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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)

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\(12.6\ \text{cm}\)

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  \(w=4y^2-5\), what is the value of \(y\) when  \(w=43\)?

  1. \(3\)
  2. \(-3\)
  3. \(\sqrt{3}\)
  4. \(-2\sqrt{3}\)
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\(D\)

Show Worked Solution

\(\text{When}\ w=43:\)

\(w\) \(=4y^2-5\)
\(43\) \(=4y^2-5\)
\(4y^2\) \(=48\)
\(y^2\) \(=\dfrac{48}{4}=12\)
\(y\) \(=\pm 2\sqrt{3}\)

 
\(y=-2\sqrt{3}\ \text{(only possibility given options)}\)

\(\Rightarrow D\)

Algebra, STD2 A1 2024 HSC 10 MC

Consider the formula  \(s=w t+\dfrac{p}{2}\).

Which of the following correctly shows \(p\) as the subject of the formula?

  1. \(p=2 w t-s\)
  2. \(p=2 w t-2 s\)
  3. \(p=2 s-w t\)
  4. \(p=2 s-2 w t\)
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\(D\)

Show Worked Solution
  \(s\) \(=w t+\dfrac{p}{2}\)
  \(\dfrac{p}{2}\) \(=s-w t\)
  \(p\) \(=2(s-w t)\)
    \(=2s-2w t\)

 
\(\Rightarrow D\)

♦♦ Mean mark 36%.

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)

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`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 2019 HSC 11 MC

Which of the following correctly expresses `y` as the subject of the formula  `3x-4y-1 = 0`?

  1.  `y = 3/4 x-1`
  2.  `y = 3/4 x + 1`
  3.  `y = (3x-1)/4`
  4.  `y = (3x + 1)/4`
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`C`

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♦ Mean mark 50%.

`3x-4y-1` `= 0`
`4y` `= 3x-1`
`:. y` `= (3x-1)/4`

 
`=> C`

Algebra, STD2 A1 2016 HSC 24 MC

Which of the following correctly expresses `Q` as the subject of  `e = iR + Q/C`?

  1. `Q = Ce + CiR`
  2. `Q = Ce-CiR`
  3. `Q = (e + iR)/C`
  4. `Q = (e-iR)/C`
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`B`

Show Worked Solution
`e` `= iR + Q/C`
`Q/C` `= e-iR`
`:. Q` `= C(e-iR)= Ce-CiR`

 
`=> B`

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 2005 HSC 24c

Make  `L`  the subject of the equation  `T = 2piL^2`.   (2 marks)

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`± sqrt(T/(2pi))`

Show Worked Solution
`T` `= 2piL^2`
`L^2` `= T/(2pi)`
`:.L` `= ±sqrt(T/(2pi))`

Algebra, STD2 A1 2006 HSC 18 MC

What is the formula for `q` as the subject of `4p =5t + 2q^2`?

  1. `q = +- sqrt(5t-4p)/2`
  2. `q = +- sqrt(4p + 5t)/2`
  3. `q = +- sqrt{(5t-4p)/2}`
  4. `q = +- sqrt{(4p-5t)/2}`
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`D`

Show Worked Solution
`4p` `= 5t + 2q^2`
`2q^2` `= 4p-5t`
`q^2` `= (4p-5t)/2`
`q` `= +- sqrt{(4p-5t)/2}`

 
`=>  D`

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`
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`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 2007 HSC 19 MC

Which of the following correctly expresses  `T`  as the subject of  `B = 2pi (R + T/2)`?

  1. `T = B/pi-2R`
  2. `T = B/pi-R`
  3. `T = 2R-B/pi`
  4. `T = B/(4pi)-R/2`
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`A`

Show Worked Solution
`B` `= 2pi (R + T/2)`
`B/(2pi)` `= R + T/2`
`T/2` `= B/(2pi)-R`
`T` `= B/pi-2R`

 
`=>  A`

Algebra, STD2 A1 2010 HSC 18 MC

Which of the following correctly express  `x`  as the subject of  `a=(nx)/5` ?

  1. `x=(an)/5`
  2. `x=(5a)/n`
  3. `x=(a-5)/n`
  4. `x=5a-n`
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`B`

Show Worked Solution
`a` `=(nx)/5`
`nx` `=5a`
`x` `=(5a)/n`

 
`=>  B`

Algebra, STD2 A1 2011 HSC 18 MC

Which of the following correctly expresses  `a`  as the subject of  `s= ut+1/2at^2 `?

  1. `a=(2(s-ut))/t^2`
  2. `a=(2s-ut)/t^2`
  3. `a=(1/2(s-ut))/t^2`
  4. `a=(1/2s-ut)/t^2`
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`A`

Show Worked Solution
`s` `=ut+1/2at^2`
`1/2at^2` `=s-ut`
`at^2` `=2(s-ut)`
`a` `=(2(s-ut))/t^2`

 
`=>A`

Algebra, STD2 A1 2012 HSC 21 MC

Which of the following correctly expresses  `c`  as the subject of  `E = mc^2 + p` ? 

  1. `c = +-sqrt(\ \ (E-p)/m)` 
  2. `c = +-sqrt(E-p)/m` 
  3. `c = +- sqrt(\ \ E/m)-p` 
  4. `c = +- sqrt(\ \ E/m-p)`
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`A`

Show Worked Solution
`E` `=\ mc^2 + p`
`mc^2` `=\ E-p`
`c^2` `=(E-p)/m`
`:.c` `= +-sqrt((E-p)/m)`

 
`=>  A`

Algebra, STD2 A1 2013 HSC 21 MC

Which equation correctly shows  `r`  as the subject of  `S=800(1-r)`?

  1. `r=(800-S)/800`
  2. `r=(S-800)/800`
  3. `r=800-S`
  4. `r=S-800`
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`A`

Show Worked Solution
♦♦♦ Mean mark 27%
`S` `=800(1-r)`
`1-r` `=S/800`
`r` `=1-S/800`
  `=(800-S)/800`

 
`=>\ A`