Questions — OCR MEI (4456 questions)

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OCR MEI FP2 2011 June Q4
18 marks Challenging +1.2
4
  1. Given that \(\cosh y = x\), show that \(y = \pm \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\) and that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { \frac { 4 } { 5 } } ^ { 1 } \frac { 1 } { \sqrt { 25 x ^ { 2 } - 16 } } \mathrm {~d} x\), expressing your answer in an exact logarithmic form.
  3. Solve the equation $$5 \cosh x - \cosh 2 x = 3$$ giving your answers in an exact logarithmic form.
OCR MEI FP2 2011 June Q5
18 marks Standard +0.8
5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).
  1. On separate diagrams, sketch the curve in each of the following cases.
    (A) \(m = 1 , n = 1\),
    (B) \(m = 2 , n = 2\),
    (C) \(m = 2 , n = 4\),
    (D) \(m = 4 , n = 2\).
  2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
  3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
  4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
  5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
  6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. }{www.ocr.org.uk}) after the live examination series.
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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 \(y = 5 - x\).
OCR MEI C1 2010 January Q4
4 marks Easy -1.8
4
  1. Describe fully the transformation which maps the curve \(y = x ^ { 2 }\) onto the curve \(y = ( x + 4 ) ^ { 2 }\).
  2. Sketch the graph of \(y = x ^ { 2 } - 4\).
OCR MEI C1 2010 January Q5
5 marks Easy -1.2
5
  1. Find the value of \(144 ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { 1 } { 5 + \sqrt { 7 } } + \frac { 4 } { 5 - \sqrt { 7 } }\). Give your answer in the form \(\frac { a + b \sqrt { 7 } } { c }\).
OCR MEI C1 2010 January Q6
5 marks Moderate -0.8
6 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).
OCR MEI C1 2010 January Q7
3 marks Easy -1.2
7 When \(x ^ { 3 } + 2 x ^ { 2 } + 5 x + k\) is divided by ( \(x + 3\) ), the remainder is 6 . Find the value of \(k\).
OCR MEI C1 2010 January Q8
4 marks Easy -1.2
8 Find the binomial expansion of \(\left( x + \frac { 5 } { x } \right) ^ { 3 }\), simplifying the terms.
OCR MEI C1 2010 January Q9
5 marks Moderate -0.5
9 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).
OCR MEI C1 2010 January Q10
13 marks Moderate -0.3
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-3_590_780_347_680} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).
  1. Verify that the lines AB and DC are parallel.
  2. Prove that the trapezium is not isosceles.
  3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
  4. Show that neither diagonal of the trapezium bisects the other.
OCR MEI C1 2010 January Q11
12 marks Moderate -0.3
11 A circle has equation \(( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  1. State the coordinates of the centre of this circle and its radius.
  2. Verify that the point A with coordinates \(( 6 , - 6 )\) lies on this circle. Show also that the point B on the circle for which AB is a diameter has coordinates \(( 0,2 )\).
  3. Find the equation of the tangent to the circle at A .
  4. A second circle touches the original circle at A . Its radius is 10 and its centre is at C , where BAC is a straight line. Find the coordinates of C and hence write down the equation of this second circle.