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OCR MEI C1 2012 June Q2
3 marks Easy -1.8
Make \(b\) the subject of the following formula. $$a = \frac{3}{5}b^2c$$ [3]
OCR MEI C1 2012 June Q3
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 2012 June Q4
3 marks Easy -1.2
Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]
OCR MEI C1 2012 June Q5
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 2012 June Q6
5 marks Moderate -0.8
  1. Evaluate \(^5C_3\). [1]
  2. Find the coefficient of \(x^3\) in the expansion of \((3 - 2x)^5\). [4]
OCR MEI C1 2012 June Q7
4 marks Moderate -0.5
Find the set of values of \(k\) for which the graph of \(y = x^2 + 2kx + 5\) does not intersect the \(x\)-axis. [4]
OCR MEI C1 2012 June Q8
5 marks Standard +0.3
The function \(f(x) = x^4 + bx + c\) is such that \(f(2) = 0\). Also, when \(f(x)\) is divided by \(x + 3\), the remainder is \(85\). Find the values of \(b\) and \(c\). [5]
OCR MEI C1 2012 June Q9
4 marks Moderate -0.8
Simplify \((n + 3)^2 - n^2\). Hence explain why, when \(n\) is an integer, \((n + 3)^2 - n^2\) is never an even number. Given also that \((n + 3)^2 - n^2\) is divisible by \(9\), what can you say about \(n\)? [4]
OCR MEI C1 2012 June Q10
11 marks Moderate -0.3
\includegraphics{figure_10} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).
  1. Show that \(AB = BC\). [3]
  2. Show that the diagonals AC and BD are perpendicular. [3]
  3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]
OCR MEI C1 2012 June Q11
12 marks Moderate -0.3
A cubic curve has equation \(y = f(x)\). The curve crosses the \(x\)-axis where \(x = -\frac{1}{2}\), \(-2\) and \(5\).
  1. Write down three linear factors of \(f(x)\). Hence find the equation of the curve in the form \(y = 2x^3 + ax^2 + bx + c\). [4]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\). State the coordinates of the point where the translated curve intersects the \(y\)-axis. [1]
  4. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give the curve \(y = g(x)\). Find an expression in factorised form for \(g(x)\) and state the coordinates of the point where the curve \(y = g(x)\) intersects the \(y\)-axis. [4]
OCR MEI C1 2012 June Q12
13 marks Moderate -0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{-1}{x - 3}\).
  1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{-1}{x - 3}\) and \(y = x^2 - 4x + 1\). [5]
  2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
  3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]
OCR MEI C1 2013 June Q1
3 marks Easy -1.2
Find the equation of the line which is perpendicular to the line \(y = 2x - 5\) and which passes through the point \((4, 1)\). Give your answer in the form \(y = ax + b\). [3]
OCR MEI C1 2013 June Q2
4 marks Easy -1.2
Find the coordinates of the point of intersection of the lines \(y = 3x - 2\) and \(x + 3y = 1\). [4]
OCR MEI C1 2013 June Q3
5 marks Easy -1.8
  1. Evaluate \((0.2)^{-2}\). [2]
  2. Simplify \((16a^{12})^{\frac{1}{4}}\). [3]
OCR MEI C1 2013 June Q4
3 marks Easy -1.2
Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac{1}{3}\pi r^2(a + b)$$ [3]
OCR MEI C1 2013 June Q5
3 marks Moderate -0.8
You are given that \(\text{f}(x) = x^5 + kx - 20\). When \(\text{f}(x)\) is divided by \((x - 2)\), the remainder is 18. Find the value of \(k\). [3]
OCR MEI C1 2013 June Q6
4 marks Moderate -0.8
Find the coefficient of \(x^3\) in the binomial expansion of \((2 - 4x)^5\). [4]
OCR MEI C1 2013 June Q7
5 marks Moderate -0.8
  1. Express \(125\sqrt{5}\) in the form \(5^t\). [2]
  2. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]
OCR MEI C1 2013 June Q8
5 marks Moderate -0.8
Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]
OCR MEI C1 2013 June Q9
4 marks Moderate -0.8
\(n - 1\), \(n\) and \(n + 1\) are any three consecutive integers.
  1. Show that the sum of these integers is always divisible by 3. [1]
  2. Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3. [3]
OCR MEI C1 2013 June Q10
12 marks Moderate -0.8
The circle \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C. [2]
  2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
  3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]
OCR MEI C1 2013 June Q11
12 marks Moderate -0.8
You are given that \(\text{f}(x) = (2x - 3)(x + 2)(x + 4)\).
  1. Sketch the graph of \(y = \text{f}(x)\). [3]
  2. State the roots of \(\text{f}(x - 2) = 0\). [2]
  3. You are also given that \(\text{g}(x) = \text{f}(x) + 15\).
    1. Show that \(\text{g}(x) = 2x^3 + 9x^2 - 2x - 9\). [2]
    2. Show that \(\text{g}(1) = 0\) and hence factorise \(\text{g}(x)\) completely. [5]
OCR MEI C1 2013 June Q12
12 marks Standard +0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).
  1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]
Edexcel C1 Q1
4 marks Easy -1.8
  1. Express \(\frac{21}{\sqrt{7}}\) in the form \(k\sqrt{7}\). [2]
  2. Express \(8^{-1}\) as an exact fraction in its simplest form. [2]
Edexcel C1 Q2
4 marks Moderate -0.5
Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]