Questions C2 (1410 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 C2 2015 June Q2
5 marks Moderate -0.8
2
  1. Use the trapezium rule, with 4 strips each of width 1.5, to estimate the value of $$\int _ { 4 } ^ { 10 } \sqrt { 2 x - 1 } \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate.
OCR C2 2015 June Q3
9 marks Standard +0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-2_576_599_1062_733} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 8 cm . The angle \(A O B\) is 1.2 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively such that \(O C = 5.2 \mathrm {~cm}\) and \(O D = 2.6 \mathrm {~cm} . C D\) is a straight line.
  1. Find the area of the shaded region \(A C D B\).
  2. Find the perimeter of the shaded region \(A C D B\).
OCR C2 2015 June Q4
7 marks Moderate -0.3
4
  1. Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
  2. In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).
OCR C2 2015 June Q5
7 marks Moderate -0.3
5 A curve has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }\) for all positive values of \(x\). The point \(P ( 4,1 )\) lies on the curve, and the gradient of the curve at \(P\) is 5 . Find the equation of the curve.
OCR C2 2015 June Q6
10 marks Moderate -0.3
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30\).
  1. Given that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = 0\), express \(\mathrm { f } ( x )\) as the product of 3 linear factors.
  2. Use integration to find the exact value of \(\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\).
  3. Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 5 \leqslant x \leqslant 3\).
OCR C2 2015 June Q7
11 marks Standard +0.3
7 In an arithmetic progression the first term is 5 and the common difference is 3 . The \(n\)th term of the progression is denoted by \(u _ { n }\).
  1. Find the value of \(u _ { 20 }\).
  2. Show that \(\sum _ { n = 10 } ^ { 20 } u _ { n } = 517\).
  3. Find the value of \(N\) such that \(\sum _ { n = N } ^ { 2 N } u _ { n } = 2750\).
OCR C2 2015 June Q8
9 marks Moderate -0.3
8
  1. Use logarithms to solve the equation $$2 ^ { n - 3 } = 18000$$ giving your answer correct to 3 significant figures.
  2. Solve the simultaneous equations $$\log _ { 2 } x + \log _ { 2 } y = 8 , \quad \log _ { 2 } \left( \frac { x ^ { 2 } } { y } \right) = 7$$
OCR C2 2015 June Q9
9 marks Standard +0.3
9
\includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-4_406_625_248_721} The diagram shows part of the curve \(y = 2 \cos \frac { 1 } { 3 } x\), where \(x\) is in radians, and the line \(y = k\).
  1. The smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\),
    (a) the next smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\),
    (b) the smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = - k\).
  2. The curve \(y = 2 \cos \frac { 1 } { 3 } x\) is shown in the Printed Answer Book. On the diagram, and for the same values of \(x\), sketch the curve of \(y = \sin \frac { 1 } { 3 } x\).
  3. Calculate the \(x\)-coordinates of the points of intersection of the curves in part (ii). Give your answers in radians correct to 3 significant figures. \section*{END OF QUESTION PAPER}
OCR C2 2016 June Q1
4 marks Moderate -0.8
1
\includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_250_611_356_721} The diagram shows triangle \(A B C\), with \(A C = 8 \mathrm {~cm}\) and angle \(C A B = 30 ^ { \circ }\).
  1. Given that the area of the triangle is \(20 \mathrm {~cm} ^ { 2 }\), find the length of \(A B\).
  2. Find the length of \(B C\), giving your answer correct to 3 significant figures.
OCR C2 2016 June Q2
5 marks Moderate -0.8
2
\includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_417_476_1030_790} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is \(54 ^ { \circ }\). The perimeter of the sector is 60 cm .
  1. Express \(54 ^ { \circ }\) exactly in radians, simplifying your answer.
  2. Find the value of \(r\), giving your answer correct to 3 significant figures.
OCR C2 2016 June Q3
6 marks Moderate -0.3
3
  1. Find the binomial expansion of \(( 3 + k x ) ^ { 3 }\), simplifying the terms.
  2. It is given that, in the expansion of \(( 3 + k x ) ^ { 3 }\), the coefficient of \(x ^ { 2 }\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form.
OCR C2 2016 June Q4
6 marks Moderate -0.8
4
  1. Express \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 )\) as a single logarithm.
  2. Hence solve the equation \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 ) = 2\).
OCR C2 2016 June Q5
8 marks Standard +0.3
5
  1. Find \(\int \left( x ^ { 2 } + 2 \right) ( 2 x - 3 ) \mathrm { d } x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 1 } ^ { a } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\), where \(a\) is a constant greater than 1 .
    2. Deduce the value of \(\int _ { 1 } ^ { \infty } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\).
OCR C2 2016 June Q6
11 marks Standard +0.8
6 An arithmetic progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 1.5\) for \(n \geqslant 1\).
  1. Given that \(u _ { k } = 140\), find the value of \(k\). A geometric progression \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { n } = 120 \times ( 0.9 ) ^ { n - 1 }\) for \(n \geqslant 1\).
  2. Find the sum of the first 16 terms of this geometric progression, giving your answer correct to 3 significant figures.
  3. Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { N } u _ { n } > \sum _ { n = 1 } ^ { \infty } w _ { n }\).
OCR C2 2016 June Q7
12 marks Standard +0.3
7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
  1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
  2. Hence find the three roots of the equation \(\mathrm { f } ( x ) = 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-3_540_718_1466_660} The diagram shows the curve \(C\) with equation \(y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9\).
  3. Show that the \(x\)-coordinates of the stationary points on \(C\) are given by \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0\).
  4. Use integration to find the exact area of the region enclosed by \(C\) and the \(x\)-axis.
OCR C2 2016 June Q8
12 marks Moderate -0.8
8
  1. The curve \(y = 3 ^ { x }\) can be transformed to the curve \(y = 3 ^ { x - 2 }\) by a translation. Give details of the translation.
  2. Alternatively, the curve \(y = 3 ^ { x }\) can be transformed to the curve \(y = 3 ^ { x - 2 }\) by a stretch. Give details of the stretch.
  3. Sketch the curve \(y = 3 ^ { x - 2 }\), stating the coordinates of any points of intersection with the axes.
  4. The point \(P\) on the curve \(y = 3 ^ { x - 2 }\) has \(y\)-coordinate equal to 180 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
  5. Use the trapezium rule, with 2 strips each of width 1.5, to find an estimate for \(\int _ { 1 } ^ { 4 } 3 ^ { x - 2 } \mathrm {~d} x\). Give your answer correct to 3 significant figures.
OCR C2 2016 June Q9
8 marks Moderate -0.3
9 A curve has equation \(y = \sin ( a x )\), where \(a\) is a positive constant and \(x\) is in radians.
  1. State the period of \(y = \sin ( a x )\), giving your answer in an exact form in terms of \(a\).
  2. Given that \(x = \frac { 1 } { 5 } \pi\) and \(x = \frac { 2 } { 5 } \pi\) are the two smallest positive solutions of \(\sin ( a x ) = k\), where \(k\) is a positive constant, find the values of \(a\) and \(k\).
  3. Given instead that \(\sin ( a x ) = \sqrt { 3 } \cos ( a x )\), find the two smallest positive solutions for \(x\), giving your answers in an exact form in terms of \(a\). \section*{END OF QUESTION PAPER}
OCR MEI C2 2009 January Q1
4 marks Easy -1.3
1 Find \(\int \left( 20 x ^ { 4 } + 6 x ^ { - \frac { 3 } { 2 } } \right) \mathrm { d } x\).
[0pt] [4]
OCR MEI C2 2009 January Q2
4 marks Easy -1.2
2 Fig. 2 shows the coordinates at certain points on a curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-2_645_1146_589_497} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Use the trapezium rule with 6 strips to calculate an estimate of the area of the region bounded by this curve and the axes.
OCR MEI C2 2009 January Q3
2 marks Easy -1.8
3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 2009 January Q4
3 marks Moderate -0.8
4 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C2 2009 January Q6
5 marks Moderate -0.8
6 An arithmetic progression has first term 7 and third term 12.
  1. Find the 20th term of this progression.
  2. Find the sum of the 21st to the 50th terms inclusive of this progression.
OCR MEI C2 2009 January Q7
5 marks Moderate -0.8
7 Differentiate \(4 x ^ { 2 } + \frac { 1 } { x }\) and hence find the \(x\)-coordinate of the stationary point of the curve \(y = 4 x ^ { 2 } + \frac { 1 } { x }\).
OCR MEI C2 2009 January Q8
5 marks Moderate -0.8
8 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } \end{aligned}$$
  1. Find the third term of this sequence and state what type of sequence it is.
  2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.
OCR MEI C2 2009 January Q9
4 marks Easy -1.8
9
  1. State the value of \(\log _ { a } a\).
  2. Express each of the following in terms of \(\log _ { a } x\).
    (A) \(\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }\)
    (B) \(\log _ { a } \frac { 1 } { x }\) Section B (36 marks)