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 2009 June Q8
8
  1. Simplify \(\frac { \sqrt { 48 } } { 2 \sqrt { 27 } }\).
  2. Expand and simplify \(( 5 - 3 \sqrt { 2 } ) ^ { 2 }\).
OCR MEI C1 2009 June Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1209192c-655e-439d-be50-8747dbbb7672-3_444_846_351_648} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).
  1. Find the equation of the line perpendicular to AB that passes through the origin, O .
  2. Find the coordinates of the point where this perpendicular meets AB .
  3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
  4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
  5. Find the area of triangle OAB .
OCR MEI C1 2009 June Q12
12
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
    Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 2009 June Q13
13 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).
  1. State the coordinates of the centre and the radius of this circle.
  2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
  3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
  4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.
OCR MEI C1 2010 June Q1
1 Find the equation of the line which is parallel to \(y = 3 x + 1\) and which passes through the point with coordinates \(( 4,5 )\).
OCR MEI C1 2010 June Q2
2
  1. Simplify \(\left( 5 a ^ { 2 } b \right) ^ { 3 } \times 2 b ^ { 4 }\).
  2. Evaluate \(\left( \frac { 1 } { 16 } \right) ^ { - 1 }\).
  3. Evaluate \(( 16 ) ^ { \frac { 3 } { 2 } }\).
OCR MEI C1 2010 June Q4
4 Solve the following inequalities.
  1. \(2 ( 1 - x ) > 6 x + 5\)
  2. \(( 2 x - 1 ) ( x + 4 ) < 0\)
OCR MEI C1 2010 June Q5
5
  1. Express \(\sqrt { 48 } + \sqrt { 27 }\) in the form \(a \sqrt { 3 }\).
  2. Simplify \(\frac { 5 \sqrt { 2 } } { 3 - \sqrt { 2 } }\). Give your answer in the form \(\frac { b + c \sqrt { 2 } } { d }\).
OCR MEI C1 2010 June Q6
6 You are given that
  • the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
  • when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .
Find the values of \(k\) and \(m\).
OCR MEI C1 2010 June Q7
7 Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 4 }\), simplifying the coefficients.
OCR MEI C1 2010 June Q8
8 Express \(5 x ^ { 2 } + 20 x + 6\) in the form \(a ( x + b ) ^ { 2 } + c\).
OCR MEI C1 2010 June Q9
9 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$
OCR MEI C1 2010 June Q10
10
  1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
  2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
  3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).
OCR MEI C1 2010 June Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7942af14-fb10-42ba-b77a-b50ce65a7bcc-3_527_1125_794_513} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).
  1. Find the equation of the line through A and B .
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.
OCR MEI C1 2010 June Q12
12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).
  1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$
OCR MEI C1 2011 June Q1
1 Solve the inequality \(6 ( x + 3 ) > 2 x + 5\).
OCR MEI C1 2011 June Q3
3
  1. Evaluate \(\left( \frac { 9 } { 16 } \right) ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { \left( 2 a c ^ { 2 } \right) ^ { 3 } \times 9 a ^ { 2 } c } { 36 a ^ { 4 } c ^ { 12 } }\).
OCR MEI C1 2011 June Q4
4 The point \(\mathrm { P } ( 5,4 )\) is on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P when the graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of
  1. \(y = \mathrm { f } ( x - 5 )\),
  2. \(y = \mathrm { f } ( x ) + 7\).
OCR MEI C1 2011 June Q5
5 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 6 }\).
OCR MEI C1 2011 June Q6
6 Expand \(( 2 x + 5 ) ( x - 1 ) ( x + 3 )\), simplifying your answer.
OCR MEI C1 2011 June Q7
7 Find the discriminant of \(3 x ^ { 2 } + 5 x + 2\). Hence state the number of distinct real roots of the equation \(3 x ^ { 2 } + 5 x + 2 = 0\).
OCR MEI C1 2011 June Q8
8 Make \(x\) the subject of the formula \(y = \frac { 1 - 2 x } { x + 3 }\).
OCR MEI C1 2011 June Q9
9 A line \(L\) is parallel to the line \(x + 2 y = 6\) and passes through the point \(( 10,1 )\). Find the area of the region bounded by the line \(L\) and the axes.
OCR MEI C1 2011 June Q10
10 Factorise \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\). Hence prove that, when \(n\) is a positive integer, \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\) is always divisible by 6 .
OCR MEI C1 2011 June Q11
11
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
  2. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
  3. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
    (A) write down the equation of the line of symmetry,
    (B) write down the minimum \(y\)-value on the curve.