Edexcel A-Level Mathematics AS P1.1.2 Use and Manipulate Surds Questions

Surd questions reward exact working: simplify roots, rationalise conjugate denominators and keep answers in exact requested forms.

Exam points

Simplify exact lengths such as QR = 2√2x from Pythagoras-style expressions.
Rationalise denominators with conjugates to give a√2 + b or p√3 + q forms.
Solve equations involving √3 terms and present x in the requested exact surd form.

[Maximum number: 5]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

(a)

Show that $\frac{\sqrt{180}-\sqrt{80}}{\sqrt{5}}$ is an integer and find its value.

[ 2 ]

Writes $\sqrt{180}$ as $6 \sqrt{5}$ or $\sqrt{80}$ as $4 \sqrt{5}$
Concludes working $\frac{6 \sqrt{5}-4 \sqrt{5}}{\sqrt{5}}=\frac{2 \sqrt{5}}{\sqrt{5}}=2$
(2)

(b)

Simplify

\frac{4 \sqrt{5}-5}{7-3 \sqrt{5}}

giving your answer in the form $a+b \sqrt{5}$ where a and b are rational numbers.

[ 3 ]

$\frac{4\sqrt5-5}{7-3\sqrt5} =\frac{4\sqrt5-5}{7-3\sqrt5}\times\frac{7+3\sqrt5}{7+3\sqrt5}$ $=\frac{28\sqrt5-35+12(\sqrt5)^2-15\sqrt5}{49-9\times5}$ $=\frac{25}{4}+\frac{13}{4}\sqrt5$
Marking guidance for this part:
M1 for a correct attempt to rationalise by multiplying by $7+3\sqrt5$ .
dM1 for a surd-free denominator and an attempt to multiply out the numerator with four terms.
A1 for $\frac{25}{4}+\frac{13}{4}\sqrt5$ .

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Edexcel A-Level Mathematics AS P1.1.2 Use and Manipulate Surds Questions question 1

Question (a)

Question (b)