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 Q8
8 Find, in the form \(y = m x + c\), the equation of the line passing through \(\mathrm { A } ( 3,7 )\) and \(\mathrm { B } ( 5 , - 1 )\).
Show that the midpoint of AB lies on the line \(x + 2 y = 10\).
OCR MEI C1 Q11
11 A cubic polynomial is given by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8\).
  1. Show that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\). Factorise \(\mathrm { f } ( x )\) fully.
    Sketch the graph of \(y = f ( x )\).
  2. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { - 3 } { 0 }\). Write down an equation for the resulting graph. You need not simplify your answer.
    Find also the intercept on the \(y\)-axis of the resulting graph.
OCR MEI C1 2009 January Q11
  1. Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
  2. Find the coordinates of the points of intersection of this circle with the \(y\)-axis. Give your answer in the form \(a \pm \sqrt { b }\).
  3. Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.
OCR MEI C1 Q10
  1. Write down the equations of the circles A and B .
  2. Find the \(x\) coordinates of the points where the two curves intersect.
  3. Find the \(y\) coordinates of these points, giving your answers in surd form.
OCR MEI C1 Q6
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9106e6b2-0b36-4ebf-ace3-a30570df73d3-2_754_780_317_763} \captionsetup{labelformat=empty} \caption{Fig. 10}
    \end{figure} Fig. 10 shows a sketch of the graph of \(y = \frac { 1 } { x }\).
    Sketch the graph of \(y = \frac { 1 } { x - 2 }\), showing clearly the coordinates of any points where it crosses the axes.
  2. Find the value of \(x\) for which \(\frac { 1 } { x - 2 } = 5\).
  3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac { 1 } { x - 2 }\). Give your answers in the form \(a \pm \sqrt { b }\). Show the position of these points on your graph in part (i).
OCR MEI C1 2007 January Q11
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 2009 January Q13
13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q6
6 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
OCR MEI C1 Q4
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\)
    (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 Q4
4 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
OCR MEI C1 2006 June Q1
1 The volume of a cone is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). Make \(r\) the subject of this formula.
OCR MEI C1 2006 June Q2
2 One root of the equation \(x ^ { 3 } + a x ^ { 2 } + 7 = 0\) is \(x = - 2\). Find the value of \(a\).
OCR MEI C1 2006 June Q3
3 A line has equation \(3 x + 2 y = 6\). Find the equation of the line parallel to this which passes through the point \(( 2,10 )\).
OCR MEI C1 2006 June Q4
4 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .
  1. P: \(x ^ { 2 } + x - 2 = 0\) Q: \(x = 1\)
  2. \(\mathrm { P } : y ^ { 3 } > 1\) Q: \(y > 1\)
OCR MEI C1 2006 June Q5
5 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\).
OCR MEI C1 2006 June Q6
6 Solve the inequality \(x ^ { 2 } + 2 x < 3\).
OCR MEI C1 2006 June Q7
7
  1. Simplify \(6 \sqrt { 2 } \times 5 \sqrt { 3 } - \sqrt { 24 }\).
  2. Express \(( 2 - 3 \sqrt { 5 } ) ^ { 2 }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
OCR MEI C1 2006 June Q8
8 Calculate \({ } ^ { 6 } \mathrm { C } _ { 3 }\).
Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 6 }\).
OCR MEI C1 2006 June Q9
9 Simplify the following.
  1. \(\frac { 16 ^ { \frac { 1 } { 2 } } } { 81 ^ { \frac { 3 } { 4 } } }\)
  2. \(\frac { 12 \left( a ^ { 3 } b ^ { 2 } c \right) ^ { 4 } } { 4 a ^ { 2 } c ^ { 6 } }\)
OCR MEI C1 2006 June Q10
10 Find the coordinates of the points of intersection of the circle \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(y = 3 x\). Give your answers in surd form.
OCR MEI C1 2006 June Q12
12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).
  1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
  3. Express \(\mathrm { f } ( x )\) in fully factorised form.
  4. Sketch the graph of \(y = \mathrm { f } ( x )\).
  5. Solve the equation \(\mathrm { f } ( x ) = 12\).
OCR MEI C1 2006 June Q13
13 Answer the whole of this question on the insert provided.
The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).