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Mechanics, EXT2* M1 2015 HSC 13a

A particle is moving along the -axis in simple harmonic motion. The displacement of the particle is metres and its velocity is ms. The parabola below shows as a function of .

2015 13a

  1. For what value(s) of is the particle at rest?  (1 mark)

  2. What is the maximum speed of the particle?  (1 mark)

  3. 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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i.   

 

ii.   

 

iii. 

♦ Mean mark 41%.

 
 

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 cm and the chord subtends an angle at the centre of the circle.
 

2015 12d
 

  1. Show that  .  (1 mark)
  2. Hence, or otherwise, show that  .  (2 marks)
  3. 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)

 

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(i)   

 
 

 

(ii)   

 

 

(iii)  
 
 
   
 
   
 
 
 

Trig Ratios, EXT1 2015 HSC 12c

A person walks 2000 metres due north along a road from point to point . The point is due east of a mountain , where is the top of the mountain. The point is directly below point and is on the same horizontal plane as the road. The height of the mountain above point is metres.

From point , the angle of elevation to the top of the mountain is 15°.

From point , the angle of elevation to the top of the mountain is 13°.
 

Trig Ratios, EXT1 2015 HSC 12c
 

  1. Show that .  (1 mark)
  2. Hence, find the value of  .  (2 marks)
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(i)    

Trig Ratios, EXT1 2015 HSC 12c Answer1

 

 

(ii)   

 

 

Trig Ratios, EXT1 2015 HSC 12c Answer2 
 
 
 

Quadratic, EXT1 2015 HSC 12b

The points    and    lie on the parabola  .

The equation of the chord    is given by  .  (Do NOT prove this.)

  1. Show that if    is a focal chord then  .  (1 mark)
  2. If    is a focal chord and    has coordinates  , what are the coordinates of    in terms of  ?  (2 marks)

 

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(i)  
 

Quadtratic, EXT1 2015 HSC 12b Answer1

 

 

 

(ii) 

 

Statistics, STD2 S4 2015 HSC 28e

The shoe size and height of ten students were recorded.

  1. Complete the scatter plot AND draw a line of fit by eye.  (2 marks)
     
     
  2. 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)

  3. 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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  2. `13\ text{cm  (or close given LOBF drawn)}


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i.    
      2UG 2015 28e Answer

ii.   ½

 

 

iii.   

♦ Mean mark (c) 39%.

Algebra, STD2 A1 2015 HSC 28d

The formula    is used to convert temperatures between degrees Fahrenheit and degrees Celsius .

Convert 3°C to the equivalent temperature in Fahrenheit.  (2 marks)

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Statistics, STD2 S1 2015 HSC 27d

In a small business, the seven employees earn the following wages per week:

  1.  Is the wage of $970 an outlier for this set of data? Justify your answer with calculations.  (3 marks)

  2.  Each employee receives a $20 pay increase.

     

     What effect will this have on the standard deviation?  (1 mark)

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i.   

ii.   

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i.   

 

 

♦ Mean mark (i) 39%.


ii.   

Measurement, STD2 M7 2015 HSC 27a

At a particular time during the day, a tower of height 19.2 metres casts a shadow. At the same time, a person who is 1.65 metres tall casts a shadow 5 metres long.

  

What is the length of the shadow cast by the tower at that time?  (2 marks)

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FS Comm, 2UG 2015 HSC 26g

Pat’s mobile phone plan is shown.

2015 26g

Last month Pat:

• made calls to the value of
• sent SMS messages
• sent MMS messages
• used GB of data.

What was the total of Pat’s phone bill for last month?  (3 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.

  1. What is the relative frequency of selecting a red jelly bean?  (1 mark)

  2. Based on this table of relative frequencies, what is the probability of NOT selecting a black jelly bean?  (1 mark)

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i. 

 

ii.  

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)

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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)

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Financial Maths, STD2 F1 2015 HSC 25 MC

An insurance company offers customers the following discounts on the basic annual premium for car insurance.

2015 25 mc

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?

  1.    $482.44
  2.    $515.50
  3.    $541.60
  4.    $557.60
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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    is the number of standard drinks consumed

is the number of hours drinking

is the person's mass in kilograms
 

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Calculus, EXT1* C1 2015 HSC 14a

In a theme park ride, a chair is released from a height of    metres and falls vertically. Magnetic brakes are applied when the velocity of the chair reaches    metres per second.
 

2015 2ua 14a
 

The height of the chair at time seconds is metres. The acceleration of the chair is given by   . At the release point,  .

  1. Using calculus, show that  .  (2 marks)

  2. How far has the chair fallen when the magnetic brakes are applied?  (2 marks)

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i. 

 
 

 
 
 

 

 

 

ii. 

 

 
 

 

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?

  1. $8919
  2. $11 156
  3. $49 173
  4. $49 316
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Measurement, 2UG 2015 HSC 14 MC

Stockholm is located at and Darwin is located at . 

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

(A)    minutes

(B)    minutes

(C)    minutes

(D)    minutes

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Statistics, STD2 S1 2015 HSC 6 MC

The times, in minutes, that a large group of students spend on exercise per day are presented in the box‑and‑whisker plot.
 

What percentage of these students spend between 40 minutes and 60 minutes per day on exercise?

  1. 17%
  2. 20%
  3. 25%
  4. 50%
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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?

  1. Categorical, ordinal
  2. Categorical, nominal
  3. Numerical, continuous
  4. Numerical, discrete
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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 hours is litres. The rate of change of the volume is given by

At time  , the volume of water in the pool is 1200 litres and is increasing.

  1. After what time does the volume of water first start to decrease?  (2 marks)

  2. Find the volume of water in the pool when  .  (2 marks)

  3. What is the greatest volume of water in the pool?  (1 mark)

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i.   

♦ Mean mark (i) 37%.

 

ii.   
 
 

 

 

 
 

 

iii. 

♦♦ Mean mark (iii) 27%.

Plane Geometry, 2UA 2015 HSC 15b

The diagram shows which has a right angle at . The point is the midpoint of the side . The point is chosen on such that . The line segment is produced to meet the line at .

Copy or trace the diagram into your writing booklet.

  1. Prove that is similar to   (2 marks)
  2. Explain why is isosceles.  (1 mark)
  3. Show that   (2 marks)
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(i) 

 

♦ Mean mark 45%.

(ii) 

 

(iii) 

♦♦ Mean mark 23%.
 
 
 
 

 

Calculus, EXT1* C1 2015 HSC 15a

The amount of caffeine, , in the human body decreases according to the equation

 

where is measured in mg and is the time in hours.

  1. Show that    is a solution to  where is a constant.

     

    When , there are 130 mg of caffeine in Lee’s body.  (1 mark)

  2. Find the value of   (1 mark)

  3. What is the amount of caffeine in Lee’s body after 7 hours?   (1 mark)

  4. What is the time taken for the amount of caffeine in Lee’s body to halve?  (2 marks)

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i.
 
   
   

 

 

ii. 

 

iii. 

 
 
 

 

 

iv. 

 
 

 

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    be the amount owing at the end of    months and    be the monthly repayment.

  1. Show that    (1 mark)

  2. Show that    (2 marks)

  3. 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)

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.

  1. After how many more months will the amount owing be completely repaid?  (3 marks)

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i. 
 
 
 

 

ii. 

 
 

 

iii. 

 
 
 
 

 

iv. 

 
 

 
 

 

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.
 

2015 2ua 14b
 

  1. Show that the probability of Saturday being dry is .  (1 mark)

  2. What is the probability of both Saturday and Sunday being wet?  (2 marks)

  3. What is the probability of at least one of Saturday and Sunday being dry?  (1 mark)

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i. 


 

ii. 


 

iii. 

Calculus, 2ADV C3 2015 HSC 13c

Consider the curve  .

  1. Find the stationary points and determine their nature.   (4 marks)

  2. Given that the point    lies on the curve, prove that there is a point of inflection at  .  (2 marks)

  3. Sketch the curve, labelling the stationary points, point of inflection and -intercept.  (2 marks)

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  3.  
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i.
 
 

 

 
 

 

 

ii. 

 

 

iii. 

Quadratic, 2UA 2015 HSC 12e

The diagram shows the parabola with focus A tangent to the parabola is drawn at

  1. Find the equation of the tangent at the point .   (2 marks)
  2. What is the equation of the directrix of the parabola?   (1 mark)
  3. The tangent and directrix intersect at .
    Show that lies on the -axis.   (1 mark)

  4. Show that is isosceles.   (1 mark)
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  3.  
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(i)   


 

 

(ii) 
 

(iii) 

 

 

(iv) 

Plane Geometry, 2UA 2006 HSC 10b

A rectangular piece of paper has sides cm and cm. The point is the midpoint of . The points and are to be chosen on and respectively, so that when the paper is folded along , the corner that was at lands on the edge at . Let cm and cm.

Copy or trace the diagram into your writing booklet.

  1. Show that .  (1 mark)
  2. Let be the point on for which is perpendicular to .
  3. By showing that , deduce that .  (3 marks)
  4. Show that the area, , of is given by
    1.   (1 mark)
    2.  
  5. Use the fact that to find the possible values of .  (2 marks)
  6. Find the minimum possible area of .  (3 marks)
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  5.  
  6. ²
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(i)     

 

 
 
 

 

(ii) 

 

   
 
 

 

(iii) 
 
 

 

(iv) 

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(v) 

 
 
 
 
 

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  ²

 

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Calculus, 2ADV C4 2006 HSC 9b

During a storm, water flows into a 7000-litre tank at a rate of litres per minute, where and is the time in minutes since the storm began.

  1. At what times is the tank filling at twice the initial rate?  (2 marks)

  2. Find the volume of water that has flowed into the tank since the start of the storm as a function of .  (1 mark)

  3. Initially, the tank contains 1500 litres of water. When the storm finishes, 30 minutes after it began, the tank is overflowing.

     

    How many litres of water have been lost?  (2 marks)

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i. 

 

ii. 
 
 

 

 

iii. 


 

 

 

 

Quadratic, 2UA 2006 HSC 7c

  1. Write down the discriminant of    where    is a constant.  (1 mark)
  2. Hence, or otherwise, find the values of    for which the parabola   does not intersect the line  .  (2 marks)

 

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(i) 

 
 
 

 

(ii) 

 

HSC quadratic

Calculus, EXT1* C1 2006 HSC 6b

A rare species of bird lives only on a remote island. A mathematical model predicts that the bird population, , is given by

where is the number of years after observations began.

  1. According to the model, how many birds were there when observations began?  (1 mark)

  2. According to the model, what will be the rate of change in the bird population ten years after observations began?  (2 marks)

  3. What does the model predict will be the limiting value of the bird population?  (1 mark)

  4. The species will become eligible for inclusion in the endangered species list when the population falls below . When does the model predict that this will occur?  (2 marks)

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i. 

 

 

 

ii. 
 

 

 
 
 

 

 

iii. 

 

iv. 

­
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Calculus, 2ADV C4 2006 HSC 5b

  1. Show that   (1 mark)


  2.   
     
    2006 5b
     
    The shaded region in the diagram is bounded by the curve    and the lines    and 

     

    Using the result of part (i), or otherwise, find the area of the shaded region.  (3 marks)

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  2. ²
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i.   
 

 

ii. 

­

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²

Calculus, 2ADV C3 2004 HSC 4b

Consider the function  .

  1. Find the coordinates of the stationary points of the curve    and determine their nature.   (3 marks)

  2. Sketch the curve showing where it meets the axes.   (2 marks)

  3. Find the values of    for which the curve    is concave up.   (2 marks)

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  2.  
    1. Geometry and Calculus, 2UA 2004 HSC 4b Answer
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(i)   
 
 

 

 

 

 

(ii)  

 Geometry and Calculus, 2UA 2004 HSC 4b Answer

(iii)  

 

Trigonometry, 2ADV T1 2004 HSC 3c

Trig Ratios, 2UA 2004 HSC 3c
 

The diagram shows a point    which is  30 km due west of the point  .

The point    is 12 km from    and has a bearing from    of  070°. 

  1. Find the distance of    from  .   (2 marks)

  2. Find the bearing of    from  .   (2 marks)

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  1.  
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i.   

Trig Ratios, 2UA 2004 HSC 3c Answer

 
 
 

 

ii.   

 
 

 

Probability, 2ADV S1 2006 HSC 4c

A chessboard has 32 black squares and 32 white squares. Tanya chooses three different squares at random.

  1. What is the probability that Tanya chooses three white squares?  (2 marks)

  2. What is the probability that the three squares Tanya chooses are the same colour?.  (1 mark)

  3. What is the probability that the three squares Tanya chooses are not the same colour?  (1 mark)

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i. 
 

 

ii. 

 

iii.