Questions — OCR MEI (4301 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 Q2
2 Find the range of values of \(x\) for which \(x ^ { 2 } - 5 x + 6 \leq 0\).
OCR MEI C1 Q3
3 Write \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 2 }\) in the form \(a + b \sqrt { 6 }\) where \(a\) and \(b\) are integers to be determined.
OCR MEI C1 Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{3b927f8b-ddf8-481d-a1ce-3b90bb1435f0-2_437_807_953_579} The graph shows a function \(y = \mathrm { f } ( x )\).
On separate graphs, sketch the graphs of the following functions:
  1. \(\quad y = \mathrm { f } ( x ) + 1\),
  2. \(y = \mathrm { f } ( x + 1 )\).
OCR MEI C1 Q5
5 Make \(u\) the subject of the formula $$\frac { 1 } { v } - \frac { 1 } { u } = \frac { 1 } { f }$$
OCR MEI C1 Q6
6 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 2 x - 8 = 0\).
Find the centre and radius of the circle.
OCR MEI C1 Q7
7 Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 4 x + 4\).
Hence solve the equation \(x ^ { 3 } - x ^ { 2 } - 4 x + 4 = 0\).
OCR MEI C1 Q8
8 Find the points where the line \(y = 2 x - 3\) cuts the curve \(y = x ^ { 2 } - 4 x + 5\).
OCR MEI C1 Q9
9
  1. Simplify \(\frac { 2 ^ { 6 } } { 8 ^ { 2 \frac { 1 } { 2 } } \times 2 ^ { - \frac { 1 } { 2 } } }\)
  2. Solve the equation \(x ^ { - \frac { 1 } { 3 } } = 8\).
OCR MEI C1 Q10
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b927f8b-ddf8-481d-a1ce-3b90bb1435f0-3_437_572_1058_538} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} In Fig.10, A has coordinates \(( 1,1 )\) and C has coordinates \(( 3,5 )\). M is the mid-point of AC . The line \(l\) is perpendicular to AC.
  1. Find the coordinates of M . Hence find the equation of \(l\).
  2. The point B has coordinates \(( - 2,5 )\). Show that B lies on the line \(l\).
    Find the coordinates of the point D such that ABCD is a rhombus.
  3. Find the lengths MC and MB . Hence calculate the area of the rhombus ABCD .
OCR MEI C1 Q11
11
  1. Multiply out \(( x - p ) ( x - q )\).
  2. You are given that \(p = 2 + \sqrt { 3 }\) and \(q = 2 - \sqrt { 3 }\) are the roots of a quadratic equation. Find \(p + q\) and \(p q\) and hence find the quadratic equation with roots \(x = p\) and \(x = q\).
  3. Solve the quadratic equation \(x ^ { 2 } + 5 x - 7 = 0\) giving the roots exactly.
  4. Show that \(x = 1\) is the only root of the equation \(x ^ { 3 } + 2 x - 3 = 0\).
  5. A quadratic equation \(x ^ { 2 } + r x + s = 0\), where \(r\) and \(s\) are integers, has two roots. One root is \(x = 3 + \sqrt { 5 }\). Without finding \(r\) or \(s\), write down the other root.
OCR MEI C1 Q12
12
  1. Expand \(( 1 + 2 x ) ^ { 6 }\), simplifying all the terms.
  2. Hence find an expression for \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { 6 } + ( 1 - 2 x ) ^ { 6 }\) in its simplest form.
  3. Substituting \(x = 0.01\) into the first two terms of \(\mathrm { f } ( x )\) gives an approximate value, z for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\). Find \(z\). By considering the value of the third term, comment on the accuracy of \(z\) as an approximation for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\).
OCR MEI C1 Q1
1 Make \(a\) the subject of the equation \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 Q2
2
  1. Find the constants \(a\) and \(b\) such that, for all values of \(x\), $$x ^ { 2 } + 4 x + 14 = ( x + a ) ^ { 2 } + b$$
  2. Write down the greatest value of \(\frac { 1 } { x ^ { 2 } + 4 x + 14 }\).
OCR MEI C1 Q3
3 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 4 }\).
OCR MEI C1 Q4
4 The coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C are ( \(- 2,2\) ), ( 1,3 ) and ( \(3 , - 3\) ) respectively.
  1. Find the gradients of the lines AB and BC .
  2. Show that the triangle ABC is a right-angled triangle.
  3. Find the area of the triangle ABC .
OCR MEI C1 Q5
5 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } - 7 x + 6\).
  1. Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI C1 Q6
6 List the integers which satisfy both of the following inequalities: $$2 x - 9 < 0 , \quad 8 - x \leq 6$$
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 \(T = 2 \pi \sqrt { \frac { l } { g } }\).