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 2006 January Q10
10 marks Moderate -0.8
A circle has equation \(x^2 + y^2 = 45\).
  1. State the centre and radius of this circle. [2]
  2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]
OCR MEI C1 2006 January Q11
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 2006 January Q12
13 marks Standard +0.3
  1. Sketch the graph of \(y = x(x - 3)^2\). [3]
  2. Show that the equation \(x(x - 3)^2 = 2\) can be expressed as \(x^3 - 6x^2 + 9x - 2 = 0\). [2]
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i). [8]
OCR MEI C1 2006 June Q1
3 marks Easy -1.2
The volume of a cone is given by the formula \(V = \frac{1}{3}\pi r^2 h\). Make \(r\) the subject of this formula. [3]
OCR MEI C1 2006 June Q2
2 marks Moderate -0.8
One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]
OCR MEI C1 2006 June Q3
3 marks Easy -1.2
A line has equation \(3x + 2y = 6\). Find the equation of the line parallel to this which passes through the point \((2, 10)\). [3]
OCR MEI C1 2006 June Q4
2 marks Easy -1.2
In each of the following cases choose one of the statements $$\text{P} \Rightarrow \text{Q} \qquad \text{P} \Leftrightarrow \text{Q} \qquad \text{P} \Leftarrow \text{Q}$$ to describe the complete relationship between P and Q.
  1. P: \(x^2 + x - 2 = 0\) Q: \(x = 1\) [1]
  2. P: \(y^3 > 1\) Q: \(y > 1\) [1]
OCR MEI C1 2006 June Q5
3 marks Easy -1.2
Find the coordinates of the point of intersection of the lines \(y = 3x + 1\) and \(x + 3y = 6\). [3]
OCR MEI C1 2006 June Q6
4 marks Moderate -0.8
Solve the inequality \(x^2 + 2x < 3\). [4]
OCR MEI C1 2006 June Q7
5 marks Moderate -0.8
  1. Simplify \(6\sqrt{2} \times 5\sqrt{3} - \sqrt{24}\). [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 2006 June Q8
4 marks Easy -1.2
Calculate \(^6C_3\). Find the coefficient of \(x^3\) in the expansion of \((1 - 2x)^6\). [4]
OCR MEI C1 2006 June Q9
5 marks Easy -1.3
Simplify the following.
  1. \(\frac{16^{\frac{1}{4}}}{81^{\frac{1}{4}}}\) [2]
  2. \(\frac{12(a^3b^2c)^4}{4a^2c^6}\) [3]
OCR MEI C1 2006 June Q10
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]
OCR MEI C1 2006 June Q11
12 marks Moderate -0.8
A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.
  1. Show that AB and BC are perpendicular. [3]
  2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
  3. BD is a diameter of the circle. Find the coordinates of D. [3]
OCR MEI C1 2006 June Q12
12 marks Moderate -0.8
You are given that \(\text{f}(x) = x^3 + 9x^2 + 20x + 12\).
  1. Show that \(x = -2\) is a root of \(\text{f}(x) = 0\). [2]
  2. Divide \(\text{f}(x)\) by \(x + 6\). [2]
  3. Express \(\text{f}(x)\) in fully factorised form. [2]
  4. Sketch the graph of \(y = \text{f}(x)\). [3]
  5. Solve the equation \(\text{f}(x) = 12\). [3]
OCR MEI C1 2006 June Q13
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 2009 June Q1
4 marks Moderate -0.8
A line has gradient \(-4\) and passes through the point \((2, 6)\). Find the coordinates of its points of intersection with the axes. [4]
OCR MEI C1 2009 June Q2
3 marks Easy -1.8
Make \(a\) the subject of the formula \(s = ut + \frac{1}{2}at^2\). [3]
OCR MEI C1 2009 June Q3
3 marks Moderate -0.8
When \(x^3 - kx + 4\) is divided by \(x - 3\), the remainder is 1. Use the remainder theorem to find the value of \(k\). [3]
OCR MEI C1 2009 June Q4
2 marks Easy -1.2
Solve the inequality \(x(x - 6) > 0\). [2]
OCR MEI C1 2009 June Q5
4 marks Easy -1.3
  1. Calculate \(^5C_3\). [2]
  2. Find the coefficient of \(x^3\) in the expansion of \((1 + 2x)^5\). [2]
OCR MEI C1 2009 June Q6
3 marks Moderate -0.8
Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]
OCR MEI C1 2009 June Q7
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 2009 June Q8
5 marks Easy -1.3
  1. Simplify \(\frac{\sqrt{48}}{2\sqrt{27}}\). [2]
  2. Expand and simplify \((5 - 3\sqrt{2})^2\). [3]
OCR MEI C1 2009 June Q9
5 marks Easy -1.2
  1. Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\). [3]
  2. Write down the coordinates of the minimum point on the graph of \(y = x^2 + 6x + 5\). [2]