Questions C1 (1442 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 Q2
5 marks Moderate -0.8
2 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).
OCR MEI C1 Q3
13 marks Moderate -0.3
3
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
    Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 Q4
12 marks Moderate -0.3
4 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).
  1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).
OCR MEI C1 Q5
12 marks Moderate -0.3
5 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).
  1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
  3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).
OCR MEI C1 Q1
3 marks Easy -1.8
1 Sketch the graph of \(y = 9 - x ^ { 2 }\).
OCR MEI C1 Q2
12 marks Moderate -0.8
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{618d118a-2557-42f3-9b55-4a55dda93a97-1_449_376_631_889} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).
  1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
  2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
  3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).
OCR MEI C1 Q3
12 marks Moderate -0.3
3 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 Q4
13 marks Moderate -0.3
4
  1. Sketch the graph of \(y = x ( x - 3 ) ^ { 2 }\).
  2. Show that the equation \(x ( x - 3 ) ^ { 2 } = 2\) can be expressed as \(x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0\).
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).
OCR MEI C1 Q5
12 marks Moderate -0.5
5
  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
12 marks Moderate -0.3
1
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes. [4] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{973ad9eb-33f2-432e-9449-e54c1728008b-1_1292_1401_887_359} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
  4. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
  5. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
  6. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation. [5]
  7. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
  8. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
  9. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
    (A) write down the equation of the line of symmetry,
    (B) write down the minimum \(y\)-value on the curve.
OCR MEI C1 Q4
12 marks Moderate -0.3
4
  1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
  2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
  3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).
OCR MEI C1 Q1
11 marks Moderate -0.8
1
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
  2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.
OCR MEI C1 Q2
11 marks Moderate -0.3
2 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 Q3
3 marks Standard +0.3
3 Factorise and hence simplify \(\frac { 3 x ^ { 2 } - 7 x + 4 } { x ^ { 2 } - 1 }\).
  1. Prove that 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all even positive integers \(n\).
  2. Determine whether 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all positive integers \(n\).
  3. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
  4. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
  5. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
  6. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.
OCR MEI C1 Q6
12 marks Moderate -0.8
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
4 marks Moderate -0.8
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
4 marks Moderate -0.5
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
13 marks Moderate -0.3
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
4 marks Easy -1.2
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 Standard +0.3
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 marks Easy -1.3
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
4 marks Easy -1.2
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
4 marks Easy -1.2
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
12 marks Moderate -0.8
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
4 marks Easy -1.2
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.