Questions — OCR MEI C1 (472 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI C1 2010 January Q7
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
8 Find the binomial expansion of \(\left( x + \frac { 5 } { x } \right) ^ { 3 }\), simplifying the terms.
OCR MEI C1 2010 January Q9
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
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
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
5 marks
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
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
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 Simplify \(\frac { \left( 3 x y ^ { 4 } \right) ^ { 3 } } { 6 x ^ { 5 } y ^ { 2 } }\).
OCR MEI C1 2011 January Q4
4 Solve the inequality \(5 - 2 x < 0\).
OCR MEI C1 2011 January Q5
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
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
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.
OCR MEI C1 2011 January Q8
8 Find the coordinates of the point of intersection of the lines \(x + 2 y = 5\) and \(y = 5 x - 1\).
OCR MEI C1 2011 January Q9
9 Fig. 9 shows a trapezium ABCD , with the lengths in centimetres of three of its sides. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{754d34e4-2f47-48b7-9fbb-6caa7ac21eb7-3_464_878_347_632} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} This trapezium has area \(140 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(x ^ { 2 } + 2 x - 35 = 0\).
  2. Hence find the length of side AB of the trapezium.
OCR MEI C1 2011 January Q10
10 Select the best statement from $$\begin{aligned} & \mathbf { P } \Rightarrow \mathbf { Q }
& \mathbf { P } \Leftarrow \mathbf { Q }
& \mathbf { P } \Leftrightarrow \mathbf { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.
  1. P : WXYZ is a quadrilateral with 4 equal sides
    \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
  2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
  3. P: \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer
OCR MEI C1 2011 January Q11
11 The points \(A ( - 1,6 ) , B ( 1,0 )\) and \(C ( 13,4 )\) are joined by straight lines.
  1. Prove that the lines AB and BC are perpendicular.
  2. Find the area of triangle ABC .
  3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
  4. Find the coordinates of the point on this circle that is furthest from B .
OCR MEI C1 2011 January Q12
12
  1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 2011 January Q13
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{754d34e4-2f47-48b7-9fbb-6caa7ac21eb7-4_686_878_936_632} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).
  1. Find the exact coordinates of the points of intersection of this curve with the axes.
  2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
  3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).
OCR MEI C1 2012 January Q1
1 Find the equation of the line which is perpendicular to the line \(y = 5 x + 2\) and which passes through the point \(( 1,6 )\). Give your answer in the form \(y = a x + b\).
OCR MEI C1 2012 January Q2
2
  1. Evaluate \(9 ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { \left( 4 x ^ { 4 } \right) ^ { 3 } y ^ { 2 } } { 2 x ^ { 2 } y ^ { 5 } }\).
OCR MEI C1 2012 January Q3
3 Expand and simplify \(( n + 2 ) ^ { 3 } - n ^ { 3 }\).
OCR MEI C1 2012 January Q4
4
  1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
  2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).
OCR MEI C1 2012 January Q5
5 Solve the following inequality. $$\frac { 2 x + 1 } { 5 } < \frac { 3 x + 4 } { 6 }$$
OCR MEI C1 2012 January Q6
6 Rearrange the following equation to make \(h\) the subject. $$4 h + 5 = 9 a - h a ^ { 2 }$$