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 Q7
7
  1. Express \(( 2 + \sqrt { 3 } ) ^ { 2 }\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers to be determined.
  2. Given that \(x\) and \(y\) are integers, prove that \(\frac { 1 } { x - \sqrt { y } } + \frac { 1 } { x + \sqrt { y } }\) can be written in the form \(\frac { p } { q }\) where \(p\) and \(q\) are both integers.
OCR MEI C1 Q8
8 Find the equation of the line that passes through the point \(( 1,2 )\) and is perpendicular to the line \(3 x + 2 y = 5\).
OCR MEI C1 Q9
9
  1. Show that \(( x - 1 ) ( x - 2 ) ( x - 3 ) - \left( x ^ { 3 } - x ^ { 2 } + 11 x - 12 \right) = 6 - 5 x ^ { 2 }\).
  2. Solve the equation \(6 - 5 x ^ { 2 } = 0\).
OCR MEI C1 Q10
10
  1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
    Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
  2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
  3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
  4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.
OCR MEI C1 Q11
11
  1. Show algebraically that the equation \(x ^ { 2 } - 6 x + 10 = 0\) has no real roots.
  2. Solve algebraically the simultaneous equations \(y = x ^ { 2 } - 6 x + 10\) and \(y + 2 x = 7\).
  3. Plot the graph of the function \(y = x ^ { 2 } - 6 x + 10\) on graph paper, taking \(1 \mathrm {~cm} = 1\) unit on each axis, with the \(x\) axis from 0 to 6 and the \(y\) axis from - 2 to 10 .
    On the same axes plot the line with equation \(y + 2 x = 7\) showing clearly where the line cuts the quadratic curve.
  4. Explain why these \(x\) coordinates satisfy the equation \(x ^ { 2 } - 4 x + 3 = 0\). Plot a graph of the function \(y = x ^ { 2 } - 4 x + 3\) on the same axes to illustrate your answer.
OCR MEI C1 Q12
12 You are given that the equation of the circle shown in Fig. 12 is $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d20d10e0-6965-4f89-8855-8c6d32f5da90-4_742_971_422_481} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Show that the centre, Q , of the circle is \(( 2,3 )\) and find the radius.
  2. The circle crosses the \(x\)-axis at B and C . Show that the coordinates of C are \(( 6,0 )\) and find the coordinates of B .
  3. Find the gradient of the line QC and hence find the equation of the tangent to the circle at C.
  4. Given that M is the mid-point of BC , find the coordinates of the point where QM meets the tangent at C .
OCR MEI C1 Q1
1 Simplify \(( 3 x - 1 ) \left( 2 x ^ { 2 } - 5 x + 3 \right)\).
OCR MEI C1 Q2
2 Make \(l\) the subject of the formula \(T = 2 \pi \sqrt { \frac { l } { g } }\).
OCR MEI C1 Q3
3 Solve the inequality \(2 x ^ { 2 } - 7 x \geq 4\).
OCR MEI C1 Q4
4 Simplify the following.
  1. \(x ^ { \frac { 5 } { 2 } } \times \sqrt { x }\)
  2. \(12 x ^ { - 5 } \div 3 x ^ { - 2 }\)
OCR MEI C1 Q5
5 The vertices of a triangle have coordinates ( 1,5 ), ( \(- 3,7\) ) and ( \(- 2 , - 1\) ).
Show that the triangle is right-angled.
OCR MEI C1 Q6
6 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q7
7 Find the coordinates of the points where the line \(y = 3 x - 2\) cuts the curve \(y = x ^ { 2 } + 4 x - 8\).
OCR MEI C1 Q8
8 The lines \(y = 5 x - a\) and \(y = 2 x + 18\) meet at the point ( \(7 , b\) ).
Find the values of \(a\) and \(b\).
OCR MEI C1 Q9
9 The graph shows the function \(y = x ^ { 2 } + b x + c\) where \(b\) and \(c\) are constants.
The point \(\mathrm { M } ( - 3 , - 16 )\) on the graph is the minimum point of the graph.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-2_478_948_1871_588}
  1. Write down the function \(y = \mathrm { f } ( x )\) in completed square form.
  2. Hence find the coordinates of the points where the curve cuts the axes.
OCR MEI C1 Q11
11 In this question \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 4 x + k\).
  1. You are asked to find the values of \(k\) which satisfy the following conditions.
    (A) The graph of \(y = \mathrm { f } ( x )\) goes through the origin.
    (B) The graph of \(y = \mathrm { f } ( x )\) intersects with the \(y\) axis at ( \(0 , - 2\) ).
    (C) ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
    (D) The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 5 .
    (E) The graph of \(y = \mathrm { f } ( x )\) is as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584}
  2. Find the solution of the equation \(\mathrm { f } ( x ) = 0\) when \(k = 8\). Sketch a graph of \(y = \mathrm { f } ( x )\) in this case.
OCR MEI C1 Q12
12 ABCD is a parallelogram. The coordinates of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are (-2, 3), (2, 4), (8, -3) and ( \(4 , - 4\) ) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-4_592_725_387_492}
  1. Prove that AB and BD are perpendicular.
  2. Find the lengths of AB and BD and hence find the area of the parallelogram ABCD
  3. Find the equation of the line CD and show that it meets the \(y\)-axis at \(\mathrm { X } ( 0 , - 5 )\).
  4. Show that the lines BX and AD bisect each other.
  5. Explain why the area of the parallelogram ABCD is equal to the area of the triangle BXC.
    Find the length of BX and hence calculate exactly the perpendicular distance of C from BX .
OCR MEI C1 Q1
1
  1. Statement P is \(a + b = 4\).
    Statement Q is \(\quad a = 1\) and \(b = 3\).
    Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
  2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$
OCR MEI C1 Q2
2 Find the equation of the straight line which is parallel to the line \(y = 3 x + 5\) and which goes through the point \(( 2,12 )\).
OCR MEI C1 Q3
3 Find the term which has the highest coefficient in the expansion of \(( 1 + x ) ^ { 8 }\).
OCR MEI C1 Q4
4 The surface area of the surface of a cylinder is given by the formula $$A = 2 \pi r ( r + h )$$ Rearrange this formula so that \(h\) is the subject.
OCR MEI C1 Q5
5 Solve the following equations.
  1. \(\quad 2 ^ { x } = \frac { 1 } { 8 }\).
  2. \(\quad x ^ { - \frac { 1 } { 2 } } = \frac { 1 } { 4 }\)
OCR MEI C1 Q6
6 Find the positive integer values of \(x\) for which $$\frac { 1 } { 2 } ( 26 - 2 x ) \geq 2 ( 3 + x )$$
OCR MEI C1 Q7
7 The remainder when \(x ^ { 3 } - 2 x + 4\) is divided by ( \(x - 2\) ) is twice the remainder when \(x ^ { 2 } + x + k\) is divided by ( \(x + 1\) ).
Find the value of \(k\).
OCR MEI C1 Q8
8 Find the values of \(a\) and \(b\) for which \(\frac { 4 } { ( 2 \sqrt { 3 } - 1 ) } = a + b \sqrt { 3 }\).