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Trigonometry, 2ADV T1 2017 HSC 11e

In the diagram, `OAB` is a sector of the circle with centre `O` and radius 6 cm, where  `/_ AOB = 30^@`.


 

  1.  Find the exact value of the area of the triangle `OAB`.  (1 mark)

  2. Find the exact value of the area of the shaded segment.  (1 mark)

Show Answers Only
  1.  `9\ text(cm²)`
  2. `3 pi – 9\ text(cm²)`
Show Worked Solution
i.   `text(Area)\ Delta OAB` `= 1/2 ab sin C`
    `= 1/2 xx 6^2 xx sin 30^@`
    `= 9\ text(cm²)`

 

ii.  `text(Area segment)` `= text(Area sector) – text(Area)\ Delta OAB`
    `= 30/360 xx pi xx 6^2 – 9`
    `= 3 pi – 9\ \ text(cm²)`

Functions, EXT1* F1 2017 HSC 8 MC

The region enclosed by  `y = 4 - x,\ \ y = x`  and  `y = 2x + 1`  is shaded in the diagram below.
 

Which of the following defines the shaded region?

A.   `y <= 2x + 1, qquad` `y <= 4-x, qquad` `y >= x`
B.   `y >= 2x + 1, qquad` `y <= 4-x, qquad` `y >= x`
C.   `y <= 2x + 1, qquad` `y >= 4-x, qquad` `y >= x`
D.   `y >= 2x + 1, qquad` `y >= 4-x, qquad` `y >= x`
Show Answers Only

`A`

Show Worked Solution

`text(Consider)\ \ y = 2x + 1,`

`text(Shading is below graph)`

`=> y <= 2x + 1`

`text(Consider)\ \ y = 4-x,`

`text(Shading is below graph)`

`=> y <= 4-x`

`=>  A`

Trigonometry, 2ADV T2 2017 HSC 7 MC

Which expression is equivalent to  `tan theta + cot theta`?

  1. `text(cosec)\ theta + sec theta`
  2. `sec theta\ text(cosec)\ theta`
  3. `2`
  4. `1`
Show Answers Only

`B`

Show Worked Solution
`tan theta + cot theta` `= (sin theta)/(cos theta) + (cos theta)/(sin theta)`
  `= (sin^2 theta + cos^2 theta)/(cos theta sin theta)`
  `= 1/(cos theta sin theta)`
  `= sec theta\ text(cosec)\ theta`

`=>  B`

Functions, 2ADV F1 2017 HSC 6 MC

The point `P` moves so that it is always equidistant from two fixed points, `A` and `B.`

What best describes the locus of `P`?

(A)  A point

(B)  A circle

(C)  A parabola

(D)  A straight line

Show Answers Only

`D`

Show Worked Solution

`text(An example of)\ \ P\ \ text(is drawn below:)`
 

`=>  D`

L&E, 2ADV E1 2017 HSC 5 MC

It is given that  `ln a = ln b-ln c`, where  `a, b, c > 0.`

Which statement is true?

  1. `a = b-c`
  2. `a = b/c`
  3. `text(ln)\ a = b/c`
  4. `text(ln)\ a = (text(ln)\ b)/(text(ln)\ c)`
Show Answers Only

`B`

Show Worked Solution
Mean mark 51%.
COMMENT: Use of log laws here proved difficult for many students.
`ln a` `= ln b-ln c`
`ln a` `= ln (b/c)`
`:. a` `= b/c`

 
`=>  B`

Calculus, 2ADV C3 2017 HSC 4 MC

The function  `f(x)` is defined for  `a <= x <= b.`

On this interval,  `f ^{′}(x) > 0 and f^{″}(x) < 0.`

Which graph best represents  `y = f(x)`?
 

(A)   (B)  
(C)   (D)  
Show Answers Only

`A`

Show Worked Solution

`text(Interpreting)\ \ f^{′}(x) > 0,`

`=>\ text(gradient is always positive)`

`text(Interpreting)\ \ f^{″}(x) < 0,`

`=>\ text(Curve is concave down)`

`=>  A`

Algebra, 2UG 2017 HSC 30d

In an investigation, students used different numbers of identical small solar panels to power model cars. The cars were then tested and their speed measured in km/h. The results are summarised in the table.
 


 

The equation of the least-squares line of best fit, relating the speed and the number of solar panels, has been calculated to be
 

`y = 2.125x + 2.0375`
 

  1. What would be the speed of a car powered by 5 solar panels, based on this equation?  (1 mark)
  2. Calculate the correlation coefficient, `r`, between the number of solar panels and the speed of a car.  (2 marks)

 

Show Answers Only
  1. `12.6625\ text(km/h)`
  2. `0.85`
Show Worked Solution
(i)    `text(Speed)(y)` `= 2.125(5) + 2.0375`
    `= 12.6625\ text(km/h)`

 

(ii)   `text(Using the formula sheet:)`

♦♦ Mean mark 23%.
`text(gradient)` `= r xx (text(std dev)(y))/(text(std dev)(x))`
`2.125` `= r xx 2/0.8`
`:. r` `= (2.125 xx 0.8)/2`
  `= 0.85`

Probability, STD2 S2 2017 HSC 29c

A group of Year 12 students was surveyed. The students were asked whether they live in the city or the country. They were also asked if they have ever waterskied.

The results are recorded in the table.
  


  

  1. A person is selected at random from the group surveyed. Calculate the probability that the person lives in the city and has never waterskied.  (2 marks)

  2. A newspaper article claimed that Year 12 students who live in the country are more likely to have waterskied than those who live in the city.

     

    Is this true, based on the survey results? Justify your answer with relevant calculations.  (2 marks)

Show Answers Only
  1. `125/176`
  2. `text(See Worked Solutions)`
Show Worked Solution
i.     `P` `= text(live in city, not skied)/text(total surveyed)`
    `= 2500/3520`
    `= 125/176`
♦ Mean mark part (ii) 47%.

 

ii.   `P(text(live in country, skied))` `= 70/((70 + 800))`
    `= 0.0804…`
    `= 8text(%)`

 

`P(text(live in city, skied))` `= 150/((150 + 2500))`
  `= 0.0566`
  `= 6text(%)`

 
`text(S)text(ince 8% > 6%, the statement is true.)`

Financial Maths, STD2 F1 2017 HSC 29b

Sabrina’s taxable income is $86 725 in a particular year.

The table below is used to calculate her tax payable. In addition, she pays the Medicare levy, which is 2% of her taxable income.
 

 
Calculate Sabrina’s net income in that year.  (3 marks)

Show Answers Only

`$65\ 257.87`

Show Worked Solution
`text(Tax payable)` `= 3572 + 0.325(86\ 725 – 37\ 000)`
  `= 19\ 732.625`
  `= $19\ 732.63`
`text(Medicare levy)` `= 2text(%) xx 86\ 725`
  `= $1734.50`

 

`:.\ text(Net Income)` `= 86\ 725 – (19\ 732.63 + 1734.50)`
  `= 65\ 257.87`
  `= $65\ 257.87`

Measurement, 2UG 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.
 


 

  1. Using TWO applications of Simpson’s rule, show that the volume of the dam is approximately  44 333 m³.  (2 marks)
  2. It is known that the catchment area for this dam is 2 km².
    Calculate how much rainfall is needed, to the nearest mm, to fill the dam.  (2 marks)

 

Show Answers Only
  1. `text(See Worked Solution)`
  2. `22\ text{mm  (nearest mm)}`
Show Worked Solution
(i)    `text(Volume)` `~~ h/3 (y_0 + 4y_1 + y_2)\ \ …\ text(applied twice)`
    `~~ 50/3(0 + 4 xx 140 + 270) + 50/3(270 + 4 xx 300 + 360)`
    `~~ 50/3(830) + 50/3(1830)`
    `~~ 44\ 333\ text(m³)\ …\ \ text(as required)`

 

(ii)   `text(Convert 2km² to m²:)`

♦♦♦ Mean mark 11%.
`text(2 km²)` `= 2 xx 1000 xx 1000`
  `= 2\ 000\ 000\ text(m²)`
`:.\ text(Rainfall needed)` `= (44\ 333)/(2\ 000\ 000)`
  `= 0.0221…\ text(m)`
  `= 22.1…\ text(mm)`
  `= 22\ text{mm  (nearest mm)}`

Algebra, STD2 A4 2017 HSC 28e

A movie theatre has 200 seats. Each ticket currently costs $8.

The theatre owners are currently selling all 200 tickets for each session. They decide to increase the price of tickets to see if they can increase the income earned from each movie session.

It is assumed that for each one dollar increase in ticket price, there will be 10 fewer tickets sold.

A graph showing the relationship between an increase in ticket price and the income is shown below.
 


 

  1. What ticket price should be charged to maximise the income from a movie session?  (1 mark)

  2. What is the number of tickets sold when the income is maximised?  (1 mark)

  3. The cost to the theatre owners of running each session is $500 plus $2 per ticket sold.

     

    Calculate the profit earned by the theatre owners when the income earned from a session is maximised.  (2 marks)

Show Answers Only
  1. `$14`
  2. `140`
  3. `$1180`
Show Worked Solution

i.   `text(Graph is highest when increase = $6)`

♦ Mean mark 50%.
`:.\ text(Ticket price)` `= 8 + 6`
  `= $14`

 

ii.   `text(Solution 1)`

♦ Mean mark 45%.
`text(Tickets sold)` `=200-(4 xx 10)`
  `=140`

 

`text(Solution 2)`

`text(Tickets)` `= text(max income)/text(ticket price)`
  `= 1960/14`
  `= 140`

 

iii.   `text(C)text(ost)` `= 140 xx $2 + $500`
    `= $780`

 

`:.\ text(Profit when income is maximised)`

`= 1960-780`

`= $1180`

Probability, 2UG 2017 HSC 28b

Five people are in a team. Two of them are selected at random to attend a competition.

  1. How many different groups of two can be selected?  (1 mark)
  2. If Mary is one of the five people in the team, what is the probability that she is selected to attend the competition?  (1 mark)

 

Show Answers Only
  1. `10`
  2. `2/5`
Show Worked Solution
(i)   `text(# Groups of 2)` `= (5 xx 4)/(2 xx 1)`
    `=10`
♦♦ Mean mark part (ii) 33%.

 

(ii)   `text(Number of pairs with Mary = 4)`

`:. Ptext{(Mary attends)}` `= 4/10`
  `= 2/5`

Algebra, STD2 A1 2017 HSC 27e

Rhys is drinking low alcohol beer at a party over a five-hour period. He reads on the label of the low alcohol beer bottle that it is equivalent to 0.8 of a standard drink.

Rhys weighs 90 kg.

The formula below  can be used to calculate a Rhys's blood alcohol content.
 

`BAC_text(Male) = (10N - 7.5H)/(6.8M)`

where    `N` is the number of standard drinks consumed

`H` is the number of hours drinking

`M` is the person's mass in kilograms
 

What is the maximum number of complete bottles of the low alcohol beer he can drink to remain under a Blood Alcohol Content (BAC) of 0.05?  (4 marks)

Show Answers Only

`8`

Show Worked Solution
`text(BAC)_text(male)` `= (10N – 7.5H)/(6.8M)`
`0.05` `= (10N – 7.5 xx 5)/(6.8 xx 90)`
`10N` `= (0.05 xx 6.8 xx 90) + 7.5 xx 5`
  `= 68.1`
`N` `= 6.81\ \ text(standard drinks)`

 

`:.\ text(Number of low alcohol bottles)`

`= 6.81/0.8`

`= 8.51`
 

`:.\ text(Max complete bottles to stay under 0.05)`

`= 8`

Financial Maths, STD2 F5 2017 HSC 27c

A table of future value interest factors for an annuity of $1 is shown.
 


 

An annuity involves contributions of $12 000 per annum for 5 years. The interest rate is 4% per annum, compounded annually.

  1. Calculate the future value of this annuity.  (1 mark)

  2. Calculate the interest earned on this annuity.  (1 mark)

Show Answers Only
  1. `$64\ 995.60`
  2. `$4995.60`
Show Worked Solution

i.   `text(FV factor = 5.4163)`

`:.\ text(FV of Annuity)` `= 12\ 000 xx 5.4163`
  `= $64\ 995.60`
♦♦ Mean mark part (ii) 22%.
COMMENT: A very poorly answered question dealing with a core concept in this area.

 

ii.   `text(Interest earned)` `=\ text(FV − total repayments)`
    `= 64\ 995.60 – (5 xx 12\ 000)`
    `= $4995.60`

Financial Maths, 2UG 2017 HSC 26g

Rachel bought a motorcycle advertised for $7990. She paid a $500 deposit and took out a flat-rate loan to repay the balance. Simple interest was charged at a rate of 7% per annum on the amount borrowed. She repaid the loan over 2 years, making equal weekly repayments.

Calculate the weekly repayment.  (3 marks)

Show Answers Only

`$82.10\ \ text{(nearest cent)}`

Show Worked Solution
`text(Loan amount)` `= 7990 – 500`
  `= $7490`
`text(Interest)` `= Prn`
  `= 7490 xx 0.07 xx 2`
  `= $1048.60`

 

`:.\ text(Weekly repayment)` `= text(total repayments)/text(total weeks)`
  `= (7490 + 1048.60)/((52 xx 2))`
  `= 82.1019…`
  `= $82.10\ \ text{(nearest cent)}`

Statistics, STD2 S1 2017 HSC 26f

The area chart shows the number of goals scored by three hockey teams, `A`, `B` and `C`, in the first 4 rounds.
 


 

  1. How many goals were scored by team `C` in round 1?  (1 mark)

  2. In which round did all three teams score the same number of goals?  (1 mark)

Show Answers Only
  1. `4`
  2. `text(Round 3)`
Show Worked Solution
i.   `text(Goals by team C)` `= 9 – 5`
    `= 4`

 

ii.   `text{In Round 3 (all teams scored 2 goals).}`

Financial Maths, STD2 F4 2017 HSC 26e

Sam purchased 500 company shares at $3.20 per share. Brokerage fees were 1.5% of the purchase price.

Sam is paid a dividend of 26 cents per share, then immediately sells the shares for $4.80 each.

If he pays no further brokerage fees, what is Sam’s total profit?  (3 marks)

Show Answers Only

`$906`

Show Worked Solution
`text(Purchase price)` `= 500 xx 3.20`
  `= $1600`
`text(Brokerage fees)` `= 1.5text(%) xx 1600`
  `= $24`
`text(Dividends)` `= 500 xx 0.26`
  `= $130`
`text(Sale price)` `= 500 xx 4.80`
  `= $2400`

 

`:.\ text(Total profit)` `= 2400 + 130 – (1600 + 24)`
  `= $906`

Measurement, STD2 M6 2017 HSC 26d

A sewer pipe needs to be placed into the ground so that it has a 2° angle of depression. The length of the pipe is 15 000 mm.
 


 

How much deeper should one end of the pipe be compared to the other end? Answer to the nearest mm.  (2 marks)

Show Answers Only

`523\ text{mm  (nearest mm)}`

Show Worked Solution

`text(Let)\ \ x = text(depth needed)`

`sin 2^@` `= x/(15\ 000)`
`x` `= 15\ 000 xx sin 2^@`
  `= 523.49…`
  `= 523\ text{mm  (nearest mm)}`

Data, 2UG 2017 HSC 26c

A farmer needed to estimate the number of goats on his property. He tagged 80 of his goats. Later, he collected a random sample of 45 goats and found that 16 of these had tags.

Estimate the number of goats the farmer has on his property.  (2 marks)

Show Answers Only

`225`

Show Worked Solution

`text(Let)\ \ P =\ text(total number of goats)`

`text(Capture) = 80/P`

`text(Recapture) = 16/45`

`:. 80/P` `= 16/45`
`16P` `= 80 xx 45`
`P` `= (80 xx 45)/16`
  `= 225`

Measurement, STD2 M7 2017 HSC 26b

Toby’s mobile phone plan costs $20 per month, plus the cost of all calls. Calls are charged at the rate of 70 cents per 30 seconds, or part thereof. There is also a call connection fee of 50c per call.

Here is a record of all his calls in July.
 

How much is Toby’s mobile phone bill for July?  (2 marks)

Show Answers Only

`$27.10`

Show Worked Solution

`text(Call cost) = 0.70 + (2 xx 0.70) + (5 xx 0.70) = $5.60`

`text(Connection fees) = 3 xx 0.50 = $1.50`

`:.\ text(Total bill)` `= 5.60 + 1.50 + 20`
  `= $27.10`

Algebra, STD2 A4 2017 HSC 17 MC

The graph of the line with equation  `y = 6 - 2x`  is shown.
 


 

When the graph of the line with equation  `y = x + 3`  is also drawn on this number plane, what will be the point of intersection of the two lines?

A.     `(0, 6)`

B.     `(1, 4)`

C.     `(2, 2)`

D.     `(3, 0)`

Show Answers Only

`text(B)`

Show Worked Solution

`text(Solution 1)`

`text(From graph, intersection at (1,4))`
 

`=>B`
 

`text(Solution 2)`

`y` `= 6 – 2x` `…\ (1)`
`y` `= x + 3` `…\ (2)`

 
`text{Substitute (2) into (1)}`

`x + 3` `= 6 – 2x`
`3x` `= 3`
`x` `= 1`

 
`text(When)\ \ x = 1, y = 4`

`=>B`

Measurement, STD2 M1 2017 HSC 22 MC

A concrete water pipe is manufactured in the shape of an annular cylinder. The dimensions are shown in the diagrams.
 


 

What is the approximate volume of concrete needed to make the water pipe?

  1. `text(0.06 m)³`
  2. `text(0.09 m)³`
  3. `text(0.70 m)³`
  4. `text(0.99 m)³`
Show Answers Only

`C`

Show Worked Solution
`text(Volume)` `= text(Area of annulus) xx h`
  `= (piR^2 – pir^2) xx 2.8`
  `= (pi xx 0.45^2 – pi xx 0.35^2) xx 2.8`
  `= 0.7037…`
  `= 0.70\ text(m)³`

  
`=>C`

Algebra, STD2 A1 2017 HSC 19 MC

Young’s formula, shown below, is used to calculate the dosage of medication for children aged 1−12 years based on the adult dosage.

`D = (yA)/(y + 12)`

where     `D`   = dosage for children aged 1−12 years
`y`   = age of child (in years)
`A`   = Adult dosage

 
A child’s dosage is calculated to be 20 mg, based on an adult dosage of 40 mg.

How old is the child in years?

A.     `6`

B.     `8`

C.     `10`

D.     `12`

Show Answers Only

`text(D)`

Show Worked Solution
`D` `= (yA)/(y + 12)`
`20` `= (40y)/(y + 12)`
`20(y + 12)` `= 40y`
`20y + 240` `= 40y`
`20y` `= 240`
`y` `= 12`

`=>\ text(D)`

Measurement, STD2 M1 2017 HSC 18 MC

A skip bin is in the shape of a trapezoidal prism, with dimensions as shown.
 

What is the volume of the skip bin?

  1. `5.4\ text(m)^3`
  2. `7.776\ text(m)^3`
  3. `10.8\ text(m)^3`
  4. `15.552\ text(m)^3`
Show Answers Only

`A`

Show Worked Solution
`text(Area of trapezoid)` `= 1/2h (a + b)`
  `= 1/2 xx 1.2 xx (3.6 + 2.4)`
  `= 3.6\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= 3.6 xx 1.5`
  `= 5.4\ text(m)^3`

 
`=>A`

Measurement, STD2 M1 2017 HSC 16 MC

The benchmark for annual greenhouse gas emissions from the residential sector is 3292 kg of carbon dioxide per person per year.

A new building, planned to house 6 people, has been designed to achieve a 25% reduction on this benchmark.

What is the maximum amount of carbon dioxide per year, to the nearest kilogram, that this building is designed to emit when fully occupied?

A.     823 kg

B.     2469 kg

C.     4938 kg

D.     14 814 kg

Show Answers Only

`D`

Show Worked Solution
`text(Benchmark emissions)` `= 6 xx 3292`
  `= 19\ 752\ text(kg)`

 
`:.\ text(Max emissions of new building)`

`= 75text(%) xx 19\ 752`

`= 14\ 814\ text(kg)`

`=> D`

Financial Maths, STD2 F1 2017 HSC 11 MC

A new car was bought for $19 900 and one year later its value had depreciated to $16 300.

What is the approximate depreciation, expressed as a percentage of the purchase price?

  1. 18%
  2. 22%
  3. 78%
  4. 82%
Show Answers Only

`A`

Show Worked Solution
`text(Net Depreciation)` `= 19\ 900-16\ 300`
  `= $3600`

 

`:. %\ text(Depreciation)` `= 3600/(19\ 900) xx 100`
  `= 18.09…text(%)`

`=>A`

Financial Maths, STD2 F4 2017 HSC 10 MC

A single amount of $10 000 is invested for 4 years, earning interest at the rate of 3% per annum, compounded monthly.

Which expression will give the future value of the investment?

  1. `10\ 000 xx (1 + 0.03)^4`
  2. `10\ 000 xx (1 + 0.03)^48`
  3. `10\ 000 xx (1 + 0.03/12)^4`
  4. `10\ 000 xx (1 + 0.03/12)^48`
Show Answers Only

`D`

Show Worked Solution

`text(Compounding rate)\ = 3/100 ÷ 12= 0.03/12`

`text(Compounding periods)` `= 4 xx 12=48`

 
`:.\ text(FV) = 10\ 000 xx (1 + 0.03/12)^48`

\(\Rightarrow D\)

Measurement, STD2 M6 2017 HSC 8 MC

The diagram shows a right-angled triangle.
 


 

What is the value of  `theta`, to the nearest minute?

  1. `70°16^{′}`
  2. ` 70°17^{′}`
  3. `70°27^{′}`
  4. `70°28^{′}`
Show Answers Only

`B`

Show Worked Solution
`tan theta` `= text(opp)/text(adj)`
  `= 5.3/1.9`
  `= 2.789…`
COMMENT: An angle that has over 30″ (seconds) is rounded up to the next minute (i.e. rounded up to 70°17′).

 

`:. theta` `= 70.277…^@`
  `=70°16^{′}39.8^{″}`
  `= 70^@17^{′}`

 
`=>B`

Statistics, STD2 S1 2017 HSC 4 MC

A factory’s quality control department has tested every 50th item produced for possible defects.

What type of sampling has been used?

A.     Random

B.     Stratified

C.     Systematic

D.     Numerical

Show Answers Only

`C`

Show Worked Solution

`text(A systematic sample divides a population)`

`text(into equal sample sizes and then selects)`

`text(equally among them.)`

`=> C`

Number and Algebra, NAP-J2-12

Trevor has poured milk into some glasses.

He makes them all exactly one quarter full.

Trevor counts by quarters to find out, in total, how many full cups of milk he has.
 

  Which number does Trevor count next?

`5` `3` `1 1/2` `2 1/4` `2 1/2`
 
 
 
 
 
Show Answers Only

`1 1/2`

Show Worked Solution

`text(Add)\ 1/4\ text(for each additional cup.)`

`:.\ text(Next number)` `= 1 1/4 + 1/4`
  `= 1 1/2`

 

CHEMISTRY, M4 2009 HSC 20

  1. Calculate the mass of ethanol \(\ce{C2H6O}\) that must be burnt to increase the temperature of 210 g of water by 65°C, if exactly half of the heat released by this combustion is lost to the surroundings.
  2. The heat of combustion of ethanol is 1367 kJ mol −1.  (3 marks)

  3. What are TWO ways to limit heat loss from the apparatus when performing a first-hand investigation to determine and compare heat of combustion of different liquid alkanols?  (1 mark)

Show Answers Only

a.    3.85 grams

b.    Answers could include two of the following:

→ Use of an insulated vessel (Styrofoam cup)

→ Place the vessel as close as safely possible to the Bunsen’s flame.

→ Use a lid for the beaker

Show Worked Solution

a.    \(q=mC \Delta T = 210 \times 4.18 \times 65 = 57\ 057\ \text{J} = 57.057\ \text{kJ}\)

\(\ce{n(C2H5OH) = \dfrac{57.057}{1367} = 0.04174\ \text{mol}}\)

\(\ce{m(C2H5OH) = n \times MM = 0.04174 \times 46.068 = 1.923\ \text{g}}\)

Since half of the heat is lost to environmental surroundings.

\(\ce{m(C2H5OH)_{\text{init}} = 2 \times 1.923= 3.85\ \text{g}}\)
 

b.    Answers could include two of the following:

→ Use of an insulated vessel (Styrofoam cup)

→ Place the vessel as close as safely possible to the Bunsen’s flame.

→ Use a lid for the beaker

CHEMISTRY, M4 2013 HSC 27

A 0.259 g sample of ethanol is burnt to raise the temperature of 120 g of an oily liquid, as shown in the graph. There is no loss of heat to the surroundings.
 

Using the information shown on the graph, calculate the specific heat capacity of the oily liquid. The heat of combustion of ethanol is 1367 kJ mol–1.   (4 marks)

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\(2.13 \times 10^{3}\ \text{J kg}^{-1}\text{K}^{-1}\)

Show Worked Solution

\(\ce{MM(C2H5OH) = 2 \times 12.01 + 6 \times 1.008 + 16 = 46.068}\)

\(\ce{n(C2H5OH) = \dfrac{\text{m}}{\text{MM}} = \dfrac{0.259}{46.068} = 5.622 \times 10^{-3}\ \text{mol}}\)

\(\text{Heat combustion}(q)\ = 1367 \times 5.622 \times 10^{-3} = 7.685\ \text{kJ} = 7685\ \text{J}\)

Mean mark 59%.
\(q\) \(=mc \Delta T\)  
\(c\) \(=\dfrac{q}{m \Delta T}\)  
  \(=\dfrac{7685}{0.120 \times (50-20)} \)  
  \(=2.13 \times 10^{3}\ \text{J kg}^{-1}\text{K}^{-1} \)  

Statistics, NAP-J3-CA01

Four students recorded the number of times they participated in different sports in one week in the table below.
 

In total, which sport was participated in the most times?

Swimming Soccer Tennis Netball
 
 
 
 
Show Answers Only

`text(Swimming)`

Show Worked Solution

`text(Times each sport was played:)`

`text(Swimming – 11)`

`text(Soccer – 10)`

`text(Tennis – 8)`

`text(Netball – 5)`

`:.\ text(Swimming was participated in the most.)`

GEOMETRY, FUR1 SM-Bank 32 MC

Makoua and Macapá are two towns on the equator.

The longitude of Makoua is 16°E and the longitude of Macapá is 52°W.

How far apart are these two towns if the radius of Earth is approximately 6400 km?

A.  `4000\ text(km)`

B.  `7600\ text(km)`

C.  `1\ 367\ 200\ text(km)`

D.  `1\ 447\ 600\ text(km)`

E.   `2\ 734\ 400\ text(km)`

Show Answers Only

`B`

Show Worked Solution

2UG-2006-19MC Answer

`text(Angular Difference)` `= 52 + 16`
  `= 68°`

 

`text(Arc)\ AB` `=\ text(Distance between two towns)`
  `= 68/360 xx 2 xx pi xx r`
  `= 68/360 xx 2 xx pi xx 6400`
  `= 7595.67…\ text(km)`

 
`=>  B`

GEOMETRY, FUR1 SM-Bank 18 MC

Stockholm is located at  59°N 18°E  and Darwin is located at  13°S 131°E. 

What is the time difference between Stockholm and Darwin? (Ignore time zones and daylight saving.)

  1. 184 minutes
  2. 188 minutes
  3. 288 minutes
  4. 452 minutes
  5. 596 minutes
Show Answers Only

`D`

Show Worked Solution

`text(Stockholm is 59°N 18°E,  Darwin is 13°S 131°E)`

`text{Angular difference (longitude)}`

`= 131^@− 18^@`

`= 113^@`
 

`:.\ text(Time difference)` `= 113 xx 4`
  `= 452\ text(minutes)`

 
`⇒ D`

GEOMETRY, FUR1 SM-Bank 15 MC

Which expression will give the shortest distance, in kilometres, between Mount Isa (20°S  140°E)  and Tokyo (35°N  140°E)?

  1. `15/360 xx 2 xx pi xx 6400`
  2. `55/360 xx 2 xx pi xx 6400`
  3. `125/360 xx 2 xx pi xx 6400`
  4. `140/360 xx 2 xx pi xx 6400`
  5. `305/360 xx 2 xx pi xx 6400`
Show Answers Only

`B`

Show Worked Solution

`text(Tokyo is 35° North of the equator, Mt Isa 20° South)`

`text(Arc length) = 55/360 xx 2 xx pi xx 6400`

`=>  B`

GEOMETRY, FUR2 SM-Bank 25

A ship sails due South from Channel-Port-aux-Basques, Canada,  47°N  59°W  to Barbados, 13°N  59°W.

How far did the ship sail, to the nearest kilometre? Assume that the radius of Earth is 6400 km.    (2 marks)

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 `3798\ text{km (nearest km)}`

Show Worked Solution
`text(Angular difference in latitude)` `=47` `-13`
  `=34^@`

 
`text{(No difference in longitude)}`
 

`:.\ text(Distance)` `=34/360` `xx 2 pi r`
  `=34/360` `xx 2` `xx pi` `xx 6400`
  `= 3797.836…`
  `= 3798` `text{km   (nearest km)}`

GEOMETRY, FUR2 SM-Bank 15

Osaka is at  34°N, 135°E, and Denver is at  40°N, 105°W. 

  1. Show that there is a 16-hour time difference between the two cities.
    (Ignore time zones.)    (1 mark)
  2. John lives in Denver and wants to ring a friend in Osaka. In Denver it is 9 pm Monday.     

     

    What time and day is it in Osaka then?   (1 mark)

  3. John’s friend in Osaka sent him a text message which happened to take 14 hours to reach him. It was sent at 10 am Thursday, Osaka time.

     

    What was the time and day in Denver when John received the text?    (1 mark)

Show Answers Only
  1. `text{Proof (See Worked Solutions)}`
  2. `1\ text(pm Tuesday)`
  3. `8\ text(am Thursday)`
Show Worked Solution
a.   `text(Longitude difference)` `= 135 + 105`
    `= 240^@`

 

`text(Time difference)` `= 240/15`
  `= 16\ text(hours   … as required)`

 

b.  `text(Denver is behind Osaka time because it is further west.)`

`:.\ text(Time in Osaka)` `= 9\ text(pm Monday plus 16 hours)`
  `= 1\ text(pm Tuesday)`

 

c.  `text(Denver is 16 hours behind Osaka)`

`:.\ text(John will receive the text at 10 am Thursday less 16)`

`text{plus 14 hours (i.e. 8 am Thursday.)}`

GEOMETRY, FUR2 SM-Bank 14

Pontianak has a longitude of 109°E, and Jarvis Island has a longitude of 160°W.

Both places lie on the Equator

  1. Find the shortest great circle distance between these two places, to the nearest kilometre. You may assume that the radius of the Earth is 6400 km.    (2 marks)
  2. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island. What is the latitude and longitude of Rabaul?     (1 mark)
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  1. `10\ 165\ text(km)\ \ \ text{(nearest km)}`
  2. `152^@ text(E)`
Show Worked Solution
a.   `text(Longitude difference)` `= 109 + 160`
    `= 269^@`

 

`=> text(Shortest distance)\ text{(by degree)}` `= 360 – 269`
  `= 91^@`

 

`:.\ text(Shortest distance)` `= 91/360 xx 2 pi r`
  `= 91/360 xx 2 xx pi xx 6400`
  `= 10\ 164.79…`
  `=10\ 165\ text(km)\ text{(nearest km)}`

 

b.   `text(Latitude)`
  `4^@\ text(South of Jarvis Island)`
  `text(S)text(ince Jarvis Island is on equator)`
  `=>\ text(Latitude is)\ 4^@ text(S)`
   
  `text(Longitude)`
  `text(Jarvis Island is)\ 160^@ text(W)`
  `text(Rubail is)\ 48^@\ text(West of Jarvis Island, or 208° West)`
  `text(which is)\ 28^@\ text{past meridian (180°)}`

 

`=>\ text(Longitude)` `= (180\ -28)^@ text(E)`
  `= 152^@ text(E)`

 

`:.\ text(Position is)\ (4^@text{S}, 152^@text{E})`

GEOMETRY, FUR2 SM-Bank 13

Melbourne is located at (38°S, 145°E) and Dubai is located at (24°N, 55°E).

  1. Calculate the difference in longitude between Melbourne and Dubai.  (1 mark)
  2. Show that the time difference between Melbourne and Dubai is 6 hours.  (1 mark)
  3. A plane leaves Melbourne on Friday at 11.30 pm. The flight time to Dubai is 15 hours.

     

    What will be the time and the day in Dubai when the plane is due to land?  (1 marks)

Show Answers Only
  1. `90^@`
  2. `text(See Worked Solutions)`
  3. `text(8:30 am on Saturday)`
Show Worked Solution

a.   `text(Difference in longitude)`

`= 145 – 55`

`= 90^@`

 

b.   `text(S)text(ince)\ 1^@ = 4\ text(min time difference,)`

`text(Time difference)`

`= 90 xx 4`

`= 360\ text(mins)`

`= 6\ text(hours  … as required)`

 

c.    `text(Arrival time)` `= 11:30 + 15\ text(hours)`
    `= 14:30\ text{(Melbourne time)}`

 

`:.\ text(Arrival time in Dubai time)`

`= 14:30 – 6\ text(hours)`

`= 8:30\ text(am on Saturday)`

GEOMETRY, FUR2 SM-Bank 12

This diagram represents Earth. `O` is at the centre, and `A` and `B` are points on the surface.
 

2UG-2005-27b
 

Find the shortest great circle distance from `A` to `B`.

Give your answer in to the nearest km.   (2 marks)

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`text{4803 km  (nearest km)}`

Show Worked Solution

`A : 35^@text(N)\ 20^@text(E)\ \ \ \ B:8^@text(S)\ 20^@text(E)`

HSC 2005 27b

`text(Angular difference)` `= 35^@ + 8^@`
  `= 43^@`

 

`:.\ text(Distance from)\ A\ text(to)\ B`

`= 43/360 xx 2 xx pi xx r`

`= 43/360 xx 2 xx pi xx 6400`

`= 4803.1…`

`= 4803\ text{km  (nearest km)}`

Probability, MET1 2012 VCAA 8b

The probability density function `f` of a random variable `X` is given by
 

`qquad qquad f(x) = {((x + 1)/12, 0 <= x <= 4), (\ \ 0, text{otherwise}):}`
 

Find the value of `b` such that  `text(Pr) (X <= b) = 5/8.`  (3 marks)

Show Answers Only

`3`

Show Worked Solution
`1/12 int_0^b (x + 1)\ dx` `= 5/8`
`[1/2 x^2 + x]_0^b` `= 15/2`
`1/2b^2 + b` `= 15/2`
`b^2 + 2b – 15` `= 0`
`(b + 5) (b – 3)` `= 0`

 

`:. b = 3,\ \ \ b in (0, 4)`

Calculus, MET1 2006 VCAA 3a

Let  `y = x tan(x)`. Evaluate  `(dy)/(dx)`  when `x = pi/6`.  (3 mark)

Show Answers Only

`(3sqrt3 + 2pi)/9`

Show Worked Solution

`text(Using the product rule:)`

MARKER’S COMMENT: A common error was not knowing the exact trig function values.

`d/(dx)(uv)` `= u′v + uv′`
`(dy)/(dx)` `= tan(x) + x/(cos^2(x))`

 

`text(When)\ x = pi/6,`

`(dy)/(dx)` `= tan(pi/6) + (pi/6)/((cos(pi/6))^2)`
  `= 1/sqrt3 + pi/6 xx (2/sqrt3)^2`
  `= sqrt3/3 + (4pi)/18`
  `= (3sqrt3 + 2pi)/9`

Calculus, MET1 2009 VCAA 1b

For  `f(x) = (cos(x))/(2x + 2)`  find  `f prime (pi).`  (3 marks)

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`1/(2 (pi + 1)^2)`

Show Worked Solution

`text(Using Quotient Rule:)`

MARKER’S COMMENT: A majority of students did not substitute  `x=pi`  correctly.
`(g/h)′` `= (g′ h – gh′)/h^2`
`f prime (x)` `= (-sin (x) (2x + 2) – 2 cos (x))/(2x + 2)^2`
`:. f prime (pi)` `= (-sin (pi) (2pi + 2) – 2 cos (pi))/(2pi + 2)^2`
  `= (0 – 2 (-1))/[2 (pi + 1)]^2`
  `= 2/(4(pi + 1)^2)`
  `= 1/(2 (pi + 1)^2)`

Calculus, MET1 SM-Bank 5

The function with rule  `f(x)`  has derivative  `f′(x) =  cos\ 3x`.

If  `f(pi/6) = 1,`  find  `f(x).` (3 marks)

Show Answers Only

`f(x)= 1/3 sin\ 3x + 2/3`

Show Worked Solution
`int f(x)\ dx` `=int cos\ 3x\ dx`
  `= 1/3 sin\ 3x + c`
`f(pi/6)`  `= 1/3\ [sin\ (3 xx pi/6) + c`
`1`  `= 1/3\ sin\ pi/2+c`
`c`  `= 2/3`

 
`:.f(x)= 1/3 sin\ 3x + 2/3`

Calculus, MET2 2007 VCAA 20 MC

The average value of the function  `y = tan (2x)`  over the interval  `[0, pi/8]`  is

  1. `2/pi log_e (2)`
  2. `pi/4`
  3. `1/2`
  4. `4/pi log_e 2`
  5. `8/pi`
Show Answers Only

`A`

Show Worked Solution
`y_text(average)` `= 1/(pi/8 – 0) int_0^(pi/8) (tan 2x)\ dx`
  `= 8/pi  int_0^(pi/8) (tan 2x)\ dx`
  `= (2 log_e (2))/pi`

`=>   A`

Probability, MET2 2007 VCAA 18 MC

The heights of the children in a queue for an amusement park ride are normally distributed with mean 130 cm and standard deviation 2.7 cm. 35% of the children are not allowed to go on the ride because they are too short.

The minimum acceptable height correct to the nearest centimetre is

  1. 126
  2. 127
  3. 128
  4. 129
  5. 130
Show Answers Only

`D`

Show Worked Solution

`X = text(height)`

`X ∼ N (130, 2.7^2)`

vcaa-2007-meth2-18

`text(Pr)(X < a)` `= 0.35\ \ \ [text(CAS: invNorm)(0.35, 130, 2.7)]`
`a` `= 128.96`

 
`=>   D`

Calculus, MET2 2007 VCAA 13 MC

For the graph of  `y = 4x^3 + 27x^2-30x + 10`  the subset of `R` for which the gradient is negative is given by the interval

  1. `(0.5, 5.0)`
  2. `(-4.99, 0.51)`
  3. `(-oo, 1/2)`
  4. `(-5, 1/2)`
  5. `(2.25, oo)`
Show Answers Only

`D`

Show Worked Solution
`y` `=4x^3 + 27x^2 – 30x + 10`
`y′` `=12x^2 + 54x – 30`

 

`text(Sketch the graph:)`

vcaa-2007-meth2-13

`:.\ text(Gradient is negative for)\ \  x in (– 5, 1/2).`

`=>   D`

Calculus, MET2 2007 VCAA 12 MC

Let  `f: R -> R`  be a differentiable function such that

  • `f prime(3) = 0`
  • `f prime(x) < 0`  when  `x < 3`  and when  `x > 3`

When  `x = 3`, the graph of  `f`  has a

  1. local minimum
  2. local maximum
  3. stationary point of inflection
  4. point of discontinuity
  5. gradient of 3
Show Answers Only

`C`

Show Worked Solution

`text(Use a gradient table:)`

vcaa-2007-meth2-12

`:.\ text(Point of inflection at)\ \ x = 3.`

`=>   C`