Sketch the region in the plane defined by
Calculus, 2ADV C3 2007 HSC 10b
The noise level,
Two sound sources, of loudness
The point
- Write down a formula for the sum of the noise levels at
in terms of . (1 mark)
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- There is a point on the line between the sound sources where the sum of the noise levels is a minimum.
Find an expression for
in terms of , and if is chosen to be this point. (4 marks)
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Calculus in the Physical World, 2UA 2007 HSC 10a
An object is moving on the
- Using Simpson’s rule, estimate the distance travelled between
and . (2 marks) - The object is initially at the origin. During which time(s) is the displacement of the object decreasing? (1 mark)
- Estimate the time at which the object returns to the origin. Justify your answer. (2 marks)
- Sketch the displacement,
, as a function of time. (2 marks)
Probability, 2ADV S1 2007 HSC 9b
A pack of 52 cards consists of four suits with 13 cards in each suit.
- One card is drawn from the pack and kept on the table. A second card is drawn and placed beside it on the table. What is the probability that the second card is from a different suit to the first? (1 mark)
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- The two cards are replaced and the pack shuffled. Four cards are chosen from the pack and placed side by side on the table. What is the probability that these four cards are all from different suits? (2 marks)
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Calculus, EXT1* C3 2007 HSC 9a
Plane Geometry, 2UA 2007 HSC 8b
Calculus, EXT1* C1 2007 HSC 8a
One model for the number of mobile phones in use worldwide is the exponential growth model,
where
- It is estimated that at the start of 2009, when
, there will be 1600 million mobile phones in use, while at the start of 2010, when , there will be 2600 million. Find and . (3 marks)
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- According to the model, during which month and year will the number of mobile phones in use first exceed 4000 million? (2 marks)
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Calculus, 2ADV C4 2007 HSC 7b
The diagram shows the graphs of
- Solve the equation
to find the -coordinates of and . (2 marks)
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- Find the area of the shaded region in the diagram. (3 marks)
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Quadratic, 2UA 2007 HSC 7a
- Find the coordinates of the focus,
, of the parabola . (2 marks) - The graphs of
and the line have only one point of intersection, . Show that the -coordinate of satisfies. . (1 mark)
- Using the discriminant, or otherwise, find the value of
. (1 mark) - Find the coordinates of
. (2 marks) - Show that
is parallel to the directrix of the parabola. (1 mark)
Calculus, 2ADV C3 2007 HSC 6b
Let
- Find the coordinates of the points where the curve crosses the axes. (2 marks)
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- Find the coordinates of the stationary points and determine their nature. (4 marks)
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- Find the coordinates of the points of inflection. (1 mark)
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- Sketch the graph of
, indicating clearly the intercepts, stationary points and points of inflection. (3 marks)
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L&E, 2ADV E1 2007 HSC 6a
Solve the following equation for
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Calculus, EXT1* C1 2007 HSC 5b
A particle is moving on the
- What is the initial velocity of the particle? (1 mark)
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- Find an expression for the acceleration of the particle. (2 marks)
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- Find the time when the acceleration of the particle is zero. (1 mark)
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- Find the position of the particle when
. (3 marks)
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Plane Geometry, 2UA 2007 HSC 5a
In the diagram,
Copy or trace this diagram into your writing booklet.
- Show that the size of
is . (1 mark) - Find the size of
. Give reasons for your answer. (2 marks) - By considering the sizes of angles, show that
is isosceles. (2 marks)
Trigonometry, 2ADV T1 2007 HSC 4c
An advertising logo is formed from two circles, which intersect as shown in the diagram.
The circles intersect at
The radius of the circle centred at
- Use Pythagoras’ theorem to show that
. (1 mark)
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- Find
and . (2 marks)
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- Find the area of the quadrilateral
. (1 mark)
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- Find the area of the major sector
. (1 mark)
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- Find the total area of the logo (the sum of all the shaded areas). (2 marks)
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Probability, 2ADV S1 2007 HSC 4b
Two ordinary dice are rolled. The score is the sum of the numbers on the top faces.
- What is the probability that the score is 10? (2 marks)
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- What is the probability that the score is not 10? (1 mark)
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Geometry and Calculus, EXT1 2005 HSC 7b
Let
- Show that
has stationary points at -
(1 mark)
-
- Show that
has exactly one zero when (2 marks)
- By observing that
, deduce that does not have a zero in the interval when (1 mark)
- Let
, where - By calculating
and applying the result in part (iii), or otherwise, show that does not have any stationary points. (3 marks)
- Hence, or otherwise, deduce that
has an inverse function. (1 mark)
Quadratic, 2UA 2005 HSC 10a
The parabola
- Explain why
and . (1 mark) - Given that
, show that the distance (2 marks) - The point
lies on the parabola between and . Show that the area of the triangle is given by (2 marks) - The point
in part (iii) is chosen so that the area of the triangle is a maximum. - Find the coordinates of
in terms of . (2 marks)
Mechanics, EXT2* M1 2015 HSC 14b
A particle is moving horizontally. Initially the particle is at the origin
The acceleration of the particle is given by
- Show that the velocity of the particle is given by
. (3 marks)
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- Find an expression for
as a function of . (2 marks)
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- Find the limiting position of the particle. (1 mark)
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Calculus, EXT1 C2 2015 HSC 13d
Let
- By considering the derivative of
, prove that is constant. (2 marks)
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- Hence deduce that
. (1 mark)
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Financial Maths, 2ADV M1 2007 HSC 3b
Heather decides to swim every day to improve her fitness level.
On the first day she swims 750 metres, and on each day after that she swims
- Write down a formula for the distance she swims on the
th day. (1 mark)
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- How far does she swim on the 10th day? (1 mark)
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- What is the total distance she swims in the first 10 days? (1 mark)
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- After how many days does the total distance she has swum equal the width of the English Channel, a distance of 34 kilometres? (2 marks)
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Calculus, 2ADV C3 2007 HSC 2c
The point
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Calculus, 2ADV C4 2007 HSC 2bii
Evaluate
Calculus, 2ADV C4 2007 HSC 2bi
Find
Combinatorics, EXT1 A1 2015 HSC 13b
Consider the binomial expansion
where
- Find an expression for
. (2 marks)
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- Find an expression for the term independent of
. (2 marks)
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Mechanics, EXT2* M1 2015 HSC 13a
A particle is moving along the
- For what value(s) of
is the particle at rest? (1 mark)
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- What is the maximum speed of the particle? (1 mark)
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- The velocity
of the particle is given by the equation
where , and are positive constants.What are the values of
, and ? (3 marks)
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Trig Ratios, EXT1 2015 HSC 12d
A kitchen bench is in the shape of a segment of a circle. The segment is bounded by an arc of length 200 cm and a chord of length 160 cm. The radius of the circle is
- Show that
. (1 mark) - Hence, or otherwise, show that
. (2 marks) - Taking
as a first approximation to the value of , use one application of Newton’s method to find a second approximation to the value of . Give your answer correct to two decimal places. (2 marks)
Trig Ratios, EXT1 2015 HSC 12c
A person walks 2000 metres due north along a road from point
From point
From point
- Show that
. (1 mark) - Hence, find the value of
. (2 marks)
Quadratic, EXT1 2015 HSC 12b
The points
The equation of the chord
- Show that if
is a focal chord then . (1 mark) - If
is a focal chord and has coordinates , what are the coordinates of in terms of ? (2 marks)
Trig Calculus, EXT1 2015 HSC 10 MC
Mechanics, EXT2* M1 2015 HSC 9 MC
Two particles oscillate horizontally. The displacement of the first is given by
What are the values of
Trig Calculus, EXT1 2015 HSC 8 MC
What is the value of
Geometry and Calculus, EXT1 2015 HSC 5 MC
What are the asymptotes of
A. | |
||
B. | |
||
C. | |
||
D. | |
Statistics, STD2 S4 2015 HSC 28e
The shoe size and height of ten students were recorded.
- Complete the scatter plot AND draw a line of fit by eye. (2 marks)
- Use the line of fit to estimate the height difference between a student who wears a size 7.5 shoe and one who wears a size 9 shoe. (1 mark)
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- A student calculated the correlation coefficient to be 1 for this set of data. Explain why this cannot be correct. (1 mark)
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Algebra, STD2 A1 2015 HSC 28d
The formula
Convert 3°C to the equivalent temperature in Fahrenheit. (2 marks)
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Measurement, 2UG 2015 HSC 28c
Measurement, STD2 M1 2015 HSC 28a
Statistics, STD2 S1 2015 HSC 27d
In a small business, the seven employees earn the following wages per week:
- Is the wage of $970 an outlier for this set of data? Justify your answer with calculations. (3 marks)
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- Each employee receives a $20 pay increase.
What effect will this have on the standard deviation? (1 mark)
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Measurement, STD2 M7 2015 HSC 27a
FS Comm, 2UG 2015 HSC 26g
Measurement, STD2 M1 2015 HSC 26f
Approximately 71% of Earth’s surface is covered by water. Assume Earth is a sphere with a radius of 6400 km.
Calculate the number of square kilometres covered by water. (2 marks)
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Probability, STD2 S2 2015 HSC 26e
The table shows the relative frequency of selecting each of the different coloured jelly beans from packets containing green, yellow, black, red and white jelly beans.
- What is the relative frequency of selecting a red jelly bean? (1 mark)
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- Based on this table of relative frequencies, what is the probability of NOT selecting a black jelly bean? (1 mark)
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Financial Maths, STD2 F4 2015 HSC 26d
A family currently pays $320 for some groceries.
Assuming a constant annual inflation rate of 2.9%, calculate how much would be paid for the same groceries in 5 years’ time. (2 marks)
Data, 2UG 2015 HSC 26a
A farmer used the ‘capture‑recapture’ technique to estimate the number of chickens he had on his farm. He captured, tagged and released 18 of the chickens. Later, he caught 26 chickens at random and found that 4 had been tagged.
What is the estimate for the total number of chickens on this farm? (2 marks)
Financial Maths, STD2 F1 2015 HSC 25 MC
An insurance company offers customers the following discounts on the basic annual premium for car insurance.
If a customer is eligible for more than one discount, subsequent discounts are applied to the already discounted premium. The combined compulsory third party (CTP) and comprehensive insurance discount is always applied last.
Jamie has three insurance policies, including combined CTP and comprehensive insurance, with this company. He has used this company for 8 years and he has never made a claim.
The basic annual premium for his car insurance is $870.
How much will Jamie need to pay after the discounts are applied?
- $482.44
- $515.50
- $541.60
- $557.60
Algebra, STD2 A1 2015 HSC 24 MC
Consider the equation
Which of the following would be a correct step in solving this equation?
Algebra, STD2 A1 2015 HSC 23 MC
The number of ‘standard drinks’ in various glasses of wine is shown.
A woman weighing 62 kg drinks three small glasses of white wine and two large glasses of red wine between 8 pm and 1 am.
Using the formula for calculating blood alcohol below, what would be her blood alcohol content (BAC) estimate at 1 am, correct to three decimal places?
where
Calculus, EXT1* C1 2015 HSC 14a
In a theme park ride, a chair is released from a height of
The height of the chair at time
- Using calculus, show that
. (2 marks)
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- How far has the chair fallen when the magnetic brakes are applied? (2 marks)
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Financial Maths, STD2 F4 2015 HSC 17 MC
What amount must be invested now at 4% per annum, compounded quarterly, so that in five years it will have grown to $60 000?
- $8919
- $11 156
- $49 173
- $49 316
Probability, STD2 S2 2015 HSC 16 MC
The probability of winning a game is
Which expression represents the probability of winning two consecutive games?
Measurement, 2UG 2015 HSC 14 MC
Stockholm is located at
What is the time difference between Stockholm and Darwin? (Ignore time zones and daylight saving.)
(A)
(B)
(C)
(D)
Financial Maths, STD2 F4 2015 HSC 10 MC
Measurement, STD2 M1 2015 HSC 8 MC
Measurement, STD2 M6 2015 HSC 7 MC
Statistics, STD2 S1 2015 HSC 6 MC
Statistics, STD2 S1 2015 HSC 4 MC
On a school report, a student’s record of completing homework is graded using the following codes.
C = consistently
U = usually
S = sometimes
R = rarely
N = never
What type of data is this?
- Categorical, ordinal
- Categorical, nominal
- Numerical, continuous
- Numerical, discrete
Calculus, 2ADV C4 2015 HSC 15c
Water is flowing in and out of a rock pool. The volume of water in the pool at time
At time
- After what time does the volume of water first start to decrease? (2 marks)
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- Find the volume of water in the pool when
. (2 marks)
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- What is the greatest volume of water in the pool? (1 mark)
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Plane Geometry, 2UA 2015 HSC 15b
The diagram shows
Copy or trace the diagram into your writing booklet.
- Prove that
is similar to (2 marks) - Explain why
is isosceles. (1 mark) - Show that
(2 marks)
Calculus, EXT1* C1 2015 HSC 15a
The amount of caffeine,
where
- Show that
is a solution to where is a constant.When
, there are 130 mg of caffeine in Lee’s body. (1 mark)
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- Find the value of
(1 mark)
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- What is the amount of caffeine in Lee’s body after 7 hours? (1 mark)
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- What is the time taken for the amount of caffeine in Lee’s body to halve? (2 marks)
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Financial Maths, 2ADV M1 2015 HSC 14c
Sam borrows $100 000 to be repaid at a reducible interest rate of 0.6% per month. Let
- Show that
(1 mark)
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- Show that
(2 marks)
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- Sam makes monthly repayments of $780. Show that after making 120 monthly repayments the amount owing is $68 500 to the nearest $100. (1 mark)
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Immediately after making the 120th repayment, Sam makes a one-off payment, reducing the amount owing to $48 500. The interest rate and monthly repayment remain unchanged.
- After how many more months will the amount owing be completely repaid? (3 marks)
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Probability, 2ADV S1 2015 HSC 14b
Weather records for a town suggest that:
- if a particular day is wet
, the probability of the next day being dry is - if a particular day is dry
, the probability of the next day being dry is .
In a specific week Thursday is dry. The tree diagram shows the possible outcomes for the next three days: Friday, Saturday and Sunday.
- Show that the probability of Saturday being dry is
. (1 mark)
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- What is the probability of both Saturday and Sunday being wet? (2 marks)
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- What is the probability of at least one of Saturday and Sunday being dry? (1 mark)
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