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Algebra, STD2 EQ-Bank 24

The area of a semicircle is given by  \(A=\dfrac{1}{2}\pi r^2\)  where \(r\) is the radius of the semicircle.

If the area of a semicircle is 250 cm², find the radius, to 1 decimal place.   (3 marks)

Show Answers Only

\(12.6\ \text{cm}\)

Show Worked Solution

\(A=\dfrac{1}{2}\pi r^2\ \ \Rightarrow \ \ r^2=\dfrac{2A}{\pi}\)

   
\(\text{When}\ \ A = 250:\)

\(r^2\) \(=\dfrac{2\times 250}{\pi}=\dfrac{500}{\pi}=159.154…\)
\( r\) \(=\sqrt{159.154…}=12.615…=12.6\ \text{cm (to 1 d.p.)}\)

Algebra, STD2 EQ-Bank 3 MC

If  \(d=\sqrt{\dfrac{h}{5}}\), what is the value of \(d\), correct to one decimal place, when  \(h=28\)?

  1. 1.1
  2. 2.4
  3. 2.8
  4. 5.6
Show Answers Only

\(B\)

Show Worked Solution

\(\text{When}\ h=28:\)

\(d\) \(=\sqrt{\dfrac{h}{5}}\)
\(d\) \(=\sqrt{\dfrac{28}{5}}\)
  \(=2.366…\)
  \(=2.4\  (\text{1 deccimal place})\)

   
\(\Rightarrow B\)

Algebra, STD2 A1 2024 HSC 1 MC

If  \(x=-2.531\), what is the value of \(x^2\) rounded to 2 decimal places?

  1. \(-6.41\)
  2. \(-6.40\)
  3. \(6.40\)
  4. \(6.41\)
Show Answers Only

\(D\)

Show Worked Solution
  \( x^2\) \(=(-2.531)^2 \)
    \(=6.405\ldots \)
    \(=6.41\)

 
\(\Rightarrow D\)

♦♦ Mean mark 29%!
COMMENT: Brain dead calculator usage will cause issues here.

Algebra, STD1 A1 2020 HSC 18

The distance, `d` metres, travelled by a car slowing down from `u` km/h to `v` km/h can be obtained using the formula

`v^2 = u^2-100 d`

What distance does a car travel while slowing down from 70 km/h to 40 km/h?   (2 marks)

Show Answers Only

`33 \ text{metres}`

Show Worked Solution

`u = 70 \ , \ v = 40`

`v^2` `= u^2-100d`
`40^2` `= 70^2-100d`
`100d` `= 70^2-40^2`
`:. d` `= frac{70^2-40^2}{100}`
  `= 33 \ text{metres}`

Algebra, STD1 A1 2019 HSC 34

Given the formula  `C = (A(y + 1))/24`, calculate the value of  `y`  when  `C = 120`  and  `A = 500`.   (3 marks)

Show Answers Only

`4.76`

Show Worked Solution

`text(Make)\ \ y\ \ text(the subject:)`

`C` `= (A(y + 1))/24`
`24C` `= A(y + 1)`
`y + 1` `= (24C)/A`
`y` `= (24C)/A-1`
  `= (24 xx 120)/500-1`
  `= 4.76`

Algebra, STD2 A1 EQ-Bank 25

For adults (18 years and older), the Body Mass Index is given by

`B = m/h^2`  where  `m = text(mass)`  in kilograms and  `h = text(height)`  in metres.

The medically accepted healthy range for  `B`  is  `21 <= B <= 25`.

What is the minimum weight for a 163 cm adult female to be considered healthy?   (2 marks)

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`55.8\ text(kg)`

Show Worked Solution

`B = m/h^2`

`h = 163\ text(cm) = 1.63\ text(m)`
  

`text(Given)\ \ 21 <= B <= 25,`

`=>B = 21\ \ text(for minimum healthy weight.)`

`21` `= m/1.63^2`
`:. m` `= 21 xx 1.63^2= 55.794…= 55.8\ text(kg)\ text{(1 d.p.)}`

Algebra, STD2 A1 2018 HSC 28b

Solve the equation  `(2x)/5 + 1 = (3x + 1)/2`, leaving your answer as a fraction.   (3 marks)

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`5/11`

Show Worked Solution

♦ Mean mark 35%.

`underbrace{(2x)/5 + 1}_text(multiply x10)` `=underbrace{(3x + 1)/2}_text(multiply x10)`
`4x + 10` `= 15x + 5`
`11x` `= 5`
`x` `= 5/11`

Algebra, STD2 A1 SM-Bank 23

What is the value of  `(a + b)/(ab)`  if  `a =-2.1`  and  `b =-3.6`, correct to 1 decimal place?   (2 marks)

Show Answers Only

`-0.8`

Show Worked Solution
`(a + b)/(ab)` `= (-2.1-3.6)/(-2.1 xx -3.6)`
  `= (-5.7)/7.56`
  `= -0.753…`
  `= -0.8`

Algebra, STD2 A1 EQ-Bank 22

If  `S = V_0 (1-r)^n`, find `S` given  `V_0 = $42\ 000, r = 0.16 and n = 4`. Give your answer to the nearest cent.   (2 marks)

Show Answers Only

`$20\ 910.60\ text{(to nearest cent)}`

Show Worked Solution
`S` `= V_0 (1-r)^n`
  `= 42\ 000 (1-0.16)^4`
  `= 42\ 000 (0.84)^4`
  `= $20\ 910.597…`
  `= $20\ 910.60\ \ text{(to nearest cent)}`

Algebra, STD2 A1 2017 HSC 9 MC

What is the value of  `x`  in the equation  `(5-x)/3 = 6`?

  1. `-13`
  2. `-3`
  3. `3`
  4. `13`
Show Answers Only

`A`

Show Worked Solution
`(5-x)/3` `= 6`
`5-x` `= 18`
`x` `= 5-18=-13`

`=>A`

Algebra, STD2 A1 2017 HSC 7 MC

It is given that  `I = 3/2 MR^2`.

What is the value of  `I`  when  `M = 26.55`  and  `R = 3.07`, correct to two decimal places?

A.     `375.35`

B.     `3246.08`

C.     `9965.45`

D.     `14\ 948.18`

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`A`

Show Worked Solution

`I= 3/2 xx 26.55 xx (3.07)^2= 375.346…`

`=> A`

Algebra, STD2 A1 2016 HSC 5 MC

Which expression is equivalent to  `2(3x-4) + 2`?

  1. `6x-2`
  2. `6x-4`
  3. `6x-6`
  4. `6x-10`
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`C`

Show Worked Solution

`2(3x-4) + 2`

`= 6x-8 + 2= 6x-6` 

`=> C`

Algebra, STD2 A1 2016 HSC 2 MC

Which of the following equations has  `x = 5`  as the solution?

  1. `x - 5 = 10`
  2. `5 - x = 10`
  3. `x/2 = 10`
  4. `2x = 10`
Show Answers Only

`D`

Show Worked Solution
`2x` `= 10`
`:. x` `= 5`

 
`=> D`

Algebra, STD2 A1 EQ-Bank 13

If   `(y-3)/3 =5`,  find  `y`.   (2 marks)

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`18`

Show Worked Solution
`(y-3)/3` `= 5`
 `y-3` `= 15`
 `y` `= 18`

Algebra, STD2 A1 EQ-Bank 3

Find the value of  `r`  given  `r/7-4 = 3`.   (1 mark)

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`49`

Show Worked Solution
`r/7-4` `= 3`
`r/7` `= 7`
`:.r` `= 49`

Algebra, STD2 A1 EQ-Bank 2

If   `A = P(1 + r)^n`, find  `A`  given  `P = $300`,  `r = 0.12`  and  `n = 3`  (give your answer to the nearest cent).   (2 marks)

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`$421.48\ \ text{(nearest cent)}`

Show Worked Solution
`A` `= P(1 + r)^n`
  `= 300(1 + 0.12)^3`
  `= 300(1.12)^3`
  `= 421.478…`
  `= $421.48\ \ text{(nearest cent)}`

Algebra, STD2 A1 EQ-Bank 21

What is the value of  `5a^2-b`, if  `a =-4`  and  `b = 3`.   (2 marks)

Show Answers Only

`77`

Show Worked Solution
`5a^2-b` `= 5(-4)^2-3`
  `= 5 xx 16-3`
  `= 77`

Algebra, STD2 A1 2015 HSC 28d

The formula  `C = 5/9 (F-32)`  is used to convert temperatures between degrees Fahrenheit `(F)` and degrees Celsius `(C)`.

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

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`37.4`

Show Worked Solution
`C` `= 5/9(F-32)`
`F-32` `= 9/5C`
`F`  `= 9/5C + 32`

 
`text(When)\ \ C = 3,`

`F= (9/5 xx 3) + 32= 37.4`

Algebra, STD2 A1 2015 HSC 24 MC

Consider the equation  `(2x)/3-4 = (5x)/2 + 1`.

Which of the following would be a correct step in solving this equation?

  1. `(2x)/3-3 = (5x)/2`
  2. `(2x)/3 = (5x)/2 + 5`
  3. `2x-4 = (15x)/2 + 3`
  4. `(4x)/6-8 = 5x + 2`
Show Answers Only

`B`

Show Worked Solution
`(2x)/3-4` `= (5x)/2 + 1`
`(2x)/3` `= (5x)/2 + 5`

 
`=>B`

Algebra, STD2 A1 2015 HSC 2 MC

Which of the following is  `4x+3y-x-5y`   in its simplest form?

  1. `3x-2y`
  2. `3x+8y`
  3. `5x-2y`
  4. `5x+8y`
Show Answers Only

`A`

Show Worked Solution

`4x+3y-x-5y= 3x-2y`

`⇒ A`

Algebra, STD2 A1 2005 HSC 14 MC

Using the formula  `d=5t^3-2`, Marcia tried to find the value of  `t`  when `d=137`.

Here is her solution. She has made one mistake.
 

2UG-2005-14MC

Which line does NOT follow correctly from the previous line?

  1. `text(Line)\ A`
  2. `text(Line)\ B`
  3. `text(Line)\ C`
  4. `text(Line)\ D`
Show Answers Only

`B`

Show Worked Solution
`d` `=5t^3-2`  
 `137` `=5t^3-2 \ \ \ ` ` …text( Line)\ A`
 `139` `=5t^3` `…text( Line)\ B`

 

`:.\ text(Line)\ B\ text(doesn’t follow on correctly.)`

`=>  B`

Algebra, STD2 A1 2006 HSC 2 MC

If  `V = 4/3 pir^3`, what is the value of  `V`  when  `r = 2`, correct to two decimal places?

  1. `8.38`
  2. `12.57`
  3. `25.13`
  4. `33.51`
Show Answers Only

`D`

Show Worked Solution

`V = 4/3 pir^3`

`text(When)\ \ r = 2,`

`V= 4/3 pi xx 2^3= 33.510\ …=33.51\ text{(2 dp)}`

`=>  D`

Algebra, STD2 A1 2005 HSC 2 MC

What is the value of  `(a-b)/4`, if  `a = 240`  and  `b = 56`?

  1. `4`
  2. `46`
  3. `226`
  4. `736`
Show Answers Only

`B`

Show Worked Solution

`(a-b)/4= (240-56)/4= 46`

`=> B`

Algebra, STD2 A1 2004 HSC 11 MC

If  `d = 6t^2`,  what is a possible value of `t` when  `d = 2400`?

  1. `0.05`
  2. `20`
  3. `120`
  4. `400`
Show Answers Only

`B`

Show Worked Solution
`d` `= 6t^2`
`t^2` `= d/6`
`t` `= +- sqrt(d/6)`

 
`text(When)\ \ d = 2400:`

`t= +- sqrt(2400/6)= +- 20`

`=> B`

Algebra, STD2 A1 2004 HSC 3 MC

If  `K = Ft^3`,  `F = 5`  and  `t = 0.715`, what is the value of `K` correct to three significant figures?

  1. `1.82`
  2. `1.827`
  3. `1.828`
  4. `1.83`
Show Answers Only

`D`

Show Worked Solution
`K` `= Ft^3`
  `= 5 xx (0.715)^3= 1.8276…`
  `= 1.83\ \ text{(3 sig figures)}`

 
`=> D`

Measurement, STD2 M1 2007 HSC 28b

This shape is made up of a right-angled triangle and a regular hexagon.
 

 

The area of a regular hexagon can be estimated using the formula  `A = 2.598H^2`  where `H` is the side-length.

Calculate the total area of the shape using this formula.   (3 marks)

Show Answers Only

`22.784\ text(cm)^2`

Show Worked Solution

`text(Area) = 2.598H^2`

`text(Using Pythagoras:)`

`H^2` `= 2^2 + 2^2= 8`
`H` `= sqrt 8`

  
`text(Area of hexagon)= 2.598 xx (sqrt 8)^2= 20.784\ text(cm)^2`

`text(Area of triangle)= 1/2 bxx  h= 1/2 xx 2 xx 2= 2\ text(cm)^2`

`:.\ text(Total Area)= 20.784 + 2= 22.784\ text(cm)^2`

Algebra, STD2 A1 2007 HSC 24b

The distance in kilometres (`D`) of an observer from the centre of a thunderstorm can be estimated by counting the number of seconds (`t`) between seeing the lightning and first hearing the thunder.

Use the formula  `D = t/3`  to estimate the number of seconds between seeing the lightning and hearing the thunder if the storm is 1.2 km away.   (1 mark)

Show Answers Only

`3.6\ text(seconds)`

Show Worked Solution

`D = t/3`

`text(When)\ \ D = 1.2,`

`t/3` `= 1.2`
`t` `= 3.6\ text(seconds)`

Algebra, STD2 A1 2008 HSC 9 MC

What is the value of  `sqrt((x+2y)/(8y))`   if  `x = 5.6`  and  `y = 3.1`, correct to 2 decimal places? 

  1. `0.69`
  2. `2.62`
  3. `2.83` 
  4. `4.77`  
Show Answers Only

`A`

Show Worked Solution
`sqrt((x+2y)/(8y))` `=sqrt((5.6+(2 xx 3.1))/((8 xx 3.1)))`
  `=sqrt(11.8/24.8)=0.6897…`

`=>  A`

Algebra, STD2 A1 2014 HSC 26c

Solve the equation  `(5x + 1)/3-4 = 5-7x`.   (3 marks)

Show Answers Only

 `x = 1`

Show Worked Solution
`(5x + 1)/3-4` `= 5-7x`
`5x + 1-3(4)` `= 3(5-7x)`
`5x + 1-12` `= 15-21x`
`26x` `= 26`
`:. x` `= 1`

Algebra, STD2 A1 2009 HSC 25a

Simplify  `5-2(x + 7)`.   (2 marks)

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 `-2x-9`

Show Worked Solution
♦ Mean mark 47%
`5-2(x + 7)` `= 5-2x-14`
  `= -2x-9`

Algebra, STD2 A1 2013 HSC 29a

Sarah tried to solve this equation and made a mistake in Line 2. 

`(W+4)/3-(2W-1)/5` `=1` `text(... Line 1)`
`5W+ 20-6W-3` `=15` `text(... Line 2)`
`17-W` `=15` `text(... Line 3)`
`W` `=2` `text(... Line 4)`

 
Copy the equation in Line 1 and continue your solution to solve this equation for `W`.

Show all lines of working.   (2 marks)

Show Answers Only
`(W+4)/3-(2W-1)/5` `=1` `text(… Line 1)`
`5W+ 20-6W+ 3` `=15` `text(… Line 2)`
`23-W` `=15` `text(… Line 3)`
`W` `=8` `text(… Line 4)`
Show Worked Solution
♦♦ Mean mark 27%
STRATEGY: The RHS of the equation increases from 1 to 15 (from Line 1 to Line 2), indicating both sides must have been multiplied by 15.
`(W+4)/3-(2W-1)/5` `=1` `text(… Line 1)`
`5W+ 20-6W+3` `=15` `text(… Line 2)`
`23-W` `=15` `text(… Line 2)`
`W` `=8` `text(… Line 4)`

Algebra, STD2 A1 2010 HSC 24a

Fred tried to solve this equation and made a mistake in Line 2. 

\begin{array}{rl}
4(y+2)-3(y+1)= -3\ & \ \ \ \text{Line 1} \\
4y+8-3y+3= -3\ &\ \ \ \text{Line 2} \\
y+11 =-3\ &\ \ \ \text{Line 3} \\
y =-14& \ \ \ \text{Line 3}
\end{array}

Copy the equation in Line 1.

  1. Rewrite Line 2 correcting his mistake.   (1 mark)

  2. Continue your solution showing the correct working for Lines 3 and 4 to solve this equation for `y`.   (1 mark)

Show Answers Only

a.   `4y+8+3y-3=-3`

b.   `y+5=-3`

`y=-8`

Show Worked Solution
a.     `4(y+2)-3(y+1)` `=-3\ \ \ \ \ \ text(Line)\ 1`
  `4y+8-3y-3` `=-3\ \ \ \ \  text(Line)\ 2`

 

b.     `y+5` `=-3\ \ \ \ \ \ text(Line)\ 3`
  `y` `=-8\ \ \ \ \ \ text(Line)\ 4`

Algebra, STD2 A1 2010 HSC 7 MC

If  `M=-9`, what is the value of   `(3M^2+5M)/6`

  1. `-250.5`
  2. `-48`
  3. `\ \ \ 33`
  4. `\ \ \ 235.5`
Show Answers Only

`C`

Show Worked Solution
 ♦♦ Only 31% of students answered correctly!
`(3M^2+5M)/6` `=(3xx(–9)^2+5xx(–9))/6`
  `=((3xx81)-45)/6`
  `=33`

`=>  C`