2ADV T3 EQ-Bank 5
The function
Describe in words how the amplitude and period have changed in this transformation. (2 marks)
Statistics, 2ADV S2 EQ-Bank 3
The table below lists the average life span (in years) and average sleeping time (in hours/day) of 9 animal species.
- Using sleeping time as the independent variable, calculate the least squares regression line. (1 mark)
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- A wallaby species sleeps for 4.5 hours, on average, each day.
Use your equation from part i to predict its expected life span, to the nearest year. (1 mark)
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Statistics, 2ADV S2 EQ-Bank 2
The table below lists the average body weight (in kilograms) and average brain weight (in grams) of nine animal species.
A least squares regression line is fitted to the data using body weight as the independent variable.
- Calculate the equation of the least squares regression line. (1 mark)
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- If dingos have an average body weight of 22.3 kilograms, calculate the predicted average brain weight of a dingo using your answer to part i. (1 mark)
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Statistics, 2ADV S2 EQ-Bank 1
The arm spans (in cm) and heights (in cm) for a group of 13 boys have been measured. The results are displayed in the table below.
The aim is to find a linear equation that allows arm span to be predicted from height.
- What will be the independent variable in the equation? (1 mark)
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- Assuming a linear association, determine the equation of the least squares regression line that enables arm span to be predicted from height. Write this equation in terms of the variables arm span and height. Give the coefficients correct to two decimal places. (2 marks)
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- Using the equation that you have determined in part b., interpret the slope of the least squares regression line in terms of the variables height and arm span. (1 mark)
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Calculus, 2ADV C1 EQ-Bank 2
When differentiating
Complete the solution. (2 marks)
Calculus, 2ADV C2 EQ-Bank 2
Differentiate with respect to
NETWORKS, FUR2-NHT 2019 VCAA 3
The zoo’s management requests quotes for parts of the new building works.
Four businesses each submit quotes for four different tasks.
Each business will be offered only one task.
The quoted cost, in $100 000, of providing the work is shown in Table 1 below.
The zoo’s management wants to complete the new building works at minimum cost.
The Hungarian algorithm is used to determine the allocation of tasks to businesses.
The first step of the Hungarian algorithm involves row reduction; that is, subtracting the smallest element in each row of Table 1 from each of the elements in that row.
The result of the first step is shown in Table 2 below.
The second step of the Hungarian algorithm involves column reduction; that is, subtracting the smallest element in each column of Table 2 from each of the elements in that column.
The results of the second step of the Hungarian algorithm are shown in Table 3 below. The values of Task 1 are given as
- Write down the values of
and . (1 mark)
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- The next step of the Hungarian algorithm involves covering all the zero elements with horizontal or vertical lines. The minimum number of lines required to cover the zeros is three.
Draw these three lines on Table 3 above. (1 mark)
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- An allocation for minimum cost is not yet possible.
When all steps of the Hungarian algorithm are complete, a bipartite graph can show the allocation for minimum cost.
Complete the bipartite graph below to show this allocation for minimum cost. (1 mark)
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- Business 4 has changed its quote for the construction of the pathways. The new cost is $1 000 000. The overall minimum cost of the building works is now reduced by reallocating the tasks.
How much is this reduction? (1 mark)
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GEOMETRY, FUR1-NHT 2019 VCAA 2 MC
Which one of the following locations is closest to the Greenwich meridian?
- 32° S, 40° E
- 32° S, 80° E
- 32° S, 60° W
- 32° S, 120° E
- 32° S, 160° W
GEOMETRY, FUR1-NHT 2019 VCAA 1 MC
Networks, STD2 N3 2019 FUR2-N 2
The construction of the new reptile exhibit is a project involving nine activities,
The directed network below shows these activities and their completion times in weeks.
- Which activities have more than one immediate predecessor? (1 mark)
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- Write down the critical path for this project. (1 mark)
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- What is the latest start time, in weeks, for activity
? (1 mark)
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NETWORKS, FUR2-NHT 2019 VCAA 1
A zoo has an entrance, a cafe and nine animal exhibits: bears
The edges on the graph below represent the paths between the entrance, the cafe and the animal exhibits. The numbers on each edge represent the length, in metres, along that path. Visitors to the zoo can use only these paths to travel around the zoo.
- What is the shortest distance, in metres, between the entrance and the seal exhibit
? (1 mark)
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- Freddy is a visitor to the zoo. He wishes to visit the cafe and each animal exhibit just once, starting and ending at the entrance.
- i. What is the mathematical term used to describe this route? (1 mark)
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- ii. Draw one possible route that Freddy may take on the graph below. (1 mark)
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A reptile exhibit
A new path of length 20 m will be built between the reptile exhibit
A second new path, of length 35 m, will be built between the reptile exhibit
- Complete the graph below with the new reptile exhibit and the two new paths added. Label the new vertex
and write the distances on the new edges. (1 mark)
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- The new paths reduce the minimum distance that visitors have to walk between the giraffe exhibit
and the cafe.By how many metres will these new paths reduce the minimum distance between the giraffe exhibit
and the cafe? (1 mark)
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MATRICES, FUR2-NHT 2019 VCAA 1
A total of six residents from two towns will be competing at the International Games.
Matrix
- How many of these athletes are residents of Haldaw? (1 mark)
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Each of the six athletes will compete in one event: table tennis, running or basketball.
Matrices
Table tennis | Running | |
|
|
- Matrix
contains the number of male and female athletes from each town who will compete in basketball.
Complete matrix
below. (1 mark)
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Matrix
-
- For which event will the total cost of uniforms for the athletes be $1030? (1 mark)
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- Write a matrix calculation, that includes matrix
, to show that the total cost of uniforms for the event named in part c.i. is contained in the matrix answer of [1030]. (1 mark)
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- For which event will the total cost of uniforms for the athletes be $1030? (1 mark)
- Matrix
and matrix are two new matrices where and:
- matrix
is a matrix - element
total cost of uniforms for all female athletes from Gillen - element
total cost of uniforms for all female athletes from Haldaw - element
total cost of uniforms for all male athletes from Gillen - element
total cost of uniforms for all male athletes from Haldaw
- Complete matrix
with the missing values. (1 mark)
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Functions, EXT1 F1 2019 MET1-N 5
Let
- State the range of
. (1 mark)
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- Find the rule for
. (2 marks)
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Functions, 2ADV F1 2019 MET1-N 2
Let
- Find
. (2 marks)
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- Express
in the form , where , and are non-zero integers. (2 marks)
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Calculus, 2ADV C2 SM-Bank 8
Let
Find
Calculus, 2ADV C2 SM-Bank 7
Let
Find
Functions, MET1-NHT 2018 VCAA 5
Let
- State the range of
. (1 mark)
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- Find the rule for
. (2 marks)
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Functions, MET1-NHT 2018 VCAA 2
Let
- Find
. (2 marks)
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Calculus, MET1-NHT 2018 VCAA 1a
Let
Find
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