Questions — OCR MEI (4333 questions)

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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.
OCR MEI C1 2010 January Q12
11 marks Moderate -0.3
12 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-4_524_885_406_628} \captionsetup{labelformat=empty} \caption{Fig. 12.1}
\end{figure}
  1. State the value of \(x\) at A , where the arch meets the road.
  2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
  3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-4_529_871_1489_678} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
    \end{figure}
  4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.
    [0pt] [5]
OCR MEI C1 2011 January Q1
3 marks Easy -1.2
1 Find the equation of the line which is parallel to \(y = 5 x - 4\) and which passes through the point (2, 13). Give your answer in the form \(y = a x + b\).
OCR MEI C1 2011 January Q2
4 marks Easy -1.8
2
  1. Write down the value of each of the following.
    (A) \(4 ^ { - 2 }\) (B) \(9 ^ { 0 }\)
  2. Find the value of \(\left( \frac { 64 } { 125 } \right) ^ { \frac { 4 } { 3 } }\).
OCR MEI C1 2011 January Q3
3 marks Easy -1.8
3 Simplify \(\frac { \left( 3 x y ^ { 4 } \right) ^ { 3 } } { 6 x ^ { 5 } y ^ { 2 } }\).
OCR MEI C1 2011 January Q4
2 marks Easy -1.8
4 Solve the inequality \(5 - 2 x < 0\).
OCR MEI C1 2011 January Q5
4 marks Moderate -0.5
5 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.
OCR MEI C1 2011 January Q6
4 marks Easy -1.2
6 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.
OCR MEI C1 2011 January Q7
5 marks Easy -1.2
7
  1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
  2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.