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OCR MEI C1 Q6
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
  2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
  3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
  4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]
OCR MEI C1 Q1
4 marks Easy -1.3
Evaluate the following.
  1. \(200^0\) [1]
  2. \(\left(\frac{25}{9}\right)^{-\frac{1}{2}}\) [3]
OCR MEI C1 Q2
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{1}{27}\right)^{\frac{2}{3}}\). [2]
  2. Simplify \(\frac{(4a^2c)^3}{32a^4c^7}\). [3]
OCR MEI C1 Q3
4 marks Easy -1.2
You are given that \(n\), \(n + 1\) and \(n + 2\) are three consecutive integers.
  1. Expand and simplify \(n^2 + (n + 1)^2 + (n + 2)^2\). [2]
  2. For what values of \(n\) will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer. [2]
OCR MEI C1 Q4
5 marks Easy -1.3
  1. Evaluate \((0.2)^{-2}\). [2]
  2. Simplify \((16a^{12})^{\frac{3}{4}}\). [3]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of each of the following.
  1. \(\left(\frac{5}{3}\right)^{-2}\) [2]
  2. \(81^{\frac{1}{4}}\) [2]
OCR MEI C1 Q6
4 marks Easy -1.8
  1. Evaluate \(\left(\frac{1}{5}\right)^{-2}\). [2]
  2. Evaluate \(\left(\frac{8}{27}\right)^{\frac{2}{3}}\). [2]
OCR MEI C1 Q7
5 marks Moderate -0.8
  1. Simplify \(\frac{10(\sqrt{6})^3}{\sqrt{24}}\). [3]
  2. Simplify \(\frac{1}{4 - \sqrt{5}} + \frac{1}{4 + \sqrt{5}}\). [2]
OCR MEI C1 Q8
5 marks Easy -1.3
  1. Evaluate \(9^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(4x^4)^3 y^2}{2x^4 y^5}\). [3]
OCR MEI C1 Q9
3 marks Moderate -0.5
Expand and simplify \((n + 2)^3 - n^3\). [3]
OCR MEI C1 Q10
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{9}{16}\right)^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(2ac^2)^3 \times 9a^2c}{36a^4c^{12}}\). [3]
OCR MEI C1 Q11
4 marks Easy -1.8
  1. Write down the value of each of the following.
    1. \(4^{-2}\) [1]
    2. \(9^0\) [1]
  2. Find the value of \(\left(\frac{64}{125}\right)^{\frac{4}{3}}\). [2]
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 Q2
5 marks Easy -1.8
  1. Simplify \((5a^2b)^2 \times 2b^4\). [2]
  2. Evaluate \(\left(\frac{4}{16}\right)^{-1}\). [1]
  3. Evaluate \((16)^{\frac{3}{2}}\). [2]
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 Q5
3 marks Easy -1.8
Find the value of each of the following.
  1. \(5^2 \times 5^{-2}\) [2]
  2. \(100^{\frac{1}{2}}\) [1]
OCR MEI C1 Q6
2 marks Easy -2.0
State the value of each of the following.
  1. \(2^{-3}\) [1]
  2. \(9^0\) [1]
OCR MEI C1 Q7
4 marks Easy -1.8
  1. Express \(125\sqrt{5}\) in the form \(5^k\). [2]
  2. Simplify \((4a^3b^5)^2\). [2]
OCR MEI C1 Q8
5 marks Easy -1.3
  1. Find the value of \(\left(\frac{1}{25}\right)^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(2x^2y^3z)^5}{4y^5z}\). [3]
OCR MEI C1 Q9
4 marks Easy -1.8
  1. Write down the value of \(\left(\frac{1}{4}\right)^0\). [1]
  2. Find the value of \(16^{-\frac{3}{2}}\). [3]
OCR MEI C1 Q10
4 marks Easy -1.8
  1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
  2. Find the value of \(\left(\frac{1}{2}\right)^{-5}\). [2]
OCR MEI C1 Q11
4 marks Easy -1.8
Find the value of each of the following, giving each answer as an integer or fraction as appropriate.
  1. \(8^{\frac{1}{3}}\) [2]
  2. \(\left(\frac{7}{3}\right)^{-2}\) [2]
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 Q13
5 marks Easy -1.3
Simplify the following.
  1. \(\frac{16^{\frac{1}{3}}}{81^{\frac{1}{4}}}\) [2]
  2. \(\frac{12(a^3b^2c)^4}{4a^2c^6}\) [3]
OCR MEI C1 Q1
5 marks Moderate -0.8
Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]