1.02b Surds: manipulation and rationalising denominators

265 questions

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OCR MEI C1 Q1
5 marks Easy -1.2
  1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
  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. [3]
OCR MEI C1 Q4
5 marks Easy -1.2
  1. Find the value of \(144^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{1}{5 + \sqrt{7}} + \frac{4}{5 - \sqrt{7}}\). Give your answer in the form \(\frac{a + b\sqrt{7}}{c}\). [3]
OCR MEI C1 Q12
5 marks Moderate -0.8
  1. Simplify \(6\sqrt{2} \times 5\sqrt{3} \times \sqrt{24}\). [2]
  2. Express \((2 - 3\sqrt{5})^2\) in the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are integers. [3]
OCR MEI C1 Q1
4 marks Moderate -0.5
You are given that \(a = \frac{3}{2}\), \(b = \frac{9 - \sqrt{17}}{4}\) and \(c = \frac{9 + \sqrt{17}}{4}\). Show that \(a + b + c = abc\). [4]
Edexcel C2 Q8
9 marks Moderate -0.3
A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).
  1. Write down an equation of the circle \(C\). [2]
  2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
  3. Find the distance between these two points. [2]
Edexcel C2 Q3
7 marks Moderate -0.8
  1. Expand (2√x + 3)². [2]
  2. Hence evaluate $$\int_1^{2^2} (2\sqrt{x} + 3)^2 \, dx$$, giving your answer in the form a + b√2, where a and b are integers. [5]
Edexcel C2 Q4
7 marks Moderate -0.3
\includegraphics{figure_1} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.
  1. Find, in surd form, the length \(AB\). [2]
  2. Find, in terms of \(\pi\), the area of the badge. [2]
  3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [3]
Edexcel C2 Q2
7 marks Moderate -0.8
  1. Expand \((2\sqrt{x} + 3)^2\). [2]
  2. Hence evaluate \(\int_1^2 (2\sqrt{x} + 3)^2 \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [5]
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Expand \((1 + x)^4\) in ascending powers of \(x\). [2]
  2. Using your expansion, express each of the following in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.
    1. \((1 + \sqrt{2})^4\)
    2. \((1 - \sqrt{2})^8\) [7]
OCR MEI C3 Q5
4 marks Moderate -0.3
Make \(x\) the subject of \(t = \ln \sqrt{\frac{5}{(x-3)}}\). [4]
OCR C3 Q3
7 marks Standard +0.8
  1. Prove the identity $$\sqrt{2} \cos (x + 45)° + 2 \cos (x - 30)° \equiv (1 + \sqrt{3}) \cos x°.$$ [4]
  2. Hence, find the exact value of \(\cos 75°\) in terms of surds. [3]
Edexcel C4 Q3
9 marks Standard +0.3
  1. Show that \((1 + \frac{1}{24})^{-\frac{1}{2}} = k\sqrt{6}\), where \(k\) is rational. [2]
  2. Expand \((1 + \frac{1}{4}x)^{-\frac{1}{2}}\), \(|x| < 2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
  3. Use your answer to part \((b)\) with \(x = \frac{1}{6}\) to find an approximate value for \(\sqrt{6}\), giving your answer to 5 decimal places. [3]
Edexcel AEA 2004 June Q1
9 marks Challenging +1.8
Solve the equation \(\cos x + \sqrt{(1 - \frac{1}{2} \sin 2x)} = 0\), in the interval \(0° \leq x < 360°\). [9]
AQA AS Paper 1 2019 June Q4
4 marks Moderate -0.8
Show that \(\frac{\sqrt{6}}{\sqrt{3} - \sqrt{2}}\) can be expressed in the form \(m\sqrt{n} + n\sqrt{m}\), where \(m\) and \(n\) are integers. Fully justify your answer. [4 marks]
AQA AS Paper 1 2021 June Q4
9 marks Moderate -0.3
\(ABCD\) is a trapezium where \(A\) is the point \((1, -2)\), \(B\) is the point \((7, 1)\) and \(C\) is the point \((3, 4)\) \(DC\) is parallel to \(AB\). \(AD\) is perpendicular to \(AB\).
    1. Find the equation of the line \(CD\). [2 marks]
    2. Show that point \(D\) has coordinates \((-1, 2)\) [3 marks]
    1. Find the sum of the length of \(AB\) and the length of \(CD\) in simplified surd form. [2 marks]
    2. Hence, find the area of the trapezium \(ABCD\). [2 marks]
AQA AS Paper 1 2024 June Q3
4 marks Moderate -0.8
Express \(\frac{\sqrt{3} + 3\sqrt{5}}{\sqrt{5} - \sqrt{3}}\) in the form \(a + b\sqrt{c}\), where \(a\) and \(b\) are integers. Fully justify your answer. [4 marks]
AQA AS Paper 1 Specimen Q4
3 marks Easy -1.2
Show that \(\frac{5\sqrt{2} + 2}{3\sqrt{2} + 4}\) can be expressed in the form \(m + n\sqrt{2}\), where \(m\) and \(n\) are integers. [3 marks]
AQA Paper 1 2024 June Q7
4 marks Standard +0.3
Show that $$\frac{3 + \sqrt{8n}}{1 + \sqrt{2n}}$$ can be written as $$\frac{4n - 3 + \sqrt{2n}}{2n - 1}$$ where \(n\) is a positive integer. [4 marks]
AQA Paper 3 2021 June Q6
4 marks Standard +0.3
Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]
OCR PURE Q1
5 marks Easy -1.3
In this question you must show detailed reasoning.
  1. Express \(3^{\frac{1}{2}}\) in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [2]
  2. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
WJEC Unit 1 2019 June Q07
6 marks Moderate -0.8
Given that \(a\), \(b\) are integers, simplify the following. Show all your working.
  1. \(\frac{2\sqrt{3} + a}{\sqrt{3} - 1}\) [3]
  2. \(\frac{2\sqrt{6b^2} - \sqrt{27} + \sqrt{192}}{\sqrt{2}}\) [3]
WJEC Unit 1 2022 June Q2
6 marks Moderate -0.3
Showing all your working, simplify the following expression. [6] $$5\sqrt{48} + \frac{2+5\sqrt{3}}{5+3\sqrt{3}} - (2\sqrt{3})^3$$
WJEC Unit 1 2024 June Q6
7 marks Moderate -0.8
  1. Find the exact value of \(x\) that satisfies the equation $$\frac{7x^{\frac{5}{4}}}{x^{\frac{1}{2}}} = \sqrt{147}.$$ [4]
  2. Show that \(\frac{(8x-18)}{(2\sqrt{x}-3)}\), where \(x \neq \frac{9}{4}\), may be written as \(2(2\sqrt{x}+3)\). [3]
WJEC Unit 1 Specimen Q10
8 marks Standard +0.8
  1. Use the binomial theorem to express \(\left(\sqrt{3} - \sqrt{2}\right)^5\) in the form \(a\sqrt{3} + b\sqrt{2}\), where \(a\), \(b\) are integers whose values are to be found. [5]
  2. Given that \(\left(\sqrt{3} - \sqrt{2}\right)^5 \approx 0\), use your answer to part (a) to find an approximate value for \(\sqrt{6}\) in the form \(\frac{c}{d}\), where \(c\) and \(d\) are positive integers whose values are to be found. [3]
SPS SPS FM 2019 Q1
3 marks Moderate -0.3
In the question you must show detailed reasoning Solve the equation below, giving your answer in the simplest form $$x\sqrt{32} - \sqrt{24} = (3\sqrt{3} - 5)(\sqrt{6} + x\sqrt{2})$$ [3]