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Topic

Higher June 2025 Paper 2R Q16

EdexcelCurrent spec3 marksSurds

16 Show that \(\dfrac{4}{3\sqrt{5} + 7}\) can be written in the form \(a - \sqrt{b}\) where \(a\) and \(b\) are integers.
Show each stage of your working.

(3)

Mark scheme

Scheme	Marks
\(\dfrac{4}{3\sqrt{5} + 7} \times \dfrac{3\sqrt{5} - 7}{3\sqrt{5} - 7}\) or \(\dfrac{4}{3\sqrt{5} + 7} \times \dfrac{-3\sqrt{5} + 7}{-3\sqrt{5} + 7}\) oe	M1
eg \(\dfrac{4(3\sqrt{5} - 7)}{45 - 21\sqrt{5} + 21\sqrt{5} - 49}\) or \(\dfrac{4(3\sqrt{5} - 7)}{45 - 7^2}\) or \(\dfrac{12\sqrt{5} - 28}{45 - 21\sqrt{5} + 21\sqrt{5} - 49}\) or \(\dfrac{12\sqrt{5} - 28}{45 - 7^2}\) or \(\dfrac{4(3\sqrt{5} - 7)}{45 - 49}\) or \(\dfrac{4(3\sqrt{5} - 7)}{-4}\) or \(\dfrac{12\sqrt{5} - 28}{45 - 49}\) or \(\dfrac{12\sqrt{5} - 28}{-4}\)	M1
Working required Answer: \(7 - \sqrt{45}\)	A1
	(3)
	(3 marks)

Notes

M1: for multiplying the numerator and denominator by \(3\sqrt{5} - 7\) or \(-3\sqrt{5} + 7\) (may be implied)

M1: for expanding the denominator in a correct fraction

denominator may be 4 terms which all need to be correct

\(\dfrac{4}{3\sqrt{5} + 7} \times \dfrac{3\sqrt{5} - 7}{3\sqrt{5} - 7} = 7 - 3\sqrt{5}\) scores M1M0

Implies the 1^st mark

A1: dep on M2

SCB1 for answer \(7 - \sqrt{45}\) with no method marks awarded

SCB2 for \(7 - \sqrt{45}\) if you would award the 1^st M1 but not the 2^nd M1 (total 2 marks)

Higher June 2025 Paper 1 Q15

EdexcelCurrent spec4 marksSurds

15 \(\left(\sqrt{3}\right)^5 = k\sqrt{3}\) where \(k\) is an integer.

(a) Find the value of \(k\) (1)

(b) Show that \(\dfrac{21}{3 - \sqrt{2}}\) can be written in the form \(c + \sqrt{d}\)
where \(c\) and \(d\) are integers.
Show each stage of your working clearly. (3)

Scheme	Marks
9	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(\dfrac{21}{3 - \sqrt{2}} \times \dfrac{3 + \sqrt{2}}{3 + \sqrt{2}}\) or \(\dfrac{21}{3 - \sqrt{2}} \times \dfrac{-3 - \sqrt{2}}{-3 - \sqrt{2}}\)	M1
eg \(\dfrac{21\left(3 + \sqrt{2}\right)}{9 - 3\sqrt{2} + 3\sqrt{2} - 2}\) or \(\dfrac{21\left(3 + \sqrt{2}\right)}{3^2 - 2}\) or \(\dfrac{21\left(3 + \sqrt{2}\right)}{9 - 2}\) or \(\dfrac{21\left(3 + \sqrt{2}\right)}{7}\) or \(\dfrac{63 + 21\sqrt{2}}{9 - 2}\) or \(\dfrac{63 + 21\sqrt{2}}{7}\)	M1
Working required Answer: \(9 + \sqrt{18}\)	A1
	(3)
	(4 marks)

Notes

M1: for explicitly multiplying the numerator and the denominator by \(3 + \sqrt{2}\) or \(-3 - \sqrt{2}\)

M1: dep on M1 (denominator may be 4 terms which all need to be correct)

\(\dfrac{21}{3 - \sqrt{2}} \times \dfrac{3 + \sqrt{2}}{3 + \sqrt{2}} = 9 + 3\sqrt{2}\) scores M1M0

A1: dep on M2

SCB1 for \(9 + \sqrt{18}\) gained with no method marks awarded

SCB2 for \(9 + \sqrt{18}\) gained if you would award 1^st M1 but not 2^nd M1 (total 2 marks)

Higher November 2024 Paper 1 Q17

EdexcelCurrent spec4 marksSurds

(a) Express \(\sqrt{675}\) in the form \(n\sqrt{27}\) where \(n\) is a positive integer. (1)

(b) Show that \(\dfrac{5 - \sqrt{2}}{\sqrt{2} - 1}\) can be written in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. (3)

Scheme	Marks
\(5\sqrt{27}\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(\dfrac{5 - \sqrt{2}}{\sqrt{2} - 1} \times \dfrac{\sqrt{2} + 1}{\sqrt{2} + 1}\) or \(\dfrac{5 - \sqrt{2}}{\sqrt{2} - 1} \times \dfrac{-\sqrt{2} - 1}{-\sqrt{2} - 1}\)	M1
eg \(\dfrac{5\sqrt{2} + 5 - 2 - \sqrt{2}}{2 - 1}\) oe or \(\dfrac{5\sqrt{2} + 5 - 2 - \sqrt{2}}{\sqrt{4} + \sqrt{2} - \sqrt{2} - 1}\) oe or \(5\sqrt{2} + 5 - 2 - \sqrt{2}\)	M1
Working required Answer: \(3 + 4\sqrt{2}\)	A1
	(3)
	(4 marks)

Notes

M1: for rationalising the denominator by multiplying numerator and denominator by \(\sqrt{2} + 1\) or \(-\sqrt{2} - 1\)

M1: (numerator must be expanded to 4 terms, denominator may be 4 terms which need to be all correct)

Accept 1 in the denominator without working

A1: or for stating \(a = 3\) and \(b = 4\)
dep on M2