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 Q6
6
  1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
  3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
  4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
OCR MEI C1 Q1
1 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
\(2 \quad\) (i) Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
(ii) Factorise \(x ^ { 2 } - 4 \quad x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.
OCR MEI C1 Q3
3 Simplify \(\left( m ^ { 2 } + 1 \right) ^ { 2 } - \left( m ^ { 2 } - 1 \right) ^ { 2 }\), showing your method.
Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer.
\includegraphics[max width=\textwidth, alt={}, center]{6978a576-e86a-48d4-93a4-268845ea6699-1_414_590_1328_772}
OCR MEI C1 Q5
5
  1. Write \(x ^ { 2 } - 7 x + 6\) in the form \(( x - a ) ^ { 2 } + b\).
  2. State the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 7 x + 6\).
  3. Find the coordinates of the points where the graph of \(y = x ^ { 2 } - 7 x + 6\) crosses the axes and sketch the graph.
  4. Show that the graphs of \(y = x ^ { 2 } - 7 x + 6\) and \(y = x ^ { 2 } - 3 x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6978a576-e86a-48d4-93a4-268845ea6699-3_643_864_390_622} \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.
OCR MEI C1 Q7
7
  1. Describe fully the transformation which maps the curve \(y = x ^ { 2 }\) onto the curve \(y = ( x + 4 ) ^ { 2 }\).
  2. Sketch the graph of \(y = x ^ { 2 } - 4\).
OCR MEI C1 Q3
4 marks
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e93e3c51-ae2b-420b-abb8-bf0c483caff8-3_1270_1219_326_463} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).
  1. Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]
OCR MEI C1 Q4
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e93e3c51-ae2b-420b-abb8-bf0c483caff8-4_679_727_357_741} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows the graph of \(y = \mathrm { g } ( x )\). Draw the graphs of the following.
  1. \(y = \mathrm { g } ( x ) + 3\)
  2. \(y = \mathrm { g } ( x + 2 )\)
OCR MEI C1 Q5
5 The point \(\mathrm { P } ( 5,4 )\) is on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P when the graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of
  1. \(y = \mathrm { f } ( x - 5 )\),
  2. \(y = \mathrm { f } ( x ) + 7\).
OCR MEI C1 Q6
6
  1. Describe fully the transformation which maps the curve \(y = x ^ { 2 }\) onto the curve \(y = ( x + 4 ) ^ { 2 }\).
  2. Sketch the graph of \(y = x ^ { 2 } - 4\).
OCR MEI C1 Q7
7
  1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
  2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
  3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
  4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.
OCR MEI C1 Q1
1
  1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
  2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.
OCR MEI C1 Q2
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91e16597-234a-4730-8c4b-765ca574e6e2-1_522_528_867_803} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 shows graphs \(A\) and \(B\).
  1. State the transformation which maps graph \(A\) onto graph \(B\).
  2. The equation of graph \(A\) is \(y = \mathrm { f } ( x )\). Which one of the following is the equation of graph \(B\) ? $$\begin{aligned} & y = \mathrm { f } ( x ) + 2
    & y = 2 \mathrm { f } ( x ) \end{aligned}$$ $$\begin{aligned} & y = \mathrm { f } ( x ) - 2
    & y = \mathrm { f } ( x + 3 ) \end{aligned}$$ $$\begin{aligned} & y = \mathrm { f } ( x + 2 )
    & y = \mathrm { f } ( x - 3 ) \end{aligned}$$
OCR MEI C1 Q3
3 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 Q4
4
  1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 Q1
1 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR MEI C1 Q2
2 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 7 }\).
OCR MEI C1 Q3
3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 2 - 4 x ) ^ { 5 }\).
OCR MEI C1 Q4
4 The binomial expansion of \(\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }\) has a term which is a constant. Find this term.
OCR MEI C1 Q5
5
  1. Evaluate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q6
6 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 6 }\).
OCR MEI C1 Q7
7 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.
OCR MEI C1 Q8
8 You are given that
  • the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
  • when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .
Find the values of \(k\) and \(m\).
OCR MEI C1 Q9
9 Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 4 }\), simplifying the coefficients.
OCR MEI C1 Q10
10 Find the binomial expansion of \(\left( x + \frac { 5 } { x } \right) ^ { 3 }\), simplifying the terms.
OCR MEI C1 Q11
11
  1. Calculate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 }\).