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OCR MEI C1 2010 June Q2
5 marks Easy -1.8
  1. Simplify \((5a^2b)^3 \times 2b^4\). [2]
  2. Evaluate \(\left(\frac{1}{16}\right)^{-1}\). [1]
  3. Evaluate \((16)^{\frac{1}{2}}\). [2]
OCR MEI C1 2010 June Q3
3 marks Easy -1.2
Make \(y\) the subject of the formula \(a = \frac{\sqrt{y} - 5}{c}\). [3]
OCR MEI C1 2010 June Q4
5 marks Easy -1.2
Solve the following inequalities.
  1. \(2(1 - x) > 6x + 5\) [3]
  2. \((2x - 1)(x + 4) < 0\) [2]
OCR MEI C1 2010 June Q5
5 marks Easy -1.3
  1. Express \(\sqrt{48} + \sqrt{27}\) in the form \(a\sqrt{3}\). [2]
  2. Simplify \(\frac{5\sqrt{7}}{3 - \sqrt{2}}\). Give your answer in the form \(\frac{b + c\sqrt{7}}{d}\). [3]
OCR MEI C1 2010 June Q6
5 marks Moderate -0.3
You are given that • the coefficient of \(x^3\) in the expansion of \((5 + 2x^2)(x^3 + kx + m)\) is 29, • when \(x^3 + kx + m\) is divided by \((x - 3)\), the remainder is 59. Find the values of \(k\) and \(m\). [5]
OCR MEI C1 2010 June Q7
4 marks Easy -1.2
Expand \((1 + \frac{1}{2}x)^4\), simplifying the coefficients. [4]
OCR MEI C1 2010 June Q8
4 marks Moderate -0.8
Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]
OCR MEI C1 2010 June Q9
2 marks Easy -1.8
Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x^2 = 25$$ [2]
OCR MEI C1 2010 June Q10
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 2010 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).
  1. Find the equation of the line through A and B. [3]
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac{32}{3}\) square units. [2]
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle. [4]
OCR MEI C1 2010 June Q12
12 marks Moderate -0.3
You are given that \(f(x) = x^3 + 6x^2 - x - 30\).
  1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]
OCR MEI C1 2011 June Q1
3 marks Easy -1.8
Solve the inequality \(6(x + 3) > 2x + 5\). [3]
OCR MEI C1 2011 June Q2
2 marks Easy -1.3
A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]
OCR MEI C1 2011 June Q3
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 2011 June Q4
4 marks Easy -1.3
The point P \((5, 4)\) is on the curve \(y = f(x)\). State the coordinates of the image of P when the graph of \(y = f(x)\) is transformed to the graph of
  1. \(y = f(x - 5)\), [2]
  2. \(y = f(x) + 7\). [2]
OCR MEI C1 2011 June Q5
4 marks Moderate -0.8
Find the coefficient of \(x^4\) in the binomial expansion of \((5 + 2x)^6\). [4]
OCR MEI C1 2011 June Q6
3 marks Easy -1.2
Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]
OCR MEI C1 2011 June Q7
3 marks Easy -1.2
Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]
OCR MEI C1 2011 June Q8
4 marks Moderate -0.8
Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]
OCR MEI C1 2011 June Q9
5 marks Moderate -0.3
A line \(L\) is parallel to the line \(x + 2y = 6\) and passes through the point \((10, 1)\). Find the area of the region bounded by the line \(L\) and the axes. [5]
OCR MEI C1 2011 June Q10
3 marks Moderate -0.8
Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]
OCR MEI C1 2011 June Q11
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
  2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
  3. For the curve \(y = 4x^2 + 24x + 31\),
    1. write down the equation of the line of symmetry, [1]
    2. write down the minimum \(y\)-value on the curve. [1]
OCR MEI C1 2011 June Q12
12 marks Moderate -0.8
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{4}{x^2}\).
  1. On the copy of Fig. 12, draw accurately the line \(y = 2x + 5\) and hence find graphically the three roots of the equation \(\frac{4}{x^2} = 2x + 5\). [3]
  2. Show that the equation you have solved in part (i) may be written as \(2x^3 + 5x^2 - 4 = 0\). Verify that \(x = -2\) is a root of this equation and hence find, in exact form, the other two roots. [6]
  3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x^3 + 2x^2 - 4 = 0\). [3]
OCR MEI C1 2011 June Q13
13 marks Moderate -0.3
\includegraphics{figure_13} Fig. 13 shows the circle with equation \((x - 4)^2 + (y - 2)^2 = 16\).
  1. Write down the radius of the circle and the coordinates of its centre. [2]
  2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form. [4]
  3. Show that the point A \((4 + 2\sqrt{2}, 2 + 2\sqrt{2})\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it. Find the equations of both these tangents. [7]
OCR MEI C1 2012 June Q1
3 marks Easy -1.2
Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [3]