1.02b Surds: manipulation and rationalising denominators

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OCR MEI Paper 3 2019 June Q4
3 marks Standard +0.3
4 In this question you must show detailed reasoning.
Show that \(\frac { 1 } { \sqrt { 10 } + \sqrt { 11 } } + \frac { 1 } { \sqrt { 11 } + \sqrt { 12 } } + \frac { 1 } { \sqrt { 12 } + \sqrt { 13 } } = \frac { 3 } { \sqrt { 10 } + \sqrt { 13 } }\).
OCR Further Pure Core AS 2023 June Q9
10 marks Standard +0.8
9 Matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b \\ 0 & 1 & 0 \\ b & 0 & a \end{array} \right)\) where \(a\) and \(b\) are constants.
  1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). The constants \(a\) and \(b\) are given by \(a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )\) and \(b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )\).
  2. By determining exact expressions for \(a b\) and \(a ^ { 2 } - b ^ { 2 }\) and using the result from part (a), show that \(\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1 \\ 0 & 2 & 0 \\ 1 & 0 & \sqrt { 3 } \end{array} \right)\) where \(k\) is a real number whose value is to be determined.
  3. Find \(\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }\) and \(\mathbf { R } ^ { 24 }\).
  4. Describe fully the transformation represented by \(\mathbf { R }\). \section*{END OF QUESTION PAPER}
Edexcel PURE 2024 October Q2
Easy -1.2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
  2. Find the exact value of \(a\) such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form \(p \sqrt { 3 } + q\) where \(p\) and \(q\) are fully simplified rational constants.
Pre-U Pre-U 9794/1 2013 November Q4
Moderate -0.8
4 Solve the equation \(x ^ { 2 } + ( \sqrt { 3 } ) x - 18 = 0\), giving each root in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers.
Pre-U Pre-U 9794/2 2015 June Q1
3 marks Easy -1.8
1 Show that \(\frac { 31 } { 6 - \sqrt { 5 } } = 6 + \sqrt { 5 }\).
Pre-U Pre-U 9794/1 2016 June Q2
2 marks Easy -1.8
2 Without using a calculator, simplify the following, giving each answer in the form \(a \sqrt { 5 }\) where \(a\) is an integer. Show all your working.
  1. \(4 \sqrt { 10 } \times \sqrt { 2 }\)
  2. \(\sqrt { 500 } + \sqrt { 125 }\)
Pre-U Pre-U 9794/2 2016 Specimen Q1
9 marks Easy -1.3
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
  3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).
Pre-U Pre-U 9794/2 2018 June Q4
12 marks Moderate -0.3
4 Solve the equation \(x + 2 \sqrt { x } - 6 = 0\), giving your answer in the form \(x = c + d \sqrt { 7 }\) where \(c\) and \(d\) are integers.
Pre-U Pre-U 9794/2 Specimen Q1
3 marks Moderate -0.3
1 Solve the equation $$x \sqrt { 32 } - \sqrt { 24 } = ( 3 \sqrt { 3 } - 5 ) ( \sqrt { 6 } + x \sqrt { 2 } )$$
WJEC Unit 1 2018 June Q1
Moderate -0.8
Showing all your working, simplify a) \(\frac { 24 \sqrt { a } } { ( \sqrt { a } + 3 ) ^ { 2 } - ( \sqrt { a } - 3 ) ^ { 2 } }\),
b) \(\frac { 3 \sqrt { 7 } + 5 \sqrt { 3 } } { \sqrt { 7 } + \sqrt { 3 } }\).
Edexcel P1 2018 Specimen Q2
5 marks Easy -1.2
  1. Given that \(3^{-1.5} = a\sqrt{3}\) find the exact value of \(a\) [2]
  2. Simplify fully \(\frac{(2x^{\frac{1}{2}})^3}{4x^2}\) [3]
Edexcel C1 Q3
6 marks Moderate -0.8
\(x^2 - 8x - 29 = (x + a)^2 + b\), where \(a\) and \(b\) are constants.
  1. Find the value of \(a\) and the value of \(b\). [3]
  2. Hence, or otherwise, show that the roots of $$x^2 - 8x - 29 = 0$$ are \(c \pm d\sqrt{5}\), where \(c\) and \(d\) are integers to be found. [3]
Edexcel C1 Q5
6 marks Easy -1.2
  1. Write \(\sqrt{45}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
  2. Express \(\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [5]
Edexcel C1 Q6
4 marks Easy -1.3
  1. Expand and simplify \((4 + \sqrt{3})(4 - \sqrt{3})\). [2]
  2. Express \(\frac{26}{4 + \sqrt{3}}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]
Edexcel C1 Q2
4 marks Easy -1.3
  1. Express \(\sqrt{108}\) in the form \(a\sqrt{3}\), where \(a\) is an integer. [1]
  2. Express \((2 - \sqrt{3})^2\) in the form \(b + c\sqrt{3}\), where \(b\) and \(c\) are integers to be found. [3]
Edexcel C1 Q6
5 marks Moderate -0.8
  1. Show that \((4 + 3\sqrt{x})^3\) can be written as \(16 + k\sqrt{x} + 9x\), where \(k\) is a constant to be found. [2]
  2. Find \(\int (4 + 3\sqrt{x})^3 \, dx\). [3]
Edexcel C1 Q1
2 marks Easy -1.8
Simplify \((3 + \sqrt{5})(3 - \sqrt{5})\). [2]
Edexcel C1 Q1
5 marks Easy -1.2
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q36
9 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\) is such that $$\frac{dy}{dx} = 3\sqrt{x} + \frac{12}{\sqrt{x}}, \quad x > 0.$$
  1. Show that, when \(x = 8\), the exact value of \(\frac{dy}{dx}\) is \(9\sqrt{2}\). [3]
The curve \(C\) passes through the point \((4, 30)\).
  1. Using integration, find \(f(x)\). [6]
Edexcel C1 Q38
7 marks Standard +0.3
A sequence is defined by the recurrence relation $$u_{n+1} = \sqrt{\frac{u_n}{2} + \frac{a}{u_n}}, \quad n = 1, 2, 3, \ldots,$$ where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u_1 = 3\), find the values of \(u_2\), \(u_3\) and \(u_4\), giving your answers to 2 decimal places. [3]
  2. Given instead that \(u_1 = u_2 = 3\),
    1. calculate the value of \(a\), [3]
    2. write down the value of \(u_5\). [1]
Edexcel C1 Q40
6 marks Moderate -0.8
Giving your answers in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational numbers, find
  1. \((3 - \sqrt{8})^2\), [3]
  2. \(\frac{1}{4 - \sqrt{8}}\). [3]
Edexcel C1 Specimen Q3
4 marks Easy -1.3
  1. Express \(\sqrt{80}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
  2. Express \((4 - \sqrt{5})^2\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [3]
Edexcel C4 2015 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$(4 + 5x)^{\frac{1}{2}}, \quad |x| < \frac{4}{5}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient in its simplest form. [5]
  2. Find the exact value of \((4 + 5x)^{\frac{1}{2}}\) when \(x = \frac{1}{10}\) Give your answer in the form \(k\sqrt{2}\), where \(k\) is a constant to be determined. [1]
  3. Substitute \(x = \frac{1}{10}\) into your binomial expansion from part (a) and hence find an approximate value for \(\sqrt{2}\) Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers. [2]
Edexcel M2 2014 January Q1
4 marks Easy -1.2
Simplify fully
  1. \((2\sqrt{x})^2\) [1]
  2. \(\frac{5 + \sqrt{7}}{2 + \sqrt{7}}\) [3]
Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]