4MA0 Higher June 2015 Paper 4R Q19

EdexcelHigherOld spec8 marksSurds

19 \(\dfrac{1}{5^3} = 5^p\) \(1 = 5^q\) \(\sqrt{5^3} = 5^r\)

(a) Write down the value of

(i) \(p\)

(ii) \(q\)

(iii) \(r\)

(3)

(b) Show that \(\dfrac{14}{\sqrt{245}} = \dfrac{2\sqrt{5}}{5}\)
You must write down each stage of your working. (2)

\(\left(e - 2\sqrt{3}\right)^2 = f - 20\sqrt{3}\) where \(e\) and \(f\) are integers.

Scheme	Marks
(i) −3	B1
(ii) 0	B1
(iii) \(\dfrac{3}{2}\)	B1
	(3)

Mark scheme (b)

Scheme	Marks
Eg \(\sqrt{245} = \sqrt{49} \times \sqrt{5}\) or \(\sqrt{245} = \sqrt{49 \times 5}\) or \(\dfrac{14}{\sqrt{49 \times 5}}\) or \(\dfrac{14}{\sqrt{49} \times \sqrt{5}}\) \(\dfrac{14}{\sqrt{245}} \times \dfrac{\sqrt{245}}{\sqrt{245}}\) or \(\dfrac{14\sqrt{245}}{245}\) or \(\dfrac{14 \times 7\sqrt{5}}{245}\)	M1
\(\dfrac{2\sqrt{5}}{5}\)	A1
	(2)

Notes

M1: For simplifying \(\sqrt{245}\) or for rationalising the denominator.

A1: For simplifying \(\sqrt{245}\) and for rationalising the denominator.

NB: The solution must be fully correct (cso)

NB: Do not accept \(\dfrac{2}{\sqrt{5}} = \dfrac{2\sqrt{5}}{5}\) as sufficient method for rationalising the denominator.

Note: To rationalise \(\dfrac{14}{\sqrt{245}}\) accept \(\dfrac{14}{\sqrt{245}} \times \dfrac{\sqrt{245}}{\sqrt{245}}\) or \(\dfrac{14\sqrt{245}}{245}\)

To rationalise \(\dfrac{2}{\sqrt{5}}\) or \(\dfrac{14}{7\sqrt{5}}\), etc, only accept \(\dfrac{2}{\sqrt{5}} \times \dfrac{\sqrt{5}}{\sqrt{5}}\) or \(\dfrac{14}{7\sqrt{5}} \times \dfrac{\sqrt{5}}{\sqrt{5}}\) etc.

Mark scheme (c)

Scheme	Marks
\(e^2 - 2e\sqrt{3} - 2e\sqrt{3} + (2\sqrt{3})^2\) or \(e^2 - 4e\sqrt{3} + 12\)	M1
\(e = 5\)	A1
\(f = 37\)	A1
	(3)

Notes

M1: Expand brackets with 3 correct terms with correct signs

Eg \(e^2 - 4e\sqrt{3}\) or \(e^2 - 2e\sqrt{3} - 2e\sqrt{3}\) or \(e^2 - 2e\sqrt{3} + 12\), etc or

all 4 correct terms with signs missing eg \(e^2\), \(4e\sqrt{3}\), 12 or \(e^2\), \(2e\sqrt{3}\), \(2e\sqrt{3}\), 12

A1: cao