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v1 Algebra, STD2 A1 2015 HSC 28d

The formula  \(C=\dfrac{5}{9}(F-32)\)  is used to convert temperatures between degrees Fahrenheit \((F)\) and degrees Celsius \((C)\).

Convert 18°C to the equivalent temperature in Fahrenheit.  (2 marks)

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\(64.4\ \text{degrees}\ F\)

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\(C\) \(=\dfrac{5}{9}(F-32)\)
\(F-32\) \(=\dfrac{9}{5}C\)
\(F\)  \(=\dfrac{9}{5}C+32\)

 
\(\text{When}\ \ C = 18,\)

\(F\)  \(=\dfrac{9}{5}\times 18+32\)
  \(=64.4\ \text{degrees}\ F\)

v1 Algebra, STD2 A1 2013 HSC 21 MC

Which equation correctly shows  \(n\)  as the subject of  \(V=600(1-n)\)?

  1. \(n=\dfrac{V-600}{600}\)
  2. \(n=\dfrac{600-V}{600}\)
  3. \(n=V-600\)
  4. \(n=600-V\)
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\(B\)

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♦♦♦ Mean mark 27%
\(V\) \(=600(1-n)\)
\(1-n\) \(=\dfrac{V}{600}\)
\(n\) \(=1-\dfrac{V}{600}\)
  \(=\dfrac{600-V}{600}\)

 
\(\Rightarrow B\)

v1 Algebra, STD2 A1 2012 HSC 21 MC

Which of the following correctly expresses  \(r\)  as the subject of  \(V=\pi r^2+x\) ?

  1. \(r=\pm\sqrt{\dfrac{V}{\pi}}-x\)
  2. \(r=\pm\sqrt{\dfrac{V}{\pi}-x}\)
  3. \(r=\pm\sqrt{\dfrac{V-x}{\pi}}\)
  4. \(r=\pm\dfrac{\sqrt{V-x}}{\pi}\)
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\(C\)

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\(V\) \(=\pi r^2+x\)
\(\pi r^2\) \(=V-x\)
\(r^2\) \(=\dfrac{V-x}{\pi}\)
\(\therefore\ r\) \(=\pm\sqrt{\dfrac{V-x}{\pi}}\)

\(\Rightarrow C\)

v1 Algebra, STD2 A1 2011 HSC 18 MC

Which of the following correctly expresses  \(b\)  as the subject of  \(y= ax+\dfrac{1}{4}bx^2\)?

  1. \(b=\dfrac{4y-ax}{x^2}\)
  2. \(b=\dfrac{4(y-ax)}{x^2}\)
  3. \(b=\dfrac{\dfrac{1}{4}y-ax}{x^2}\)
  4. \(b=\dfrac{\dfrac{1}{4}(y-ax)}{x^2}\)
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\(B\)

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\(y\) \(= ax+\dfrac{1}{4}bx^2\)
\(\dfrac{1}{4}bx^2\) \(=y-ax\)
\(bx^2\)  \(=4(y-ax)\)
\(b\) \(=\dfrac{4(y-ax)}{x^2}\)

 
\(\Rightarrow B\)

v1 Algebra, STD2 A1 2004 HSC 11 MC

If  \(m = 8n^2\), what is a possible value of  \(n\)  when  \(m=7200\)?

  1. \(0.0\dot{3}\)
  2. \(30\)
  3. \(240\)
  4. \(900\)
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\(B\)

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\(m\) \(=8n^2\)
\(n^2\) \(=\dfrac{m}{8}\)
\(n\) \(=\pm\sqrt{\dfrac{m}{8}}\)

 
\(\text{When}\ m=7200:\)

\(n\) \(=\pm\sqrt{\dfrac{7200}{8}}\)
  \(=\pm 30\)

 
\(\Rightarrow B\)

v1 Algebra, STD2 A1 2006 HSC 18 MC

What is the formula for \(g\) as the subject of \(7d=8e+5g^2\)?

  1. \(g =\pm\sqrt{\dfrac{8e-7d}{5}}\)
  2. \(g =\pm\sqrt{\dfrac{7d-8e}{5}}\)
  3. \(g =\pm\dfrac{\sqrt{7d+8e}}{5}\)
  4. \(g =\pm\dfrac{\sqrt{8e-7d}}{5}\)
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\(B\)

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\(7d\) \(=8e+5g^2\)
\(5g^2\) \(=7d-8e\)
\(g^2\) \(=\dfrac{7d-8e}{5}\)
\(g\) \(=\pm\sqrt{\dfrac{7d-8e}{5}}\)

\(\Rightarrow B\)

v1 Algebra, STD2 A1 2007 HSC 19 MC

Which of the following correctly expresses  \(X\)  as the subject of  \(Y=4\pi\Bigg(\dfrac{X}{4}+L\Bigg)\)?

  1. \(X=\dfrac{Y}{\pi}-L\)
  2. \(X=\dfrac{Y}{\pi}-4L\)
  3. \(X=4L-\dfrac{Y}{2\pi}\)
  4. \(X=\dfrac{Y}{8\pi}-\dfrac{L}{4}\)
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\(B\)

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\(Y\) \(=4\pi\Bigg(\dfrac{X}{4}+L\Bigg)\)
\(\dfrac{Y}{4\pi}\) \(=\dfrac{X}{4}+L\)
\(\dfrac{X}{4}\) \(=\dfrac{Y}{4\pi}-L\)
\(X\) \(=4\Bigg(\dfrac{Y}{4\pi}-L\Bigg)\)
\(X\) \(=\dfrac{Y}{\pi}-4L\)

 
\(\Rightarrow B\)

v1 Algebra, STD2 A1 2016 HSC 24 MC

Which of the following correctly expresses \(M\) as the subject of  \(y=\dfrac{M}{V}+cX\)?

  1. \(M=Vy-VcX\)
  2. \(M=Vy+VcX\)
  3. \(M=\dfrac{y-cX}{V}\)
  4. \(M=\dfrac{y+cX}{V}\)
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\(A\)

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\(y\) \(=\dfrac{M}{V}+cX\)
\(\dfrac{M}{V}\) \(=y-cX\)
\(\therefore\ M\) \(=V(y-cX)\)
  \(=Vy-VcX\)

 
\(\Rightarrow A\)

v1 Algebra, STD2 A1 2019 HSC 11 MC

Which of the following correctly expresses \(y\) as the subject of the formula  \(5x-2y-9=0\)?

  1.  \(y=\dfrac{5}{2}x-9\)
  2.  \(y=\dfrac{5}{2}x+9\)
  3.  \(y=\dfrac{5x+9}{2}\)
  4.  \(y=\dfrac{5x-9}{2}\)
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\(D\)

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♦ Mean mark 50%.

\(5x-2y-9\) \(=0\)
\(2y\) \(=5x-9\)
\(\therefore\ y\) \(=\dfrac{5x-9}{2}\)

 
\(\Rightarrow D\)

v1 Algebra, STD2 A1 SM-Bank 11

Make  \(V\)  the subject of the equation  \(E=\dfrac{3}{2}mV^3\).  (3 marks)

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\(v=\sqrt[3]{\dfrac{2E}{3m}}\)

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\(E\) \(=\dfrac{3}{2}mV^3\)
\(2E\) \(=3mV^3\)
\(\dfrac{2E}{3}\) \(=mV^3\)
\(V^3\) \(=\dfrac{2E}{3m}\)
\(V\) \(=\sqrt[3]{\dfrac{2E}{3m}}\)

v1 Algebra, STD1 A1 2019 HSC 34

Given the formula  \(D=\dfrac{B(x+1)}{18}\), calculate the value of  \(x\)  when  \(D=90\)  and  \(B=400\).  (3 marks)

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\(3.05\)

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\(\text{Make}\ x\ \text{the subject:}\)

\(D\) \(=\dfrac{B(x+1)}{18}\)
\(18D\) \(=B(x+1)\)
\(x+1\) \(=\dfrac{18D}{B}\)
\(x\) \(=\dfrac{18D}{B}-1\)
\(\text{When }\) \(D=90, B=400\)
\(\therefore\ x\) \(=\dfrac{18\times 90}{400}-1=3.05\)

v1 Algebra, STD2 A2 2022 HSC 14 MC

Which of the following correctly expresses \(x\) as the subject of  \(y=\dfrac{mx-c}{3}\) ?

  1. \(x=\dfrac{3y}{m}+c\)
  2. \(x=\dfrac{y}{3m}+c\)
  3. \(x=\dfrac{y+c}{3m}\)
  4. \(x=\dfrac{3y+c}{m}\)
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\(D\)

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\(y\) \(=\dfrac{mx-c}{3}\)
\(3y\) \(=mx-c\)
\(mx\) \(=3y+c\)
\(\therefore\ x\) \(=\dfrac{3y+c}{m}\)

 
\(\Rightarrow D\)

v1 Algebra, STD2 A1 SM-Bank 9

The volume of a sphere is given by  \(V=\dfrac{4}{3}\pi r^3\)  where  \(r\)  is the radius of the sphere.

If the volume of a sphere is  \(385\ \text{cm}^3\), find the radius, to 1 decimal place.  (3 marks)

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\(4.5\ \text{cm  (to 1 d.p.)}\)

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\(V\) \(=\dfrac{4}{3}\pi r^3\)
\(3V\) \(= 4\pi r^3\)
\(r^3\) \(=\dfrac{3V}{4\pi}\)

 

\(\text{When}\ \ V =385\)

\(r^3\) \(=\dfrac{3\times 385}{4\pi}\)
  \(=91.911\dots\)
\(\therefore\ r\) \(=\sqrt[3]{91.911\dots}\)
  \(=4.512\dots\ \ \text{(by calc)}\)
  \(=4.5\ \text{cm   (to 1 d.p.)}\)