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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]
OCR MEI C1 2009 June Q10
4 marks Moderate -0.3
Find the real roots of the equation \(x^4 - 5x^2 - 36 = 0\) by considering it as a quadratic equation in \(x^2\). [4]
OCR MEI C1 2009 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows the line joining the points A \((0, 3)\) and B \((6, 1)\).
  1. Find the equation of the line perpendicular to AB that passes through the origin, O. [2]
  2. Find the coordinates of the point where this perpendicular meets AB. [4]
  3. Show that the perpendicular distance of AB from the origin is \(\frac{9\sqrt{10}}{10}\). [2]
  4. Find the length of AB, expressing your answer in the form \(a\sqrt{10}\). [2]
  5. Find the area of triangle OAB. [2]
OCR MEI C1 2009 June Q12
13 marks Moderate -0.8
  1. You are given that \(\text{f}(x) = (x + 1)(x - 2)(x - 4)\).
    1. Show that \(\text{f}(x) = x^3 - 5x^2 + 2x + 8\). [2]
    2. Sketch the graph of \(y = \text{f}(x)\). [3]
    3. The graph of \(y = \text{f}(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). State an equation for the resulting graph. You need not simplify your answer. Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis. [3]
  2. Show that 3 is a root of \(x^3 - 5x^2 + 2x + 8 = -4\). Hence solve this equation completely, giving the other roots in surd form. [5]
OCR MEI C1 2009 June Q13
11 marks Moderate -0.8
A circle has equation \((x - 5)^2 + (y - 2)^2 = 20\).
  1. State the coordinates of the centre and the radius of this circle. [2]
  2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
  3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
  4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]
OCR MEI C1 2010 June Q1
3 marks Easy -1.2
Find the equation of the line which is parallel to \(y = 3x + 1\) and which passes through the point with coordinates \((4, 5)\). [3]