smc-5232-20-Non-Linear

Algebra, STD2 A1 2012 HSC 21 MC v1

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

\(r=\pm\sqrt{\dfrac{V}{\pi}}-x\)
\(r=\pm\sqrt{\dfrac{V}{\pi}-x}\)
\(r=\pm\sqrt{\dfrac{V-x}{\pi}}\)
\(r=\pm\dfrac{\sqrt{V-x}}{\pi}\)

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\(C\)

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

\(\Rightarrow C\)

Algebra, STD2 A1 2011 HSC 18 MC v1

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

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

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\(B\)

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

\(\Rightarrow B\)

Algebra, STD2 A1 2004 HSC 11 MC v1

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

\(0.03\)
\(30\)
\(240\)
\(900\)

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\(B\)

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

\(\text{When}\ m=7200:\)

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

\(\Rightarrow B\)

Algebra, STD2 A1 2006 HSC 18 MC v1

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

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

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\(B\)

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

\(\Rightarrow B\)

Algebra, STD2 A1 2005 HSC 24c v1

Make \(r\) the subject of the equation \(V=4\pi r^2\). (2 marks)

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\(r=\pm\sqrt{\dfrac{V}{4\pi}}\)

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\(V\)	\(=4\pi r^2\)
\(r^2\)	\(=\dfrac{V}{4\pi}\)
\(\therefore\ r\)	\(=\pm\sqrt{\dfrac{V}{4\pi}}\)

Algebra, STD2 A1 2017 HSC 28d v1

Make \(b\) the subject of the equation \(a=\sqrt{bc-4}\). (2 marks)

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\(b=\dfrac{a^2+4}{c}\)

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♦ Mean mark 46%.

\(a\)	\(=\sqrt{bc-4}\)
\(a^2\)	\(=bc-4\)
\(bc\)	\(=a^2+4\)
\(\therefore\ b\)	\(=\dfrac{a^2+4}{c}\)

Algebra, STD2 A1 EO-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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\(E\)	\(=\dfrac{3}{2}mV^3\)
\(2E\)	\(=3mV^3\)
\(\dfrac{2E}{3}\)	\(=mV^3\)
\(V^3\)	\(=\dfrac{2E}{3m}\)
\(V\)	\(=\sqrt[3]{\dfrac{2E}{3m}}\)

Algebra, STD2 A1 EO-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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\(V\)	\(=\dfrac{4}{3}\pi r^3\)
\(3V\)	\(= 4\pi r^3\)
\(r^3\)	\(=\dfrac{3V}{4\pi}\)

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

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