Questions C1 (1562 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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OCR MEI C1 Q2
11 marks Moderate -0.3
Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\).
  1. On the insert, on the same axes, plot the graph of \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]
OCR MEI C1 Q3
3 marks Moderate -0.8
Factorise and hence simplify \(\frac{3x^2 - 7x + 4}{x^2 - 1}\). [3]
OCR MEI C1 Q4
5 marks Moderate -0.3
  1. Prove that 12 is a factor of \(3n^2 + 6n\) for all even positive integers \(n\). [3]
  2. Determine whether 12 is a factor of \(3n^2 + 6n\) for all positive integers \(n\). [2]
OCR MEI C1 Q5
12 marks Moderate -0.3
  1. Write \(x^2 - 5x + 8\) in the form \((x - a)^2 + b\) and hence show that \(x^2 - 5x + 8 > 0\) for all values of \(x\). [4]
  2. Sketch the graph of \(y = x^2 - 5x + 8\), showing the coordinates of the turning point. [3]
  3. Find the set of values of \(x\) for which \(x^2 - 5x + 8 > 14\). [3]
  4. If \(f(x) = x^2 - 5x + 8\), does the graph of \(y = f(x) - 10\) cross the \(x\)-axis? Show how you decide. [2]
OCR MEI C1 Q6
12 marks Moderate -0.8
  1. Write \(4x^2 - 24x + 27\) in the form \(a(x - b)^2 + c\). [4]
  2. State the coordinates of the minimum point on the curve \(y = 4x^2 - 24x + 27\). [2]
  3. Solve the equation \(4x^2 - 24x + 27 = 0\). [3]
  4. Sketch the graph of the curve \(y = 4x^2 - 24x + 27\). [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]
OCR MEI C1 Q2
5 marks Easy -1.3
  1. Simplify \(3a^3b \times 4(ab)^2\). [2]
  2. Factorise \(x^2 - 4\) and \(x^2 - 5x + 6\). Hence express \(\frac{x^2 - 4}{x^2 - 5x + 6}\) as a fraction in its simplest form. [3]
OCR MEI C1 Q3
4 marks Moderate -0.8
Simplify \((m^2 + 1)^2 - (m^2 - 1)^2\), showing your method. Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. [4] \includegraphics{figure_3}
OCR MEI C1 Q4
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 Q5
13 marks Moderate -0.8
  1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
  2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
  3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
  4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]
OCR MEI C1 Q6
13 marks Moderate -0.8
\includegraphics{figure_6} Fig. 11 shows a sketch of the curve with equation \(y = (x - 4)^2 - 3\).
  1. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point. [2]
  2. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary. [4]
  3. The curve is translated by \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\). Show that the equation of the translated curve may be written as \(y = x^2 - 12x + 33\). [2]
  4. Show that the line \(y = 8 - 2x\) meets the curve \(y = x^2 - 12x + 33\) at just one point, and find the coordinates of this point. [5]
OCR MEI C1 Q7
4 marks Easy -1.2
  1. Describe fully the transformation which maps the curve \(y = x^2\) onto the curve \(y = (x + 4)^2\). [2]
  2. Sketch the graph of \(y = x^2 - 4\). [2]