`- 25 = 15` |
Geometry, NAP-F2-05
Number and Algebra, NAP-F2-04
Five surfers owned a total of 58 surfboards between them.
Two of the surfers owned 20 surfboards each.
The other three, Lane, Gordon and Shelly, owned the same number of surfboards.
How many surfboards does Gordon own?
`6` | `8` | `12` | `38` |
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Probability, NAP-F2-03
Geometry, NAP-E2-09
Number and Algebra, NAP-G2-10
In which one of these numbers does the numeral 7 represent 7 tens?
`7209` | `457` | `712` | `2072` |
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Statistics, NAP-E2-13
A combined year 5 and year 6 class were asked a question in a survey.
The first seven answers were
`9, 12, 10, 11, 11, 12, 11`
Which one of these questions could have been asked?
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Are you left or right handed? |
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How many centimetres tall are you? |
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What is your favourite colour? |
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How many years old are you? |
Measurement, NAP-E2-07
Measurement, NAP-E2-05
Richard started his jog at 1:25 pm. He finished at 2:08 pm.
How long did Richard jog for?
`text(35 minutes)` | `text(43 minutes)` | `text(73 minutes)` | `text(83 minutes)` |
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Probability, NAP-H2-11
Number and Algebra, NAP-G2-07
Wayne needs to buy 14 pairs of socks for the players in his soccer team.
The socks are sold in packets of 3.
How many packets does Wayne need to buy?
`4` | `5` | `17` | `42` |
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Number and Algebra, NAP-H2-7
`3 xx` |
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`= 51` |
What number makes the above number sentence correct?
`17` | `27` | `48` | `153` |
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Geometry, NAP-H2-06
Number and Algebra, NAP-H2-5
What number is 11 less than 1007?
`906` | `996` | `1018` | `1096` |
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Statistics, NAP-G2-05
Statistics, NAP-D2-13
Benson asked his friends what their favourite ice cream flavour was.
He recorded their answers in the table below but left off same labels.
Benson remembered that
- strawberry was the least liked flavour
- chocolate was the most liked flavour
- vanilla was more liked than caramel
How many of Benson's friends liked caramel the most?
Number and Algebra, NAP-I2-11
George has no money in his bank account.
He deposits $6 in his account in week 1.
He then deposits twice the amount into his account each week than he did the previous week.
The total amount in his account is?
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always odd. |
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always even. |
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sometimes odd and sometimes even. |
Number and Algebra, NAP-I2-9
Olivia was the top scorer in her soccer team over two seasons.
In season 1, she scored 25 goals.
In season 2, she scored 35 goals.
Which number sentence could be used to find the total number of goals she scored over both seasons?
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`20 + 30 = 50` |
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`30 + 40 = 70` |
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`20 + 30 + 5 = 55` |
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`20 + 30 + 10 = 60` |
Geometry, NAP-I2-08
Number and Algebra, NAP-I2-6
Number and Algebra, NAP-I2-05 SA
Geometry, NAP-I2-04
Statistics, NAP-I2-02 SA
Number and Algebra, NAP-I2-01
Bojacks lives in South Australia and receives 10 cents for every glass bottle that he takes to the recycling depot.
Bojacks recycles 28 glass bottles.
How much money will he receive?
`28 ¢` | `$2.80` | `$28` | `$280` |
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Number and Algebra, NAP-D2-10
Number and Algebra, NAP-D2-08
Bronte deducted 3 from each number to get the next number.
Which is Bronte's number pattern?
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`5,9,13,17,…` |
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`4,7,10,13,…` |
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`43,33,23,13,…` |
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`22,19,16,13,…` |
Measurement, NAP-D2-07
Geometry, NAP-D2-06
Number, NAP-C4-CA01
`5 xx` |
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`\ + 15 = 95` |
What value would make this number sentence correct?
`10` | `14` | `16` | `20` |
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Statistics, NAP-C4-NC02
Geometry, NAP-B3-CA01
Statistics, NAP-D3-CA02
Number, NAP-D3-CA03
Srinath spends $12 a month on coffee.
After how many months will his total spending on coffee amount to $180?
`text(3 months)` | `text(9 months)` | `text(15 months)` | `text(20 months)` |
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Geometry, NAP-D3-CA01
Geometry, NAP-D3-NC02
Statistics, NAP-D3-NC01
Algebra, NAP-B3-NC01
`4.25,\ 4.0,\ 3.75,\ 3.5,\ 3.25,\ …`
What is the rule to continue this decimal number pattern?
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`text(increase by 0.5)` |
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`text(increase by 0.25)` |
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`text(decrease by 0.5)` |
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`text(decrease by 0.25)` |
Measurement, NAP-E3-NC03
Richard started his jog at 1:25. He finished at 2:08.
How long did Richard jog for?
`text(35 minutes)` | `text(43 minutes)` | `text(73 minutes)` | `text(83 minutes)` |
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Number, NAP-E3-NC02
Which number sentence is correct when 7 is placed in the box?
`17 + ` |
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`= 10` |
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`- 8 = 1` | `10 -` |
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`= 3` | `12 + 5 =` |
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Measurement, NAP-G3-CA03
Number, NAP-G3-CA01
In Australia, 288 701 children were enrolled in kindergarten in 2013.
Of these children, 150 125 were boys.
How many girls were enrolled in kindergarten in 2013?
`122\ 896` | `138\ 576` | `288\ 701` | `438\ 826` |
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Number, NAP-F4-CA01
A return trip from Brodie's house to the beach is 5.78 kilometres.
How far does Brodie travel if he does this 14 times?
`8.09\ text(km)` | `23.12\ text(km)` | `42.12\ text(km)` | `69.36\ text(km)` | `80.92\ text(km)` |
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Number, NAP-G4-CA01
In Africa, a national park estimated the population of flamingos was 183 409 in 2021.
Of these, 87 396 were male.
How many female flamingos were there?
`96\ 013` | `123\ 586` | `212\ 786` | `270\ 805` |
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Number, NAP-I4-CA01
Byron earns $16 per hour.
How much will he be paid for working 8 hours?
`$2` | `$32` | `$128` | `$216` |
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Quadratic, EXT1 2016 HSC 14c
The point `T(2at,at^2)` lies on the parabola `P_1` with the equation `x^2=4ay`.
The tangent to the parabola `P_1` at `T` meets the directrix at `D`.
The normal to the parabola `P_1` at `T` meets the vertical line through `D` at the point `R`, as shown in the diagram.
- Show that the point `D` has coordinates `(at - a/t, −a)`. (1 mark)
- Show that the locus of `R` lies on another parabola `P_2`. (3 marks)
- State the focal length of the parabola `P_2`. (1 mark)
It can be shown that the minimum distance between `R` and `T` occurs when the normal to `P_1` at `T` is also the normal to `P_2` at `R`. (Do NOT prove this.)
- Find the values of `t` so that the distance between `R` and `T` is a minimum. (2 marks)
Binomial, EXT1 2016 HSC 14b
Consider the expansion of `(1 + x)^n`, where `n` is a positive integer.
- Show that `2^n = ((n),(0)) + ((n),(1)) + ((n),(2)) + ((n),(3)) + … + ((n),(n))`. (1 mark)
- Show that `n2^(n - 1) = ((n),(1)) + 2((n),(2)) + 3((n),(3)) + … + n((n),(n))`. (1 mark)
- Hence, or otherwise, show that `sum_(r = 1)^n ((n),(r))(2r - n) = n`. (2 marks)
Proof, EXT1 P1 2016 HSC 14a
- Show that `4n^3 + 18n^2 + 23n + 9` can be written as
`qquad (n + 1)(4n^2 + 14n + 9)`. (1 marks)
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- Using the result in part (i), or otherwise, prove by mathematical induction that, for `n >= 1`,
`qquad 1 × 3 + 3 × 5 + 5 × 7 + … + (2n - 1)(2n + 1) = 1/3 n(4n^2 + 6n - 1)`. (3 marks)
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Plane Geometry, EXT1 2016 HSC 13c
The circle centred at `O` has a diameter `AB`. From the point `M` outside the circle the line segments `MA` and `MB` are drawn meeting the circle at `C` and `D` respectively, as shown in the diagram. The chords `AD` and `BC` meet at `E`. The line segment `ME` produced meets the diameter `AB` at `F`.
Copy or trace the diagram into your writing booklet.
- Show that `CMDE` is a cyclic quadrilateral. (2 marks)
- Hence, or otherwise, prove that `MF` is perpendicular to `AB`. (2 marks)
Mechanics, EXT2* M1 2016 HSC 13b
The trajectory of a projectile fired with speed `u\ text(ms)^-1` at an angle `theta` to the horizontal is represented by the parametric equations
`x = utcostheta` and `y = utsintheta - 5t^2`,
where `t` is the time in seconds.
- Prove that the greatest height reached by the projectile is `(u^2 sin^2 theta)/20`. (2 marks)
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A ball is thrown from a point `20\ text(m)` above the horizontal ground. It is thrown with speed `30\ text(ms)^-1` at an angle of `30^@` to the horizontal. At its highest point the ball hits a wall, as shown in the diagram.
- Show that the ball hits the wall at a height of `125/4\ text(m)` above the ground. (2 marks)
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The ball then rebounds horizontally from the wall with speed `10\ text(ms)^-1`. You may assume that the acceleration due to gravity is `10\ text(ms)^-2`.
- How long does it take the ball to reach the ground after it rebounds from the wall? (2 marks)
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- How far from the wall is the ball when it hits the ground? (1 mark)
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Mechanics, EXT2* M1 2016 HSC 13a
The tide can be modelled using simple harmonic motion.
At a particular location, the high tide is 9 metres and the low tide is 1 metre.
At this location the tide completes 2 full periods every 25 hours.
Let `t` be the time in hours after the first high tide today.
- Explain why the tide can be modelled by the function `x = 5 + 4cos ((4pi)/25 t)`. (2 marks)
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- The first high tide tomorrow is at 2 am.
What is the earliest time tomorrow at which the tide is increasing at the fastest rate? (2 marks)
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Polynomials, EXT1 2016 HSC 12c
The graphs of `y = tan x` and `y = cos x` meet at the point where `x = α`, as shown.
- Show that the tangents to the curves at `x = α` are perpendicular. (2 marks)
- Use one application of Newton’s method with `x_1 = 1` to find an approximate value for `α`. Give your answer correct to two decimal places. (2 marks)
Calculus, EXT1 C1 2016 HSC 12b
In a chemical reaction, a compound `X` is formed from a compound `Y`. The mass in grams of `X` and `Y` are `x(t)` and `y(t)` respectively, where `t` is the time in seconds after the start of the chemical reaction.
Throughout the reaction the sum of the two masses is 500 g. At any time `t`, the rate at which the mass of compound `X` is increasing is proportional to the mass of compound `Y`.
At the start of the chemical reaction, `x = 0` and `(dx)/(dt) = 2`.
- Show that `(dx)/(dt) = 0.004(500 - x)`. (3 marks)
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- Show that ` x = 500 - Ae^(−0.004t)` satisfies the equation in part (i), and find the value of `A`. (2 marks)
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Calculus, EXT1 C1 2016 HSC 12a
The diagram shows a conical soap dispenser of radius 5 cm and height 20 cm.
At any time `t` seconds, the top surface of the soap in the container is a circle of radius `r` cm and its height is `h` cm.
The volume of the soap is given by `v = 1/3 pir^2h`.
- Explain why `r = h/4`. (1 mark)
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- Show that `(dv)/(dh) = pi/16 h^2`. (1 mark)
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The dispenser has a leak which causes soap to drip from the container. The area of the circle formed by the top surface of the soap is decreasing at a constant rate of `0.04\ text(cm² s)^-1`.
- Show that `(dh)/(dt) = (−0.32)/(pih)`. (2 marks)
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- What is the rate of change of the volume of the soap, with respect to time, when `h = 10`? (2 marks)
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Statistics, EXT1 S1 2016 HSC 11f
A darts player calculates that when she aims for the bullseye the probability of her hitting the bullseye is `3/5` with each throw.
- Find the probability that she hits the bullseye with exactly one of her first three throws. (1 mark)
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- Find the probability that she hits the bullseye with at least two of her first six throws. (2 marks)
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Functions, EXT1 F1 2016 HSC 11e
Solve `3/(2x + 5) - x > 0`. (3 marks)
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Calculus, EXT1 C2 2016 HSC 11c
Differentiate `3tan^(−1)(2x)`. (2 marks)
Functions, EXT1 F2 2016 HSC 10 MC
Combinatorics, EXT1 A1 2016 HSC 8 MC
A team of 11 students is to be formed from a group of 18 students. Among the 18 students are 3 students who are left-handed.
What is the number of possible teams containing at least 1 student who is left-handed?
(A) `19\ 448`
(B) `30\ 459`
(C) `31\ 824`
(D) `58\ 344`
Mechanics, EXT2* M1 2016 HSC 7 MC
The displacement `x` of a particle at time `t` is given by
`x = 5 sin 4t + 12 cos 4t`.
What is the maximum velocity of the particle?
(A) `13`
(B) `28`
(C) `52`
(D) `68`
Plane Geometry, EXT1 2016 HSC 4 MC
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