1.15 Rationalising the denominator
9 exam questions worth 23 marks. Write your answer, open the mark scheme, then give yourself the marks you earned. Your answers are saved on this device.
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What is \dfrac{1}{\sqrt{5}} with a rational denominator?
Choose one answer.
Correct Not quite: the answer is \( \dfrac{\sqrt{5}}{5} \)
Mark scheme
A: \dfrac{\sqrt{5}}{5}
Model answer
\dfrac{\sqrt{5}}{5} (A). Multiply the top and bottom by √5.
Rationalise the denominator of \dfrac{6}{\sqrt{3}}. Give your answer in its simplest form.
Mark scheme
M1: multiplies top and bottom by √3; A1: 2√3
Model answer
6√3 / 3 = 2√3.
Rationalise the denominator of \dfrac{10}{\sqrt{2}}.
Mark scheme
M1: multiplies top and bottom by √2; A1: 5√2
Model answer
10√2 / 2 = 5√2.
Rationalise the denominator of \dfrac{3}{2\sqrt{3}}. Give your answer in its simplest form.
Mark scheme
M1: multiplies top and bottom by √3; M1: 3√3 / 6; A1: √3 / 2
Model answer
3√3 / (2 × 3) = 3√3 / 6 = √3 / 2.
Rationalise the denominator of \dfrac{1}{2 + \sqrt{3}}.
Mark scheme
M1: multiplies by (2 - √3) / (2 - √3); M1: denominator 4 - 3; A1: 2 - √3
Model answer
(2 - √3) / ((2 + √3)(2 - √3)) = (2 - √3) / (4 - 3) = 2 - √3.
Rationalise the denominator of \dfrac{4}{3 - \sqrt{5}}. Give your answer in the form a + \sqrt{b}.
Mark scheme
M1: multiplies by (3 + √5); M1: denominator 9 - 5 = 4; A1: 3 + √5
Model answer
4(3 + √5) / (9 - 5) = 4(3 + √5) / 4 = 3 + √5.
(a) Write down the conjugate of 5 - \sqrt{2}. (b) Expand and simplify (5 - \sqrt{2})(5 + \sqrt{2}).
Mark scheme
B1: 5 + √2; B1: 23
Model answer
(a) 5 + √2. (b) 25 - 2 = 23.
Simplify \dfrac{\sqrt{2} + 1}{\sqrt{2} - 1}. Give your answer in the form a + b\sqrt{2}.
Mark scheme
M1: multiplies by (√2 + 1); M1: top 3 + 2√2 and bottom 1; A1: 3 + 2√2
Model answer
Top: (√2 + 1)² = 2 + 2√2 + 1 = 3 + 2√2. Bottom: 2 - 1 = 1. The answer is 3 + 2√2.
Show that \dfrac{3 + \sqrt{5}}{3 - \sqrt{5}} = \dfrac{7 + 3\sqrt{5}}{2}.
Mark scheme
M1: multiplies top and bottom by (3 + √5); M1: top = 14 + 6√5; M1: bottom = 4; A1: shows division by 2 to give (7 + 3√5)/2
Model answer
Multiply by (3 + √5). Top: (3 + √5)² = 9 + 6√5 + 5 = 14 + 6√5. Bottom: 9 - 5 = 4. So (14 + 6√5)/4 = (7 + 3√5)/2.
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