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 2009 June Q6
8 marks Moderate -0.8
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a\), where \(a\) is a constant. The curve passes through the points \(( - 1,2 )\) and \(( 2,17 )\). Find the equation of the curve.
OCR C2 2009 June Q7
9 marks Moderate -0.8
7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
  2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
  4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.
OCR C2 2009 June Q8
11 marks Moderate -0.3
8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_378_467_269_840} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Fig. 1 shows a sector \(A O B\) of a circle, centre \(O\) and radius \(O A\). The angle \(A O B\) is 1.2 radians and the area of the sector is \(60 \mathrm {~cm} ^ { 2 }\).
  1. Find the perimeter of the sector. A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector \(A O B\) from Fig. 1, and the area of each successive sector is \(\frac { 3 } { 5 }\) of the area of the previous one. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. (a) Find the area of the fifth sector in the pattern.
    (b) Find the total area of the first ten sectors in the pattern.
    (c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.
OCR C2 2009 June Q9
12 marks Standard +0.3
9
  1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
  2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
  3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).
OCR C2 2010 June Q1
5 marks Moderate -0.8
1 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x - 14\), where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. Using this value of \(a\), find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ).
OCR C2 2010 June Q2
5 marks Moderate -0.8
2
  1. Use the trapezium rule, with 3 strips each of width 3 , to estimate the area of the region bounded by the curve \(y = \sqrt [ 3 ] { 7 + x }\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 10\). Give your answer correct to 3 significant figures.
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate of the area.
OCR C2 2010 June Q3
7 marks Standard +0.3
3
  1. Find and simplify the first four terms in the binomial expansion of \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 + 4 x + 2 x ^ { 2 } \right) \left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\).
OCR C2 2010 June Q4
7 marks Moderate -0.8
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 n + 1\).
  1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Evaluate \(\sum _ { n = 1 } ^ { 40 } u _ { n }\). Another sequence \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { 1 } = 2\) and \(w _ { n + 1 } = 5 w _ { n } + 1\).
  3. Find the value of \(p\) such that \(u _ { p } = w _ { 3 }\).
OCR C2 2010 June Q5
8 marks Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{570435e0-5685-4c5b-9ed8-f2bc22bdfb24-02_396_1070_1768_536} The diagram shows two congruent triangles, \(B C D\) and \(B A E\), where \(A B C\) is a straight line. In triangle \(B C D , B D = 8 \mathrm {~cm} , C D = 11 \mathrm {~cm}\) and angle \(C B D = 65 ^ { \circ }\). The points \(E\) and \(D\) are joined by an arc of a circle with centre \(B\) and radius 8 cm .
  1. Find angle \(B C D\).
  2. (a) Show that angle \(E B D\) is 0.873 radians, correct to 3 significant figures.
    (b) Hence find the area of the shaded segment bounded by the chord \(E D\) and the arc \(E D\), giving your answer correct to 3 significant figures.
OCR C2 2010 June Q6
11 marks Moderate -0.8
6
  1. Use integration to find the exact area of the region enclosed by the curve \(y = x ^ { 2 } + 4 x\), the \(x\)-axis and the lines \(x = 3\) and \(x = 5\).
  2. Find \(\int ( 2 - 6 \sqrt { y } ) \mathrm { d } y\).
  3. Evaluate \(\int _ { 1 } ^ { \infty } \frac { 8 } { x ^ { 3 } } \mathrm {~d} x\).
OCR C2 2010 June Q7
8 marks Standard +0.3
7
  1. Show that \(\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv \tan ^ { 2 } x - 1\).
  2. Hence solve the equation $$\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5 - \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR C2 2010 June Q8
9 marks Moderate -0.3
8
  1. Use logarithms to solve the equation \(5 ^ { 3 w - 1 } = 4 ^ { 250 }\), giving the value of \(w\) correct to 3 significant figures.
  2. Given that \(\log _ { x } ( 5 y + 1 ) - \log _ { x } 3 = 4\), express \(y\) in terms of \(x\).
OCR C2 2010 June Q9
12 marks Standard +0.3
9 A geometric progression has first term \(a\) and common ratio \(r\), and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.
  1. Show that \(r ^ { 3 } - 2 r + 1 = 0\).
  2. Given that the geometric progression converges, find the exact value of \(r\).
  3. Given also that the sum to infinity of this geometric progression is \(3 + \sqrt { 5 }\), find the value of the integer \(a\).
OCR C2 2011 June Q1
7 marks Standard +0.3
1 The diagram shows triangle \(A B C\), with \(A B = 9 \mathrm {~cm} , A C = 17 \mathrm {~cm}\) and angle \(B A C = 40 ^ { \circ }\).
  1. Find the length of \(B C\).
  2. Find the area of triangle \(A B C\).
  3. \(D\) is the point on \(A C\) such that angle \(B D A = 63 ^ { \circ }\). Find the length of \(B D\).
OCR C2 2011 June Q2
6 marks Moderate -0.8
2
  1. Find \(\int \left( 6 x ^ { \frac { 1 } { 2 } } - 1 \right) \mathrm { d } x\).
  2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 1\) and which passes through the point \(( 4,17 )\).
OCR C2 2011 June Q3
5 marks Moderate -0.8
3
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-2_515_501_1439_822} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius 8 cm . The perimeter of the sector is 23.2 cm .
  1. Find angle \(A O B\) in radians.
  2. Find the area of the sector.
OCR C2 2011 June Q4
7 marks Moderate -0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-3_588_1136_255_502} The diagram shows the curve \(y = - 1 + \sqrt { x + 4 }\) and the line \(y = 3\).
  1. Show that \(y = - 1 + \sqrt { x + 4 }\) can be rearranged as \(x = y ^ { 2 } + 2 y - 3\).
  2. Hence find by integration the exact area of the shaded region enclosed between the curve, the \(y\)-axis and the line \(y = 3\).
OCR C2 2011 June Q5
8 marks Moderate -0.8
5 The first four terms in the binomial expansion of \(( 3 + k x ) ^ { 5 }\), in ascending powers of \(x\), can be written as \(a + b x + c x ^ { 2 } + d x ^ { 3 }\).
  1. State the value of \(a\).
  2. Given that \(b = c\), find the value of \(k\).
  3. Hence find the value of \(d\).
OCR C2 2011 June Q6
8 marks Moderate -0.8
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10\).
  1. Use the factor theorem to find a factor of \(\mathrm { f } ( x )\).
  2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form.
OCR C2 2011 June Q7
9 marks Moderate -0.8
7
  1. The first term of a geometric progression is 7 and the common ratio is - 2 .
    1. Find the ninth term.
    2. Find the sum of the first 15 terms.
  2. The first term of an arithmetic progression is 7 and the common difference is - 2 . The sum of the first \(N\) terms is - 2900 . Find the value of \(N\).
OCR C2 2011 June Q8
11 marks Moderate -0.8
8
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_417_931_255_607} The diagram shows the curve \(y = 2 ^ { x } - 3\).
  1. Describe the geometrical transformation that transforms the curve \(y = 2 ^ { x }\) to the curve \(y = 2 ^ { x } - 3\).
  2. State the \(y\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(y\)-axis.
  3. Find the \(x\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(x\)-axis, giving your answer in the form \(\log _ { a } b\).
  4. The curve \(y = 2 ^ { x } - 3\) passes through the point ( \(p , 62\) ). Use logarithms to find the value of \(p\), correct to 3 significant figures.
  5. Use the trapezium rule, with 2 strips each of width 0.5 , to find an estimate for \(\int _ { 3 } ^ { 4 } \left( 2 ^ { x } - 3 \right) \mathrm { d } x\). Give your answer correct to 3 significant figures.
OCR C2 2011 June Q9
11 marks Moderate -0.8
9

  1. \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_362_979_1505_625} The diagram shows part of the curve \(y = \cos 2 x\), where \(x\) is in radians. The point \(A\) is the minimum point of this part of the curve.
    1. State the period of \(y = \cos 2 x\).
    2. State the coordinates of \(A\).
    3. Solve the inequality \(\cos 2 x \leqslant 0.5\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
  2. Solve the equation \(\cos 2 x = \sqrt { 3 } \sin 2 x\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
OCR C2 2012 June Q1
6 marks Moderate -0.8
1
  1. Find the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), simplifying the terms.
  2. Hence find the binomial expansion of \(( 3 + 2 x ) ^ { 5 } + ( 3 - 2 x ) ^ { 5 }\).
OCR C2 2012 June Q2
6 marks Moderate -0.8
2
  1. Find \(\int \left( x ^ { 2 } - 2 x + 5 \right) \mathrm { d } x\).
  2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 2 x + 5\) and which passes through the point \(( 3,11 )\).
OCR C2 2012 June Q3
6 marks Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-2_436_526_762_767} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B\) is \(72 ^ { \circ }\).
  1. Express \(72 ^ { \circ }\) exactly in radians, simplifying your answer. The area of the sector \(A O B\) is \(45 \pi \mathrm {~cm} ^ { 2 }\).
  2. Find the value of \(r\).
  3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\), giving your answer correct to 3 significant figures.