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 2012 June Q4
6 marks Moderate -0.3
4 Solve the equation $$4 \cos ^ { 2 } x + 7 \sin x - 7 = 0$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2012 June Q5
8 marks Moderate -0.8
5
  1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 4 \quad \text { and } \quad u _ { n + 1 } = \frac { 2 } { u _ { n } } \quad \text { for } n \geqslant 1 .$$
    1. Write down the values of \(u _ { 2 }\) and \(u _ { 3 }\).
    2. Describe the behaviour of the sequence.
  2. In an arithmetic progression the ninth term is 18 and the sum of the first nine terms is 72. Find the first term and the common difference.
OCR C2 2012 June Q6
9 marks Moderate -0.8
6
  1. Use the trapezium rule, with 2 strips each of width 4 , to show that an approximate value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is \(32 + 16 \sqrt { 5 }\).
  2. Use a sketch graph to explain why the actual value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is greater than \(32 + 16 \sqrt { 5 }\).
  3. Use integration to find the exact value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\).
OCR C2 2012 June Q7
8 marks Moderate -0.8
7
    1. Given that \(\alpha\) is the acute angle such that \(\tan \alpha = \frac { 2 } { 5 }\), find the exact value of \(\cos \alpha\).
    2. Given that \(\beta\) is the obtuse angle such that \(\sin \beta = \frac { 3 } { 7 }\), find the exact value of \(\cos \beta\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-3_316_662_955_700} The diagram shows a triangle \(A B C\) with \(A C = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A B C = \gamma\). Find the exact value of \(\sin \gamma\), simplifying your answer.
OCR C2 2012 June Q8
11 marks Standard +0.3
8 Two cubic polynomials are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + ( a - 3 ) x + 2 b , \quad \mathrm {~g} ( x ) = 3 x ^ { 3 } + x ^ { 2 } + 5 a x + 4 b$$ where \(a\) and \(b\) are constants.
  1. Given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have a common factor of ( \(x - 2\) ), show that \(a = - 4\) and find the value of \(b\).
  2. Using these values of \(a\) and \(b\), factorise \(\mathrm { f } ( x )\) fully. Hence show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have two common factors.
OCR C2 2012 June Q9
12 marks Standard +0.3
9
  1. An arithmetic progression has first term \(\log _ { 2 } 27\) and common difference \(\log _ { 2 } x\).
    1. Show that the fourth term can be written as \(\log _ { 2 } \left( 27 x ^ { 3 } \right)\).
    2. Given that the fourth term is 6, find the exact value of \(x\).
  2. A geometric progression has first term \(\log _ { 2 } 27\) and common ratio \(\log _ { 2 } y\).
    1. Find the set of values of \(y\) for which the geometric progression has a sum to infinity.
    2. Find the exact value of \(y\) for which the sum to infinity of the geometric progression is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR C2 2013 June Q1
4 marks Easy -1.2
1 Use the trapezium rule, with 3 strips each of width 2 , to estimate the value of $$\int _ { 5 } ^ { 11 } \frac { 8 } { x } \mathrm {~d} x .$$
OCR C2 2013 June Q2
6 marks Moderate -0.3
2 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. \(\sin \frac { 1 } { 2 } x = 0.8\)
  2. \(\sin x = 3 \cos x\)
OCR C2 2013 June Q3
7 marks Moderate -0.3
3
  1. Find and simplify the first three terms in the expansion of \(( 2 + 5 x ) ^ { 6 }\) in ascending powers of \(x\).
  2. In the expansion of \(( 3 + c x ) ^ { 2 } ( 2 + 5 x ) ^ { 6 }\), the coefficient of \(x\) is 4416. Find the value of \(c\).
OCR C2 2013 June Q4
8 marks Standard +0.3
4
  1. Find \(\int \left( 5 x ^ { 3 } - 6 x + 1 \right) \mathrm { d } x\).
    1. Find \(\int 24 x ^ { - 3 } \mathrm {~d} x\).
    2. Given that \(\int _ { a } ^ { \infty } 24 x ^ { - 3 } \mathrm {~d} x = 3\), find the value of the positive constant \(a\).
OCR C2 2013 June Q5
8 marks Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-2_405_688_1535_685} The diagram shows a sector \(B A C\) of a circle with centre \(A\) and radius 16 cm . The angle \(B A C\) is 0.8 radians. The length \(A D\) is 7 cm .
  1. Find the area of the region \(B D C\).
  2. Find the perimeter of the region \(B D C\).
OCR C2 2013 June Q6
9 marks Moderate -0.3
6 Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she uses 7.8 g of the chemical.
  1. Given that the amounts of the chemical used form an arithmetic progression, find the total amount of chemical used in the first 30 experiments.
  2. Instead it is given that the amounts of the chemical used form a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that \(N\), the greatest number of experiments possible, satisfies the inequality $$1.3 ^ { N } \leqslant 91 ,$$ and use logarithms to calculate the value of \(N\).
OCR C2 2013 June Q7
9 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-3_519_611_1087_712} The diagram shows the curve \(y = x ^ { \frac { 3 } { 2 } } - 1\), which crosses the \(x\)-axis at \(( 1,0 )\), and the tangent to the curve at the point \(( 4,7 )\).
  1. Show that \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) \mathrm { d } x = 9 \frac { 2 } { 5 }\).
  2. Hence find the exact area of the shaded region enclosed by the curve, the tangent and the \(x\)-axis.
OCR C2 2013 June Q8
9 marks Moderate -0.8
8
\includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-4_524_822_274_609} The diagram shows the curves \(y = a ^ { x }\) and \(y = 4 b ^ { x }\).
  1. (a) State the coordinates of the point of intersection of \(y = a ^ { x }\) with the \(y\)-axis.
    (b) State the coordinates of the point of intersection of \(y = 4 b ^ { x }\) with the \(y\)-axis.
    (c) State a possible value for \(a\) and a possible value for \(b\).
  2. It is now given that \(a b = 2\). Show that the \(x\)-coordinate of the point of intersection of \(y = a ^ { x }\) and \(y = 4 b ^ { x }\) can be written as $$x = \frac { 2 } { 2 \log _ { 2 } a - 1 } .$$
OCR C2 2013 June Q9
12 marks Standard +0.3
9 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ).
  2. Show that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and hence factorise \(\mathrm { f } ( x )\) completely.
  3. Solve the equation $$4 \cos ^ { 3 } \theta - 7 \cos \theta - 3 = 0$$ for \(0 \leqslant \theta \leqslant 2 \pi\). Give each solution for \(\theta\) in an exact form.
OCR C2 2014 June Q1
6 marks Standard +0.3
1
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_426_1244_280_413} The diagram shows triangle \(A B C\), with \(A B = 8 \mathrm {~cm}\), angle \(B A C = 65 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The point \(D\) is on \(A C\) such that \(A D = 10 \mathrm {~cm}\).
  1. Find the area of triangle \(A B D\).
  2. Find the length of \(B D\).
  3. Find the length of \(B C\).
OCR C2 2014 June Q2
5 marks Easy -1.2
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 3 n - 1\), for \(n \geqslant 1\).
  1. Find the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).
OCR C2 2014 June Q3
7 marks Moderate -0.8
3
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_350_597_1695_735} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 12 cm . The angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians.
  1. Find the exact length of the \(\operatorname { arc } A B\).
  2. Find the exact area of the shaded segment enclosed by the arc \(A B\) and the chord \(A B\).
OCR C2 2014 June Q4
6 marks Standard +0.3
4
  1. Show that the equation $$\sin x - \cos x = \frac { 6 \cos x } { \tan x }$$ can be expressed in the form $$\tan ^ { 2 } x - \tan x - 6 = 0 .$$
  2. Hence solve the equation \(\sin x - \cos x = \frac { 6 \cos x } { \tan x }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR C2 2014 June Q5
6 marks Moderate -0.3
5 Solve the equation \(2 ^ { 4 x - 1 } = 3 ^ { 5 - 2 x }\), giving your answer in the form \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).
OCR C2 2014 June Q6
9 marks Moderate -0.8
6
  1. Find the binomial expansion of \(\left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 } \mathrm {~d} x\).
OCR C2 2014 June Q7
9 marks Moderate -0.8
7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Show that ( \(3 - x\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  4. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in simplified surd form where appropriate.
OCR C2 2014 June Q8
12 marks Standard +0.3
8
  1. The first term of a geometric progression is 50 and the common ratio is 0.8 . Use logarithms to find the smallest value of \(k\) such that the value of the \(k\) th term is less than 0.15 .
  2. In a different geometric progression, the second term is - 3 and the sum to infinity is 4 . Show that there is only one possible value of the common ratio and hence find the first term. \section*{Question 9 begins on page 4.}
OCR C2 2014 June Q9
12 marks Standard +0.3
9
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-4_387_624_287_717} The diagram shows part of the curve \(y = - 3 + 2 \sqrt { x + 4 }\). The point \(P ( 5,3 )\) lies on the curve. Region \(A\) is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 5\). Region \(B\) is bounded by the curve, the \(y\)-axis and the line \(y = 3\).
  1. Use the trapezium rule, with 2 strips each of width 2.5 , to find an approximate value for the area of region \(A\), giving your answer correct to 3 significant figures.
  2. Use your answer to part (i) to deduce an approximate value for the area of region \(B\).
  3. By first writing the equation of the curve in the form \(x = \mathrm { f } ( y )\), use integration to show that the exact area of region \(B\) is \(\frac { 14 } { 3 }\). \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {N } }\)}
OCR C2 2015 June Q1
5 marks Moderate -0.8
1 A geometric progression has first term 3 and second term - 6 .
  1. State the value of the common ratio.
  2. Find the value of the eleventh term.
  3. Find the sum of the first twenty terms.