1.02b Surds: manipulation and rationalising denominators

265 questions

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OCR C1 2009 January Q1
3 marks Easy -1.2
1 Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
OCR C1 2010 January Q5
7 marks Standard +0.3
5 Solve the equation \(x - 8 \sqrt { x } + 13 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2012 January Q1
4 marks Easy -1.2
1 Express \(\frac { 15 + \sqrt { 3 } } { 3 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
OCR C1 2012 January Q4
5 marks Easy -1.8
4 Evaluate
  1. \(3 ^ { - 2 }\),
  2. \(16 ^ { \frac { 3 } { 4 } }\),
  3. \(\frac { \sqrt { 200 } } { \sqrt { 8 } }\).
OCR C1 2009 June Q2
4 marks Easy -1.2
2 Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.
OCR C1 2010 June Q3
5 marks Easy -1.2
3
  1. Express \(\frac { 12 } { 3 + \sqrt { 5 } }\) in the form \(a - b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
  2. Express \(\sqrt { 18 } - \sqrt { 2 }\) in simplified surd form.
OCR C1 2011 June Q5
6 marks Easy -1.3
5
  1. Express \(\sqrt { 300 } - \sqrt { 48 }\) in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  2. Express \(\frac { 15 + \sqrt { 40 } } { \sqrt { 5 } }\) in the form \(a \sqrt { 5 } + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
OCR C1 2012 June Q7
6 marks Standard +0.3
7 Solve the equation \(x - 6 x ^ { \frac { 1 } { 2 } } + 2 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2015 June Q1
3 marks Easy -1.2
1 Express \(\frac { 8 } { \sqrt { 3 } - 1 }\) in the form \(a \sqrt { 3 } + b\), where \(a\) and \(b\) are integers.
OCR C1 2016 June Q2
4 marks Easy -1.2
2 Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
OCR MEI C1 2007 January Q7
4 marks Standard +0.8
7 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 - \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
OCR MEI C1 2010 January Q1
3 marks Easy -1.8
1 Rearrange the formula \(c = \sqrt { \frac { a + b } { 2 } }\) to make \(a\) the subject.
OCR MEI C1 2010 January Q5
5 marks Easy -1.2
5
  1. Find the value of \(144 ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { 1 } { 5 + \sqrt { 7 } } + \frac { 4 } { 5 - \sqrt { 7 } }\). Give your answer in the form \(\frac { a + b \sqrt { 7 } } { c }\).
OCR MEI C1 2011 January Q5
4 marks Moderate -0.5
5 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.
OCR MEI C1 2011 January Q7
5 marks Easy -1.2
7
  1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
  2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2012 January Q4
5 marks Easy -1.2
4
  1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
  2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).
OCR MEI C1 2013 January Q7
5 marks Moderate -0.8
7
  1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
  2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2014 June Q4
5 marks Easy -1.2
4
  1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 2016 June Q5
5 marks Easy -1.2
5
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Express \(\frac { 5 + 2 \sqrt { 3 } } { 4 - \sqrt { 3 } }\) in the form \(c + d \sqrt { 3 }\), where \(c\) and \(d\) are integers.
OCR C4 2011 January Q5
9 marks Standard +0.3
5 In this question, \(I\) denotes the definite integral \(\int _ { 2 } ^ { 5 } \frac { 5 - x } { 2 + \sqrt { x - 1 } } \mathrm {~d} x\). The value of \(I\) is to be found using two different methods.
  1. Show that the substitution \(u = \sqrt { x - 1 }\) transforms \(I\) to \(\int _ { 1 } ^ { 2 } \left( 4 u - 2 u ^ { 2 } \right) \mathrm { d } u\) and hence find the exact value of \(I\).
  2. (a) Simplify \(( 2 + \sqrt { x - 1 } ) ( 2 - \sqrt { x - 1 } )\).
    (b) By first multiplying the numerator and denominator of \(\frac { 5 - x } { 2 + \sqrt { x - 1 } }\) by \(2 - \sqrt { x - 1 }\), find the exact value of \(I\).
OCR MEI C4 Q5
Standard +0.8
5 Justify the statement in line 87 that $$\frac { 1 } { \phi } = \frac { \sqrt { 5 } - 1 } { 2 }$$
OCR H240/01 2020 November Q2
8 marks Easy -1.3
2 Simplify fully.
  1. \(\sqrt { 12 a } \times \sqrt { 3 a ^ { 5 } }\)
  2. \(\left( 64 b ^ { 3 } \right) ^ { \frac { 1 } { 3 } } \times \left( 4 b ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\)
  3. \(7 \times 9 ^ { 3 c } - 4 \times 27 ^ { 2 c }\)
OCR H240/01 2023 June Q2
8 marks Moderate -0.8
2
    1. Show that \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }\) can be written in the form \(\frac { a } { b + c x }\), where \(a , b\) and \(c\) are constants to be determined.
    2. Hence solve the equation \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2\).
  2. In this question you must show detailed reasoning. Solve the equation \(2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0\).
OCR H240/02 Q1
4 marks Easy -1.3
1 Simplify fully.
  1. \(\sqrt { a ^ { 3 } } \times \sqrt { 16 a }\)
  2. \(\quad \left( 4 b ^ { 6 } \right) ^ { \frac { 5 } { 2 } }\)
Edexcel AS Paper 1 2019 June Q2
8 marks Moderate -0.8
  1. Find, using algebra, all real solutions to the equation
    1. \(16 a ^ { 2 } = 2 \sqrt { a }\)
    2. \(b ^ { 4 } + 7 b ^ { 2 } - 18 = 0\)