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 2007 June Q6
6 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 2007 June Q7
7 Solve the equation \(\frac { 4 x + 5 } { 2 x } = - 3\).
OCR MEI C1 2007 June Q8
8
  1. Simplify \(\sqrt { 98 } - \sqrt { 50 }\).
  2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
OCR MEI C1 2007 June Q9
9
  1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
  2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.
OCR MEI C1 2007 June Q10
10 The triangle shown in Fig. 10 has height \(( x + 1 ) \mathrm { cm }\) and base \(( 2 x - 3 ) \mathrm { cm }\). Its area is \(9 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d8caf0f-7594-42cb-bd40-e6c11e2b6832-3_444_1088_351_715} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Show that \(2 x ^ { 2 } - x - 21 = 0\).
  2. By factorising, solve the equation \(2 x ^ { 2 } - x - 21 = 0\). Hence find the height and base of the triangle. Section B (36 marks)
OCR MEI C1 2007 June Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d8caf0f-7594-42cb-bd40-e6c11e2b6832-3_442_1102_1384_717} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) as shown in Fig. 11.
  1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
  2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
  3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.
OCR MEI C1 2007 June Q12
12
  1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
  3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
  4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
OCR MEI C1 2007 June Q13
13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).
  1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
  3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).
OCR MEI C1 2008 June Q1
1 Solve the inequality \(3 x - 1 > 5 - x\).
OCR MEI C1 2008 June Q2
2
  1. Find the points of intersection of the line \(2 x + 3 y = 12\) with the axes.
  2. Find also the gradient of this line.
OCR MEI C1 2008 June Q3
3
  1. Solve the equation \(2 x ^ { 2 } + 3 x = 0\).
  2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 x - k = 0\) has no real roots.
OCR MEI C1 2008 June Q4
4 Given that \(n\) is a positive integer, write down whether the following statements are always true (T), always false (F) or could be either true or false (E).
  1. \(2 n + 1\) is an odd integer
  2. \(3 n + 1\) is an even integer
  3. \(n\) is odd \(\Rightarrow n ^ { 2 }\) is odd
  4. \(n ^ { 2 }\) is odd \(\Rightarrow n ^ { 3 }\) is even
OCR MEI C1 2008 June Q5
5 Make \(x\) the subject of the equation \(y = \frac { x + 3 } { x - 2 }\).
OCR MEI C1 2008 June Q6
6
  1. Find the value of \(\left( \frac { 1 } { 25 } \right) ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { \left( 2 x ^ { 2 } y ^ { 3 } z \right) ^ { 5 } } { 4 y ^ { 2 } z }\).
OCR MEI C1 2008 June Q7
7
  1. Express \(\frac { 1 } { 5 + \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
  2. Expand and simplify \(( 3 - 2 \sqrt { 7 } ) ^ { 2 }\).
OCR MEI C1 2008 June Q8
8 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 5 - 2 x ) ^ { 5 }\).
OCR MEI C1 2008 June Q9
9 Solve the equation \(y ^ { 2 } - 7 y + 12 = 0\).
Hence solve the equation \(x ^ { 4 } - 7 x ^ { 2 } + 12 = 0\). Section B (36 marks)
OCR MEI C1 2008 June Q10
10
  1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
  2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
  3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
  4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).
OCR MEI C1 2008 June Q11
11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).
  1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).
OCR MEI C1 2008 June Q12
12
  1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
  2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
  3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
  4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.
OCR MEI C1 2015 June Q1
1 Make \(r\) the subject of the formula \(A = \pi r ^ { 2 } ( x + y )\), where \(r > 0\).
OCR MEI C1 2015 June Q2
2 A line \(L\) is parallel to \(y = 4 x + 5\) and passes through the point \(( - 1,6 )\). Find the equation of the line \(L\) in the form \(y = a x + b\). Find also the coordinates of its intersections with the axes.
OCR MEI C1 2015 June Q3
3 Evaluate the following.
  1. \(200 ^ { \circ }\)
  2. \(\left( \frac { 25 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\)
OCR MEI C1 2015 June Q4
4 Solve the inequality \(\frac { 4 x - 5 } { 7 } > 2 x + 1\).
OCR MEI C1 2015 June Q5
5 Find the coordinates of the point of intersection of the lines \(y = 5 x - 2\) and \(x + 3 y = 8\).