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 2014 June Q4
4
  1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 2014 June Q5
5 Make \(a\) the subject of \(3 ( a + 4 ) = a c + 5 f\).
OCR MEI C1 2014 June Q6
6 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).
OCR MEI C1 2014 June Q7
7 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 7 }\).
OCR MEI C1 2014 June Q8
8 You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6\), where \(k\) is a constant. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 42 . Use the remainder theorem to find the value of \(k\). Hence find a root of \(\mathrm { f } ( x ) = 0\).
OCR MEI C1 2014 June Q9
9 You are given that \(n , n + 1\) and \(n + 2\) are three consecutive integers.
  1. Expand and simplify \(n ^ { 2 } + ( n + 1 ) ^ { 2 } + ( n + 2 ) ^ { 2 }\).
  2. For what values of \(n\) will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer. Section B (36 marks)
OCR MEI C1 2014 June Q10
10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-3_680_800_1146_628} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Write down the coordinates of B .
  2. Find the radius of the circle and hence write down the equation of the circle.
  3. AD is a diameter of the circle. Find the coordinates of D .
  4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
  5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
  6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
  7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
  8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).
  9. Use the intersections with both axes to express the equation of the curve in a factorised form.
  10. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
  11. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
  12. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0 .$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. \section*{END OF QUESTION PAPER}
OCR MEI C1 2016 June Q1
1 Find the value of each of the following.
  1. \(3 ^ { 0 }\)
  2. \(9 ^ { \frac { 3 } { 2 } }\)
  3. \(\left( \frac { 4 } { 5 } \right) ^ { - 2 }\)
OCR MEI C1 2016 June Q2
2 Find the coordinates of the point of intersection of the lines \(2 x + 3 y = 12\) and \(y = 7 - 3 x\).
OCR MEI C1 2016 June Q3
3
  1. Solve the inequality \(\frac { 1 - 2 x } { 4 } > 3\).
  2. Simplify \(\left( 5 c ^ { 2 } d \right) ^ { 3 } \times \frac { 2 c ^ { 4 } } { d ^ { 5 } }\).
OCR MEI C1 2016 June Q4
4 You are given that \(a = \frac { 3 c + 2 a } { 2 c - 5 }\). Express \(a\) in terms of \(c\).
OCR MEI C1 2016 June Q5
5
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Express \(\frac { 5 + 2 \sqrt { 3 } } { 4 - \sqrt { 3 } }\) in the form \(c + d \sqrt { 3 }\), where \(c\) and \(d\) are integers.
OCR MEI C1 2016 June Q6
6 Find the binomial expansion of \(( 1 - 5 x ) ^ { 4 }\), expressing the terms as simply as possible.
OCR MEI C1 2016 June Q7
7
  1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
  2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.
OCR MEI C1 2016 June Q8
8 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + a x + c\) and that \(\mathrm { f } ( 2 ) = 11\). The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 8 . Find the values of \(a\) and \(c\).
OCR MEI C1 2016 June Q9
9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-3_1255_1470_434_299} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
  2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
  3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
    (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
    (B) Find the equation of the translated curve, simplifying your answer.
OCR MEI C1 2016 June Q10
10 Fig. 10 shows a sketch of the points \(\mathrm { A } ( 2,7 ) , \mathrm { B } ( 0,3 )\) and \(\mathrm { C } ( 8 , - 1 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-4_579_748_301_657} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Prove that angle ABC is \(90 ^ { \circ }\).
  2. Find the equation of the circle which has AC as a diameter.
  3. Find the equation of the tangent to this circle at A . Give your answer in the form \(a y = b x + c\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2016 June Q11
11
  1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
  2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
  3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}
OCR MEI C1 Q1
1 Solve the inequality \(2 ( x - 3 ) < 6 x + 15\).
OCR MEI C1 Q2
2 Make \(r\) the subject of \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).
OCR MEI C1 Q3
3 In each case, choose one of the statements $$\mathbf { P } \Rightarrow \mathbf { Q } \quad \mathbf { P } \Leftarrow \mathbf { Q } \quad \mathbf { P } \Leftrightarrow \mathbf { Q }$$ to describe the complete relationship between P and Q .
  1. For \(n\) an integer: P: \(n\) is an even number
    Q: \(n\) is a multiple of 4
  2. For triangle ABC : P: \(\quad \mathrm { B }\) is a right-angle
    Q: \(\quad \mathrm { AB } ^ { 2 } + \mathrm { BC } ^ { 2 } = \mathrm { AC } ^ { 2 }\)
OCR MEI C1 Q4
4 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 2 + 3 x ) ^ { 5 }\).
OCR MEI C1 Q5
5 Find the value of the following.
  1. \(\left( \frac { 1 } { 3 } \right) ^ { - 2 }\)
  2. \(16 ^ { \frac { 3 } { 4 } }\)
OCR MEI C1 Q6
6 The line \(L\) is parallel to \(y = - 2 x + 1\) and passes through the point \(( 5,2 )\).
Find the coordinates of the points of intersection of \(L\) with the axes.
OCR MEI C1 Q7
7 Express \(x ^ { 2 } - 6 x\) in the form \(( x - a ) ^ { 2 } - b\).
Sketch the graph of \(y = x ^ { 2 } - 6 x\), giving the coordinates of its minimum point and the intersections with the axes.