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Complex Numbers, EXT2 N1 2009 HSC 2c

The points    and    on the Argand diagram represent the complex numbers    and   respectively.
 


 

Copy the diagram into your writing booklet, and mark on it the following points:

  1. the point    representing     (1 mark)
  2. the point    representing     (1 mark)
  3. the point    representing     (1 mark)

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i, ii, and iii.

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i, ii, and iii.

Polynomials, EXT2 2010 HSC 6c

  1. Expand    using the binomial theorem.   (1 mark)

  2. Expand    using de Moivre’s theorem, and hence show that

    1. .   (3 marks)

  3. Deduce that

  4.  is one of the solutions to

    1. .   (1 mark)

  5. Find the polynomial    such that  .   (1 mark)

  6. Find the value of    such that  .   (1 mark)

  7. Hence find an exact value for
    1. .   (1 mark)

 

 

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(i)   

 

(ii) 

 
 
 
 

 

(iii)  

 

 

(iv) 

 

(v)   

 

(vi)  

♦♦ Mean mark part (vi) 27%.
 

 

Conics, EXT2 2010 HSC 5a

The diagram shows two circles,  and  , centred at the origin with radii    and  , where  .

The point    lies on    and has coordinates  .

The point    is the intersection of    and  .

The point    is the intersection of the horizontal line through    and the vertical line through  .

Conics, EXT2 2010 HSC 5a

  1. Write down the coordinates of  .   (1 mark)
  2. Show that    lies on the ellipse
    .   (1 mark)

  3. Find the equation of the tangent to the ellipse
     at  .   (2 marks)

  4. Assume that is not on the -axis.
  5. Show that the tangent to the circle    at  , and the tangent to the ellipse
     at  , intersect at a point on the -axis.   (2 marks)
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(i)  

 

(ii) 

 
 
 

 

 

(iii)  

 

 

 

 
 

 

 

(iv) 

 

 

 

 

Conics, EXT2 2010 HSC 3d

The diagram shows the rectangular hyperbola  , with  .

Conics, EXT2 2010 HSC 3d

The points  ,  and    are points on the hyperbola,
with  .

  1. The line    is the line through    perpendicular to  .
  2. Show that the equation of    is
    1. .   (2 marks)

  4. The line    is the line through    perpendicular to  .
  5. Write down the equation of  .   (1 mark)
  6. Let    be the point of intersection of the lines    and  .
  7. Show that    is the point  .   (2 marks)
  8. Give a geometric description of the locus of  .   (1 mark)
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(i) 

Conics, EXT2 2010 HSC 3d Answer1
 
 
 

 

 

 

 

(ii)  
   
   
   

 

 

 

 

(iii) 

 

 

 

(iv) 

♦♦ Mean mark part (iv) 1%.
MARKER’S COMMENT: Most students found the correct locus, although very few indicated that it was only the branch in the 1st quadrant.

 

 

Complex Numbers, EXT2 N1 2010 HSC 2a

Let  .

  1. Find    in the form  .   (1 mark)

  2. Find    in the form  .   (1 mark)

  3. Find   in the form  .   (2 marks)

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i.
 
   
   

 

ii.   
   

 

iii.   
   
   
   

Conics, EXT2 2011 HSC 3d

The equation    represents a hyperbola.

  1. Find the eccentricity    (1 mark)
  2. Find the coordinates of the foci.  (1 mark)
  3. State the equations of the asymptotes.  (1 mark)
  4. Sketch the hyperbola.  (1 mark)
  5. For the general hyperbola  
  6. ,
  8. describe the effect on the hyperbola as    (1 mark)

 

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(i)  

 

 

 

(ii)  

 

(iii) 

 

(iv)   HSC 2011 3di

 

♦♦♦ Mean mark part (v) 24%.

 

(v)  

Complex Numbers, EXT2 N2 2011 HSC 2d

  1. Use the binomial theorem to expand    (1 mark)

  2. Use de Moivre’s theorem and your result from part (i) to prove that
     
          (3 marks)

  3. Hence, or otherwise, find the smallest positive solution of
     
          (2 marks)

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i.   

 

ii. 


 

 

 

iii.   

 

Complex Numbers, EXT2 N1 2011 HSC 2a

Let    and  

  1.  Find     (1 mark)

  2.  Find     (1 mark)

  3.  Express    in the form  , where    and    are real numbers.   (2 marks)

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i.   

 

 

ii.   
 

 

 iii.   
 
 
 

Conics, EXT2 2012 HSC 12b

The diagram shows the ellipse    with  . The ellipse has focus    and eccentricity  . The tangent to the ellipse at    meets the -axis at  . The normal at    meets the -axis at  .  

Conics, EXT2 2012 HSC 12b

  1. Show that the tangent to the ellipse at    is given by the equation
    1. .   (2 marks) 
  2. Show that the -coordinate of    is  .   (2 marks)
  3. Show that    (2 marks) 
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(i)   

(ii) 

(iii) 

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(i)   

 

 

 

(ii) 

 

 

(iii)

   

 

 

 
 

Complex Numbers, EXT2 N1 2012 HSC 11d

  1. Write    in modulus-argument form.  (2 marks)

  2. Hence express    in the form  , where and are real.  (1 mark)

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i.
 

 

   
   

 

ii.  
   
   
   
   

Conics, EXT2 2013 HSC 12d

The points and , where , lie on the rectangular hyperbola with equation  

The tangent to the hyperbola at intersects the -axis at and the -axis at . Similarly, the tangent to the hyperbola at intersects the -axis at and the - axis at .

  1. Show that the equation of the tangent at is   (2 marks)
  2. Show that are on a circle with centre   (2 marks)
  3. Prove that is parallel to   (1 mark)
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(i) 

 

 

 

(ii)   

 

 

 

(iii) 

 
 
 
 

 

Complex Numbers, EXT2 N1 2013 HSC 11a

Let    and 

  1. Find    (1 mark)
  2. Express in modulus–argument form.  (2 marks)
  3. Write in its simplest form.  (2 marks)
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i.   

 

 

ii.   
 
 
 
MARKER’S COMMENT: The directive “in its simplest form” required students to convert to 1.

 

iii.   
 
 

Conics, EXT2 2014 HSC 13c

The point   is the focus of the hyperbola   on the positive -axis.

The points    and    lie on the asymptotes of the hyperbola, where .

The point   is the midpoint of .

Conics, EXT2 2014 HSC 13c

  1. Show that lies on the hyperbola.  (1 mark)
  2. Prove that the line through and is a tangent to the hyperbola at .  (3 marks)
  3. Show that  .  (2 marks)
  4. If    and    have the same -coordinate, show that    is parallel to one of the asymptotes of the hyperbola.  (2 marks)
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(i)  

 
 
 
 

 

 

(ii)

 
 

 

   

 

 

 

 

(iii)  
   
 
 
 
 

 

 

(iv) 

 
 
 
 

Calculus, EXT2 C1 2014 HSC 12d

Let  , where    is an integer and  .

  1. Show that  .  (1 mark)

  2. Show that
     
         .  (2 marks)

  3. Hence, or otherwise, find
     
         .  (2 marks)

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i.   

 
   
 
 

 

♦ Mean mark 41%.
STRATEGY: The denominator of the proof, , should flag the potential existence of in the integral.
ii.  
 

 

 
 
 
 
 

 

iii.  
 
 

Harder Ext1 Topics, EXT2 2014 HSC 12b

It can be shown that  .   (Do NOT prove this.)

Assume that   is a solution of . 

  1. Show that
    1. .  (1 mark)
  2. Hence, or otherwise, find the three real solutions of  .  (2 marks)
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(i)  

 

 

MARKER’S COMMENT: Many students failed to complete the final step of part (ii) and lost an easy mark!
(ii)  
 
 
 

Complex Numbers, EXT2 N1 2014 HSC 11a

Consider the complex numbers    and  .

  1. Express    in modulus–argument form.  (2 marks)

  2. Express in the form  , where    and    are real numbers.  (2 marks)

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i.  
    
 
 
   

 

ii.  
   
   
   
   

Linear Functions, 2UA 2007 HSC 3a

2007 3a

In the diagram, , and are the points , and respectively.

Copy or trace this diagram into your writing booklet.

  1. Find the distance .  (1 mark)
  2. Find the midpoint of .  (1 mark)
  3. Show that .  (2 marks)
  4. Find the midpoint of and hence explain why is a rhombus.  (2 marks)
  5. Hence, or otherwise, find the area of .  (1 mark)
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  5. ²
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(i)  

 
 
 
 

 

(ii) 
 
 

 

(iii) 

 
 
 
 

 

(iv) 
 

 

(v) 

²

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 
 
 
 

Plane Geometry, EXT1 2015 HSC 12a

In the diagram, the points , , and are on the circumference of a circle, whose centre lies on . The chord intersects the diameter at . The tangent at passes through the point .

It is given that    and  .

 

 

Copy or trace the diagram into your writing booklet.

  1. What is the size of  ?  (1 mark)
  2. What is the size of  ?  (1 mark)
  3. Find, giving reasons, the size of  .  (2 marks)
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(i)   

Plane Geometry, EXT1 2015 HSC 12a Answer
 

 

(ii)    
   

 

(iii) 
   
   
   
   
   

Functions, EXT1 F2 2015 HSC 11f

Consider the polynomials    and  .

  1. Given that    is divisible by  , show that  .  (1 mark)

  2. Find all the zeros of    when  .  (2 marks)

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i.   
 

 

 

ii. 


 

 
 

 

Financial Maths, STD2 F1 2015 HSC 3 MC

Gayle’s gross pay each week is $952.25 .

The following deductions are taken from her gross pay each week:

• tax $180.93
• superannuation $85.70
• union membership $21.40
• health fund $38.15.

What is Gayle’s net pay each week?

  1.    $326.18
  2.    $626.07
  3.    $771.32
  4.    $952.25
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Calculus, 2ADV C4 2015 HSC 16a

The diagram shows the curve with equation  . The curve intersects the -axis at points . The point on the curve has the same -coordinate as the -intercept of the curve.
 


 

  1. Find the -coordinates of points   (1 mark)

  2. Write down the coordinates of   (1 mark)

  3. Evaluate    (1 mark)

  4. Hence, or otherwise, find the area of the shaded region.  (2 marks)

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  4. ²
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i.   
 

 

 

ii.   


 

 

 

 

iii. 

 

iv.  

♦ Mean mark 49%.

²

²

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. 

 
 

 

Data, 2UG 2006 HSC 23d

The graph shows the amounts charged by Company and Company to deliver parcels of various weights.

2UG-2006-23d

  1. How much does Company charge to deliver a kg parcel?  (1 mark)
  2. Give an example of the weight of a parcel for which both Company and Company charge the same amount.  (1 mark)
  4. For what weight(s) is it cheaper to use Company ?  (2 marks)
  5. What is the rate per kilogram charged by Company for parcels up to kg?  (1 mark)
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(i)  

 

(ii) 

 

(iii) 

 

(iv)  

 

Linear Functions, 2UA 2004 HSC 2a

The diagram shows the points    and  .

The line    is drawn perpendicular to the  -axis through the point  .
 

Linear Functions, 2UA 2004 HSC 2a 
 

  1. Calculate the length of the interval  .   (1 mark)
  2. Find the gradient of the line  .   (1 mark)
  3. What is the size of the acute angle between the line    and the line  ?   (1 mark)
  4. Show that the equation of the line    is  .    (1 mark)
  5. Copy the diagram into your writing booklet and shade the region defined by  .   (1 mark)
  6. Write down the equation of the line  .   (1 mark)
  7. The point    is on the line    such that    is perpendicular to  . Find the coordinates of  .   (2 marks)

 

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  5. Linear Functions, 2UA 2004 HSC 2a Answer
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(i)   
 
 
 
 

 

(ii)  
   
   

 

(iii)  

 

(iv)  

 

 

(v)

Linear Functions, 2UA 2004 HSC 2a Answer

 

(vi)  
 

 

(vii)  
 
 

 

 

GRAPHS, FUR1 2006 VCAA 3-4 MC

GRAPHS, FUR1 2006 VCAA 3-4 MC

A gas-powered camping lamp is lit and the gas is left on for six hours. During this time the lamp runs out of gas.

The graph shows how the mass, , of the gas container (in grams) changes with time, (in hours), over this period. 

 

Part 1

Assume that the loss in weight of the gas container is due only to the gas being burnt.

From the graph it can be seen that the lamp runs out of gas after

A.     

B.     

C.     

D.     

E. 

 

Part 2

Which one of the following rules could be used to describe the graph above?

A.  

B.  

C.  

D.  

E.  

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♦ Mean mark 50%.

 
 
 

 

GRAPHS, FUR1 2007 VCAA 3 MC

GRAPHS, FUR1 2007 VCAA 3 MC

The graph above represents the temperature, in degrees Celsius, over a 16-hour period.

During this period, the minimum temperature occurred at

A.     

B.     

C.     

D.     

E.   

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GRAPHS, FUR1 2008 VCAA 2 MC

Initially there are 5000 litres of water in a tank. Water starts to flow out of the tank at the constant rate of 2 litres per minute until the tank is empty.

After  minutes, the number of litres of water in the tank, , will be

A.   

B.   

C.   

D.   

E.    

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