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OCR C1 2016 June Q2
4 marks Easy -1.2
2 Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
OCR C1 2016 June Q3
5 marks Moderate -0.5
3 Solve the simultaneous equations $$x ^ { 2 } + y ^ { 2 } = 34 , \quad 3 x - y + 4 = 0$$
OCR C1 2016 June Q4
5 marks Standard +0.3
4 Solve the equation \(2 y ^ { \frac { 1 } { 2 } } - 7 y ^ { \frac { 1 } { 4 } } + 3 = 0\).
OCR C1 2016 June Q5
5 marks Easy -1.3
5 Express the following in the form \(2 ^ { p }\).
  1. \(\left( 2 ^ { 5 } \div 2 ^ { 7 } \right) ^ { 3 }\)
  2. \(5 \times 4 ^ { \frac { 2 } { 3 } } + 3 \times 16 ^ { \frac { 1 } { 3 } }\)
OCR C1 2016 June Q6
6 marks Moderate -0.8
6
  1. Express \(4 + 12 x - 2 x ^ { 2 }\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the coordinates of the maximum point of the curve \(y = 4 + 12 x - 2 x ^ { 2 }\).
OCR C1 2016 June Q7
7 marks Moderate -0.3
7
  1. Sketch the curve \(y = x ^ { 2 } ( 3 - x )\) stating the coordinates of points of intersection with the axes.
  2. The curve \(y = x ^ { 2 } ( 3 - x )\) is translated by 2 units in the positive direction parallel to the \(x\)-axis. State the equation of the curve after it has been translated.
  3. Describe fully a transformation that transforms the curve \(y = x ^ { 2 } ( 3 - x )\) to \(y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )\).
OCR C1 2016 June Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 2 }\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates 5 and \(5 + h\) respectively, where \(h > 0\).
  1. Show that the gradient of the line \(A B\) is \(20 + 2 h\).
  2. Explain how the answer to part (i) relates to the gradient of the curve at \(A\).
  3. The normal to the curve at \(A\) meets the \(y\)-axis at the point \(C\). Find the \(y\)-coordinate of \(C\).
OCR C1 2016 June Q9
7 marks Standard +0.3
9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.
OCR C1 2016 June Q10
14 marks Moderate -0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).
  1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
  2. Find the equation of the tangent to the circle at \(A\).
  3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
  4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).
OCR C1 2016 June Q11
8 marks Standard +0.3
11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .
OCR MEI C1 2007 January Q1
3 marks Easy -1.8
1 Find, in the form \(y = a x + b\), the equation of the line through \(( 3,10 )\) which is parallel to \(y = 2 x + 7\).
OCR MEI C1 2007 January Q2
3 marks Easy -1.8
2 Sketch the graph of \(y = 9 - x ^ { 2 }\).
OCR MEI C1 2007 January Q3
3 marks Easy -1.8
3 Make \(a\) the subject of the equation $$2 a + 5 c = a f + 7 c$$
OCR MEI C1 2007 January Q4
3 marks Moderate -0.8
4 When \(x ^ { 3 } + k x + 5\) is divided by \(x - 2\), the remainder is 3 . Use the remainder theorem to find the value of \(k\).
OCR MEI C1 2007 January Q5
3 marks Easy -1.2
5 Calculate the coefficient of \(x ^ { 4 }\) in the expansion of \(( x + 5 ) ^ { 6 }\).
OCR MEI C1 2007 January Q6
4 marks Easy -1.5
6 Find the value of each of the following, giving each answer as an integer or fraction as appropriate.
  1. \(25 ^ { \frac { 3 } { 2 } }\)
  2. \(\left( \frac { 7 } { 3 } \right) ^ { - 2 }\)
OCR MEI C1 2007 January Q7
4 marks Standard +0.8
7 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 - \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
OCR MEI C1 2007 January Q8
4 marks Moderate -0.5
8 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 2 = 0\) has no real roots.
OCR MEI C1 2007 January Q9
5 marks Easy -1.3
9
  1. Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
  2. Factorise \(x ^ { 2 } - 4\) and \(x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.
OCR MEI C1 2007 January Q10
4 marks Moderate -0.8
10 Simplify \(\left( m ^ { 2 } + 1 \right) ^ { 2 } - \left( m ^ { 2 } - 1 \right) ^ { 2 }\), showing your method.
Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-3_414_593_452_735} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
OCR MEI C1 2007 January Q12
12 marks Moderate -0.3
12 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks. \(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.
  1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
  2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_451_483_776_790} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Not to scale
OCR MEI C1 2007 January Q13
12 marks Moderate -0.8
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_456_387_1539_833} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).
  1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
  2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
  3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).
OCR MEI C1 2010 January Q1
3 marks Easy -1.8
1 Rearrange the formula \(c = \sqrt { \frac { a + b } { 2 } }\) to make \(a\) the subject.
OCR MEI C1 2010 January Q2
3 marks Easy -1.8
2 Solve the inequality \(\frac { 5 x - 3 } { 2 } < x + 5\).
OCR MEI C1 2010 January Q3
4 marks Easy -1.2
3
  1. Find the coordinates of the point where the line \(5 x + 2 y = 20\) intersects the \(x\)-axis.
  2. Find the coordinates of the point of intersection of the lines \(5 x + 2 y = 20\) and \(y = 5 - x\).