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 2015 June Q6
6
  1. Expand and simplify \(( 3 + 4 \sqrt { 5 } ) ( 3 - 2 \sqrt { 5 } )\).
  2. Express \(\sqrt { 72 } + \frac { 32 } { \sqrt { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 2015 June Q7
7 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR MEI C1 2015 June Q8
8 Fig. 8 shows a right-angled triangle with base \(2 x + 1\), height \(h\) and hypotenuse \(3 x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55e1f96-670a-4bc3-9e77-92d28769b7f5-2_317_593_1653_543} \captionsetup{labelformat=empty} \caption{Not to scale}
\end{figure} Fig. 8
  1. Show that \(h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1\).
  2. Given that \(h = \sqrt { 7 }\), find the value of \(x\), giving your answer in surd form.
OCR MEI C1 2015 June Q9
9 Explain why each of the following statements is false. State in each case which of the symbols ⇒, ⟸ or ⇔ would make the statement true.
  1. ABCD is a square \(\Leftrightarrow\) the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
  2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer
OCR MEI C1 2015 June Q10
10 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
  3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
  4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 2015 June Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55e1f96-670a-4bc3-9e77-92d28769b7f5-3_700_751_906_641} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows a sketch of the circle with equation \(( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 125\) and centre C . The points \(\mathrm { A } , \mathrm { B }\), D and E are the intersections of the circle with the axes.
  1. Write down the radius of the circle and the coordinates of C .
  2. Verify that B is the point \(( 21,0 )\) and find the coordinates of \(\mathrm { A } , \mathrm { D }\) and E .
  3. Find the equation of the perpendicular bisector of BE and verify that this line passes through C .
OCR MEI C1 2015 June Q12
12
  1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
  2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
  3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).
OCR MEI C1 Q1
1 Find the equation of the line which passes through \(( 1,3 )\) and ( 4,9 ).
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: $$2 x - 9 < 0 , \quad 8 - x \leq 6$$