1.02b Surds: manipulation and rationalising denominators

265 questions

Sort by: Default | Easiest first | Hardest first
Edexcel AS Paper 1 2020 June Q3
6 marks Moderate -0.8
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Solve the equation $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
  2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$
Edexcel AS Paper 1 Specimen Q12
8 marks Moderate -0.3
12.
[diagram]
Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.
  1. Using algebra, find the \(x\) coordinate of point \(A\).
  2. [diagram]
    The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).
Edexcel PMT Mocks Q5
6 marks Standard +0.3
  1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } ( x )\) The values of \(y\) are given to 2 decimal places as appropriate.
\(x\)34.567.59
\(y\)11.371.631.832
a. Obtain an estimate for \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( x ) \mathrm { d } x\), giving your answer to two decimal places. Use your answer to part (a) and making your method clear, estimate
b. i) \(\int _ { 3 } ^ { 9 } \log _ { 3 } \sqrt { x } \mathrm {~d} x\) ii) \(\int _ { 3 } ^ { 18 } \log _ { 3 } \left( 9 x ^ { 3 } \right) \mathrm { d } x\)
Edexcel PMT Mocks Q8
5 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-14_1090_1205_274_456} \captionsetup{labelformat=empty} \caption{Figure 2
Figure 2 shows a sketch of part of the curve with equation $$y = \frac { 12 x - x ^ { 2 } } { \sqrt { x } } , \quad x > 0$$ The region \(R\), shows shaded in figure 2, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 8\).
Show that the area of the shaded region \(R\) is \(\frac { 128 } { 5 } ( 3 \sqrt { 2 } - 2 )\).}
\end{figure} (5)
Edexcel Paper 1 2018 June Q13
7 marks Standard +0.3
  1. Show that
$$\int _ { 0 } ^ { 2 } 2 x \sqrt { x + 2 } \mathrm {~d} x = \frac { 32 } { 15 } ( 2 + \sqrt { 2 } )$$
Edexcel Paper 1 2021 October Q6
5 marks Moderate -0.3
6. Figure 1 Figure 1 shows a sketch of triangle \(A B C\).
Given that
  • \(\overrightarrow { A B } = - 3 \mathbf { i } - 4 \mathbf { j } - 5 \mathbf { k }\)
  • \(\overrightarrow { B C } = \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
    1. find \(\overrightarrow { A C }\)
    2. show that \(\cos A B C = \frac { 9 } { 10 }\)
Edexcel Paper 1 2021 October Q13
3 marks Standard +0.3
  1. A curve \(C\) has parametric equations
$$x = \frac { t ^ { 2 } + 5 } { t ^ { 2 } + 1 } \quad y = \frac { 4 t } { t ^ { 2 } + 1 } \quad t \in \mathbb { R }$$ Show that all points on \(C\) satisfy $$( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$$
OCR MEI AS Paper 1 2018 June Q1
2 marks Easy -1.2
1 Write \(\frac { 8 } { 3 - \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers to be found.
OCR MEI AS Paper 1 2019 June Q3
4 marks Moderate -0.8
3 Given that \(k\) is an integer, express \(\frac { 3 \sqrt { 2 } - k } { \sqrt { 8 } + 1 }\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational expressions in terms of \(k\).
OCR MEI AS Paper 1 2022 June Q1
3 marks Easy -1.2
1 Rationalise the denominator of the fraction \(\frac { 2 + \sqrt { n } } { 3 + \sqrt { n } }\), where \(n\) is a positive integer.
OCR MEI AS Paper 2 2019 June Q3
3 marks Moderate -0.8
3 Without using a calculator, prove that \(3 \sqrt { 2 } > 2 \sqrt { 3 }\).
OCR MEI AS Paper 2 2024 June Q4
3 marks Easy -1.2
4 In this question you must show detailed reasoning.
Express \(\frac { 1 + 4 \sqrt { 3 } } { 2 + \sqrt { 3 } }\) in the form \(\mathrm { a } + \mathrm { b } \sqrt { 3 }\), where \(a\) and \(b\) are integers to be determined.
OCR MEI Paper 1 2020 November Q2
3 marks Easy -1.2
2 Express \(\frac { a + \sqrt { 2 } } { 3 - \sqrt { 2 } }\) in the form \(\mathrm { p } + \mathrm { q } \sqrt { 2 }\), giving \(p\) and \(q\) in terms of \(a\).
OCR MEI Paper 2 2018 June Q1
2 marks Easy -1.8
1 Show that \(\sqrt { 27 } + \sqrt { 192 } = a \sqrt { b }\), where \(a\) and \(b\) are prime numbers to be determined.
OCR MEI Paper 2 2024 June Q1
2 marks Easy -1.8
1 Calculate the exact distance between the points ( \(2 , - 1\) ) and ( 6,1 ). Give your answer in the form \(\mathrm { a } \sqrt { \mathrm { b } }\), where \(a\) and \(b\) are prime numbers.
OCR MEI Paper 3 2023 June Q3
3 marks Moderate -0.3
3 In this question you must show detailed reasoning.
Find the value of \(k\) such that \(\frac { 1 } { \sqrt { 5 } + \sqrt { 6 } } + \frac { 1 } { \sqrt { 6 } + \sqrt { 7 } } = \frac { k } { \sqrt { 5 } + \sqrt { 7 } }\).
OCR MEI Paper 3 2021 November Q6
4 marks Moderate -0.8
6 In this question you must show detailed reasoning.
Show that \(\sum _ { r = 1 } ^ { 3 } \frac { 1 } { \sqrt { r + 1 } + \sqrt { r } } = 1\).
OCR MEI Paper 3 Specimen Q14
3 marks Standard +0.8
14 Show that the two values of \(b\) given on line 36 are equivalent.
AQA C1 2005 January Q5
7 marks Easy -1.2
5
  1. Simplify \(( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )\).
  2. Express \(\sqrt { 12 }\) in the form \(m \sqrt { 3 }\), where \(m\) is an integer.
  3. Express \(\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
AQA C1 2006 January Q1
4 marks Easy -1.8
1
  1. Simplify \(( \sqrt { 5 } + 2 ) ( \sqrt { 5 } - 2 )\).
  2. Express \(\sqrt { 8 } + \sqrt { 18 }\) in the form \(n \sqrt { 2 }\), where \(n\) is an integer.
AQA C1 2009 January Q3
7 marks Easy -1.2
3
  1. Express \(\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
  2. Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
AQA C1 2010 January Q4
7 marks Easy -1.2
4
  1. Show that \(\frac { \sqrt { 50 } + \sqrt { 18 } } { \sqrt { 8 } }\) is an integer and find its value.
    (3 marks)
  2. Express \(\frac { 2 \sqrt { 7 } - 1 } { 2 \sqrt { 7 } + 5 }\) in the form \(m + n \sqrt { 7 }\), where \(m\) and \(n\) are integers.
    (4 marks)
AQA C1 2011 January Q2
5 marks Easy -1.2
2
  1. Simplify \(( 3 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 4 \sqrt { 3 } + 3 \sqrt { 7 } } { 3 \sqrt { 3 } + \sqrt { 7 } }\) in the form \(\frac { m + \sqrt { 21 } } { n }\), where \(m\) and \(n\) are integers.
AQA C1 2012 January Q3
9 marks Easy -1.2
3
    1. Simplify \(( 3 \sqrt { 2 } ) ^ { 2 }\).
    2. Show that \(( 3 \sqrt { 2 } - 1 ) ^ { 2 } + ( 3 + \sqrt { 2 } ) ^ { 2 }\) is an integer and find its value.
  2. Express \(\frac { 4 \sqrt { 5 } - 7 \sqrt { 2 } } { 2 \sqrt { 5 } + \sqrt { 2 } }\) in the form \(m - \sqrt { n }\), where \(m\) and \(n\) are integers.
AQA C1 2013 January Q3
8 marks Easy -1.3
3
    1. Express \(\sqrt { 18 }\) in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
    2. Simplify \(\frac { \sqrt { 8 } } { \sqrt { 18 } + \sqrt { 32 } }\).
  2. Express \(\frac { 7 \sqrt { 2 } - \sqrt { 3 } } { 2 \sqrt { 2 } - \sqrt { 3 } }\) in the form \(m + \sqrt { n }\), where \(m\) and \(n\) are integers.