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v1 Measurement, STD2 M1 2016 HSC 30c

A landscape artist was commissioned to design a garden consisting of part of a circle, with centre `O`, and a rectangle, as shown in the diagram. The radius `OC` of the circle is 20 m, the width `BC` of the rectangle is 10 m, and `DOC` is 100°.
 

What is the area of the whole garden, correct to the nearest square metre?  (5 marks)

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`6281\ text{m²  (nearest m²)}`

Show Worked Solution

`text(In)\ \triangle ODC,`

`sin50^@` `= (ED)/20`
`ED` `= 20 \times \sin50^@`
  `= 34.472`
`:. DC` `= 2 \times 34.472 = 68.944\ \text{m}`

 

`cos50^@` `= (OE)/20`
`:. OE` `= 20 \times \cos50^@ = 28.925`

 

`text(Area of)\ \triangle ODC`

`= \frac{1}{2} \times 68.944 \times 28.925 = 997.12\ \text{m}^2`

 

`text(Area of rectangle ABCD)` `= 10 \times 68.944 = 689.44\ \text{m}^2`

 

`text(Area of major sector DOAC)`

`= \pi \times 20^2 \times \frac{260}{360} = 4594.58\ \text{m}^2`

 

`:.\ \text{Area of garden}`

`= 997.12 + 689.44 + 4594.58 = 6281.14`

`= 6281\ \text{m² (nearest m²)}`

Advanced Trigonometry, 2ADV T2 SM-Bank 33

Given  \(\cos\,\theta = -\dfrac{12}{37}\)  for  \(0^{\circ} \lt \theta \lt 180^{\circ}\),

find the exact value of \(\sin\,\theta\).   (2 marks)

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\(\dfrac{35}{37}\)

Show Worked Solution

\(\cos\,\theta = -\dfrac{12}{37}\)

\(\text{Graphically:}\)

\(x=\sqrt{37^2-12^2}=35\) 

\(\therefore\ \sin\,\theta=\dfrac{35}{37}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 31

Given  \(\tan\,\theta = -\dfrac{7}{24}\)  for  \(-90^{\circ} \lt \theta \lt 90^{\circ}\), find the exact value of

  1.  \(\dfrac{1}{\cos\,\theta}\)   (2 marks)

  2.  \(\sin\,\theta\)   (1 mark)

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a.   \(\dfrac{25}{24}\)

b.   \(-\dfrac{7}{25}\)

Show Worked Solution

a.   \(\tan\,\theta = -\dfrac{7}{24}\)

\(\text{Graphically, given}\ \ -90^{\circ} \lt \theta \lt 90^{\circ}:\)

\(x= \sqrt{24^2 + 7^2}=25\)

\(\dfrac{1}{\cos\,\theta}=\dfrac{1}{\frac{24}{25}}=\dfrac{25}{24}\)
  

b.   \(\sin\,\theta = -\dfrac{7}{25}\)

Functions and Graphs, SMB-020

A circle has centre `(5,3)` and radius 3.

  1.  Describe, with inequalities, the region that consists of the interior of the circle and more than 2 units above the `x`-axis.  (2 marks)

  2.  Sketch the region.  (1 mark)

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i.    `(x-5)^2 + (y-3)^2 < 9\ ∩\ y > 2`

ii.    
       

Show Worked Solution

i.   `text(Equation of circle:)`

`(x-5)^2 + (y-3)^2 = 3^2`
 

`:.\ text(Region is:)`

`(x-5)^2 + (y- 3)^2 < 9\ ∩\ y > 2`

COMMENT: The broken line on the graph represents an excluded boundary.

 

ii.   

Circles and Hyperbolas, SMB-005

Sketch the graph of  `f(x) = (2x+1)/(x-1)`. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation.  (4 marks)

  

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Show Worked Solution
`(2x+1)/(x-1)` `=(2x-2+3)/(x-1)`  
  `=(2(x-1)+3)/(x-1)`  
  `=2 + 3/(x-1)`  

COMMENT: Manipulation of the equation (as shown) makes graphing much easier.

 
`text(Asymptotes:)\ \ x = 1,\ \ y = 2`

`text(As)\ \ x->oo,\ \ y->2(+)`

`text(As)\ \ x->-oo,\ \ y->2(-)`

`text(As)\ \ x->-1 (-),\ \ y->-oo`

`text(As)\ \ x->-1 (+),\ \ y->oo`

Indices, SMB-032

Rationalise the denominator of the surd fraction  `(sqrt(12))/(sqrt(6)-2)`.   (3 marks)

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`3sqrt(2)+2sqrt(3)`

Show Worked Solution
`(sqrt(12))/(sqrt(6)-2)` `=(2sqrt(3))/(sqrt(6)-2) xx (sqrt(6)+2)/(sqrt(6)+2)`  
  `=(2sqrt(3)(sqrt(6)+2))/((sqrt(6))^2-2^2)`  
  `=(2sqrt(18)+4sqrt(3))/(2)`  
  `=(6sqrt(2)+4sqrt(3))/(2)`  
  `=3sqrt(2)+2sqrt(3)`  

Indices, SMB-031

Rationalise the denominator of the surd fraction  `(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))`.   (3 marks)

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`6sqrt(2)-14/3sqrt(3)`

Show Worked Solution

`(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))`

`=(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))xx(3sqrt(2)-2sqrt(3))/(3sqrt(2)-2sqrt(3))`

`=((8-2sqrt(6))(3sqrt(2)-2sqrt(3)))/((3sqrt(2))^2-(2sqrt(3))^2)`

`=(24sqrt(2)-16sqrt(3)-6sqrt(12)+4sqrt(18))/(18-12)`

`=(24sqrt(2)-16sqrt(3)-12sqrt(3)+12sqrt(2))/6`

`=(36sqrt(2)-28sqrt(3))/6`

`=6sqrt(2)-14/3sqrt(3)`

Algebraic Techniques, SMB-074

Worker A picks a bucket of blueberries in `a` hours. Worker B picks a bucket of blueberries in `b` hours.

  1.  Write an algebraic expression for the fraction of a bucket of blueberries that could be picked in one hour if A and B worked together.  (2 marks)

  2.  What does the reciprocal of this fraction represent?  (1 mark)

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i.    `(a + b)/(ab)`

ii.    `text(The reciprocal represents the number of hours it would)`

`text(take to fill one bucket, with A and B working together.)`

Show Worked Solution

i.    `text(In one hour:)`

COMMENT: Note that the question asks for “a fraction”.

`text(Worker A picks)\ 1/a\ text(bucket.)`

`text(Worker B picks)\ 1/b\ text(bucket.)`
 

`:.\ text(Fraction picked in 1 hour working together)`

`= 1/a + 1/b`

`= (a + b)/(ab)`
 

ii.   `text(The reciprocal represents the number of hours it would)`

`text(take to fill one bucket, with A and B working together.)`

Functions, EXT1 F2 2022 HSC 3 MC

Let `P(x)` be a polynomial of degree 5. When `P(x)` is divided by the polynomial `Q(x)`, the remainder is `2x+5`.

Which of the following is true about the degree of `Q`?

  1. The degree must be 1.
  2. The degree could be 1.
  3. The degree must be 2.
  4. The degree could be 2.
Show Answers Only

`D`

Show Worked Solution

`text{Given}\ \ P(x)\ \ text{has degree 5}`

`P(x) -: Q(x)\ \ text{has remainder}\ \ 2x+5`

`text{Consider examples to resolve possibilities:}`

`text{eg.}\ \ x^5+2x+5 -: x^3 = x^2+\ text{remainder}\ 2x+5`

`:.\ text{Degree must be 2 is incorrect}`

`Q(x)\ \ text{can have a degree of 2, 3 or 4}`

`=>D`


♦ Mean mark 51%.

Functions, 2ADV F1 2020 HSC 24

The circle of  `x^2-6x + y^2 + 4y-3 = 0`  is reflected in the `x`-axis.

Sketch the reflected circle, showing the coordinates of the centre and the radius.  (3 marks)

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Show Worked Solution
`x^2-6x + y^2 + 4y-3` `= 0`
`x^2-6x + 9 + y^2 + 4y + 4-16` `= 0`
`(x-3)^2 + (y + 2)^2` `= 16`

 
`=>\  text{Original circle has centre (3, − 2), radius = 4}`

`text(Reflect in)\ xtext(-axis):`

♦ Mean mark 48%.

`text{Centre (3, − 2) → (3, 2)}`
 

Networks, STD2 N2 SM-Bank 37

The map of Australia shows the six states, the Northern Territory and the Australian Capital Territory (ACT).
  

In the network diagram below, each of the vertices `A` to `H` represents one of the states or territories shown on the map of Australia. The edges represent a border shared between two states or between a state and a territory.
 

  1. In the network diagram, what is the order of the vertex that represents the Australian Capital Territory (ACT)?  (1 mark)

  2. In the network diagram, Queensland is represented by which letter? Explain why.  (2 marks)

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i.    `1`

ii.   `text{NSW is Vertex B (it is connected to the ACT – Vertex D)}`

`=> C\ text{is Victoria as it has degree 2}`

`:.\ text(Queensland is vertex)\ A\ text(as it is connected to)\ B\ text(and has degree 3.)`

Show Worked Solution

i.     `text {ACT has 1 border (with NSW)}`

`:.\ text(Degree of ACT’s vertex = 1)`
 

ii.   `text{NSW is Vertex B (it is connected to the ACT – Vertex D)}`

`=> C\ text{is Victoria as it has degree 2}`

`:.\ text(Queensland is vertex)\ A\ text(as it is connected to)\ B\ text(and has degree 3.)`

Measurement, STD2 M1 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre `O`, has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

Using the formula  `V=1/3 pi r^2h`  where `r`  is the radius of the cone's circular base and `h` is the perpendicular height of the cone, find the volume of the cone, correct to the nearest cm³?  (3 marks)

Show Answers Only

`113\ text{cm}^3`

Show Worked Solution

`V = 1/3 xx text(base of cone × height)`

`text(Consider the circular base area of the cone,)`

`text(Find)\ x\ \ text{(using Pythagoras):}`

`x^2` `= 4^2-2^2 = 16-4 = 12`
`x` `= sqrt12\ text(cm)`

 

`:. V` `= 1/3 xx pi xx (sqrt12)^2 xx (15-6)`
  `= 1/3 xx pi xx 12 xx 9`
  `= 113.097…`
  `= 113\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 2016 HSC 30c

A school playground consists of part of a circle, with centre `O`, and a rectangle as shown in the diagram. The radius `OB` of the circle is 45 m, the width `BC` of the rectangle is 20 m and `AOB` is 100°.
 

What is the area of the whole playground, correct to the nearest square metre?  (5 marks)

Show Answers Only

`6971\ text{m²  (nearest m²)}`

Show Worked Solution

`text(In)\ DeltaOEB,`

`sin50^@` `= (EB)/45`
`EB` `= 45 xx sin50^@`
  `= 34.47…`
`:. AB` `= 2 xx 34.47…`
  `= 68.944\ \ (text(3 d.p.))`

 

`cos50^@` `= (OE)/45`
`:. OE` `= 45 xx cos50^@`
  `= 28.925\ \ (text(3 d.p.))`

 

`text(Area of)\ DeltaOAB`

`= 1/2 xx AB xx OE`

`= 1/2 xx 68.944 xx 28.925`

`= 997.12\ text(m²)`

 

`text(Area)\ ABCD` `= 20 xx 68.944`
  `= 1378.88\ text(m²)`

 

`text(Area of major sector)\ OAB`

`= pi xx 45^2 xx 260/360`

`= 4594.58\ text(m²)`

 

`:.\ text(Area of playground)`

`= 997.12 + 1378.88 + 4594.58`

`= 6970.58`

`= 6971\ text{m²  (nearest m²)}`

Functions, EXT1 F1 2010 HSC 1d

Solve  `3/(x+2) < 4`.   (3 marks)

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 `x < -2\ \ text(or)\ \ x > -5/4`

Show Worked Solution

`text(Solution 1)`

`3/(x + 2) < 4`

`text(Multiply b.s. by)\ \ (x + 2)^2`

`3(x + 2)` `< 4(x + 2)^2`
`3x + 6` `< 4 (x^2 + 4x + 4)`
`3x + 6` `< 4x^2 + 16x + 16`
`4x^2 + 13x + 10` `> 0`
`(4x + 5)(x + 2)` `> 0`

 
`text(LHS)\ = 0\ \ text(when)\ \ x = -5/4\ \ text(or)\ \ -2`

Algebra, EXT1 2010 HSC 1d Answer

`text(From graph)`

`x < -2\ \ text(or)\ \  x > -5/4`

 
`text(Alternate Solution)`

`text(If)\ \ x + 2 > 0\ \ \ \ text{(i.e.}\ \ x > –2 text{)}`

`3` `< 4(x + 2)`
`3` `< 4x + 8`
`4x` `> -5`
`x` `> -5/4`

 
`text(If)\ \ x + 2 < 0\ \ \ \ text{(i.e.}\ x < –2 text{)}`

`3` `> 4 (x + 2)`
`3` `> 4x + 8`
`4x` `< -5`
`x` `< -5/4`
`:. x` `< –2\ \ \ \ text{(satisfies both)}`

 
`:.\ x < –2\ \ text(or)\ \ x > –5/4`

Functions, EXT1 F1 2011 HSC 1c

Solve  `(4-x)/x <1`.  (3 marks)

Show Answers Only

 `x<0\ \ text(or)\ \ x>2`

Show Worked Solution

`text(Solution 1)`

`(4-x)/x < 1`

`text(If)\ x<0,\ \ \ \ \ 4-x` `> x`
`2x` `< 4`
`x` `<2`

`=> x<0\ \ \ text{(satisfies both)}`
 

`text(If)\ x>0,\ \ \ \ \ 4-x` `<x`
`2x` `>4`
`x` `>2`

`=> x>2\ \ \ text{(satisfies both)}`

`:.\ x < 0\ \ text(or)\ \ x > 2`

 
`text(Solution 2)`

`text(Multiply both sides by)\ \ x^2`

`x(4-x)` `< x^2`
`4x-x^2` `< x^2`
`2x^2-4x` `>0`
`2x(x-2)` `>0`

 

 EXT1 2011 1c

`text(From graph)`

`x<0\ \ text(or)\ \ x >2`

Functions, EXT1* F1 2009 HSC 3c

Shade the region in the plane defined by  `y >= 0`  and  `y <= 4-x^2`.   (2 marks)

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`text(Shaded area is region where)`

`y >= 0\ text(and)\ y >= 4-x^2`

Show Worked Solution

COMMENT: This past “Advanced” HSC question now fits into the Ext1 (new) syllabus.

`text(Shaded area is region where)`

`y >= 0\ \ text(and)\ \ y >= 4-x^2`

Functions, EXT1* F1 2012 HSC 8 MC

The diagram shows the region enclosed by  `y = x- 2`  and  `y^2 = 4-x`. 
  

Which of the following pairs of inequalities describes the shaded region in the diagram? 

  1. `y^2 <= 4-x\ \ and\ \ y <= x-2`  
  2. `y^2 <= 4-x\ \ and\ \ y >= x-2`  
  3. `y^2 >= 4-x\ \ and\ \ y<= x-2`  
  4. `y^2 >= 4-x\ \ and\ \ y >= x-2`
Show Answers Only

`A`

Show Worked Solution
♦  Mean mark 44%.

`text(Using information from diagram)`

`(3,0)\ text(is in the shaded region)`

`text{Substituting (3,0) into}\ \ \ y^2<=4-x,\ \ \ 0 <= 4-3 => text(true)`

`:.\ text(Cannot be)\ C\ text(or)\ D`
 

`text(Similarly)`

`(3,0)\ text(must satisfy other inequality)`

`text(i.e.)\ \ y <= x-2\ \ text(becomes)\ \ 0<= 3-2 =>\ text(true)`

`=>  A`

Functions, EXT1* F1 2013 HSC 11g

Sketch the region defined by  `(x-2)^2 + ( y-3)^2 >= 4`.    (3 marks)

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Show Worked Solution

`text(The region is the exterior of a circle,)`

COMMENT: This past “Advanced” HSC question now fits into the Ext1 (new) syllabus.

`text(centre)\ text{(2,3)}\ text(and radius 2.)`
 

Measurement, STD2 M6 2010 HSC 26d

Find the area of triangle `ABC`, correct to the nearest square metre.   (3 marks)

Show Answers Only

`717\ text(m²)`    `text{(nearest m²)}`

Show Worked Solution
♦♦ Mean mark 32%.
TIP: The allocation of 3 marks to this question should flag the need for more than 1 step.
`cos/_C` `=(AC^2 + CB^2-AB^2)/(2 xx AC xx CB)`
  `=(50^2 + 40^2-83^2)/(2 xx 50 xx 40)`
  `= -0.69725…`
`/_C` `=134.2067…^@`

 

`text(Using Area) = 1/2 ab\ sinC :`
`text(Area)\ Delta ABC` `=1/2 xx 50 xx 40 xx sin134.2067…^@`
  `=716.828…`
  `=717\ text(m²)\ \ \ \ text{(nearest m²)}`