4MA0 Higher June 2016 Paper 4R Q21

21 \(y = at^2 - 2at\)
\(x = 2a\sqrt{t}\)

Express \(y\) in terms of \(x\) and \(a\).

Give your answer in the form

\[y = \dfrac{x^p}{ma^3} - \dfrac{x^q}{na}\]

where \(p\), \(q\), \(m\) and \(n\) are integers.

(4)

Mark scheme

Scheme	Marks
\(\sqrt{t} = \dfrac{x}{2a}\) or \(x^2 = (2a\sqrt{t})^2\) or \(x^4 = (2a\sqrt{t})^4\) oe	M1
\(t = \left(\dfrac{x}{2a}\right)^2\) oe or \(t^2 = \dfrac{x^4}{16a^4}\) oe	M1
\(y = a\left[\left(\dfrac{x}{2a}\right)^2\right]^2 - 2a\left(\dfrac{x}{2a}\right)^2\) oe	M1
\(y = \dfrac{x^4}{16a^3} - \dfrac{x^2}{2a}\)	A1
	(4)

M1: Correct rearrangement for \(\sqrt{t}\) or correct expression for \(x^2\) or \(x^4\)

M1: Correct expressions for \(t\) or \(t^2\) or for \(at^2\) or \(2at\) in terms of \(x\) and \(a\)

M1: For correct substitution of \(t\) and \(t^2\) into expression for \(y\)

A1: Fully correct answer in required form