Questions C3 (1200 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 C3 2009 June Q4
4 It is given that \(\int _ { a } ^ { 3 a } \left( \mathrm { e } ^ { 3 x } + \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.
  1. Show that \(a = \frac { 1 } { 9 } \ln \left( 300 + 3 \mathrm { e } ^ { a } - 2 \mathrm { e } ^ { 3 a } \right)\).
  2. Use an iterative process, based on the equation in part (i), to find the value of \(a\) correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process.
OCR C3 2009 June Q5
5 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 3 x - 2 \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + 7$$ Find the exact coordinates of the point at which
  1. the graph of \(y = \operatorname { fg } ( x )\) meets the \(x\)-axis,
  2. the graph of \(y = \mathrm { g } ( x )\) meets the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\),
  3. the graph of \(y = | \mathrm { f } ( x ) |\) meets the graph of \(y = | \mathrm { g } ( x ) |\).
OCR C3 2009 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-3_590_606_1197_772} The diagram shows the curve with equation \(x = \left( 37 + 10 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
  1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  2. Hence find the equation of the tangent to the curve at the point ( 7,3 ), giving your answer in the form \(y = m x + c\).
  3. Express \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  4. Hence
    (a) solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(8 \sin \theta - 6 \cos \theta = 9\),
    (b) find the greatest possible value of $$32 \sin x - 24 \cos x - ( 16 \sin y - 12 \cos y )$$ as the angles \(x\) and \(y\) vary.
OCR C3 2009 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-4_648_1132_262_504} The diagram shows the curves \(y = \ln x\) and \(y = 2 \ln ( x - 6 )\). The curves meet at the point \(P\) which has \(x\)-coordinate \(a\). The shaded region is bounded by the curve \(y = 2 \ln ( x - 6 )\) and the lines \(x = a\) and \(y = 0\).
  1. Give details of the pair of transformations which transforms the curve \(y = \ln x\) to the curve \(y = 2 \ln ( x - 6 )\).
  2. Solve an equation to find the value of \(a\).
  3. Use Simpson's rule with two strips to find an approximation to the area of the shaded region.
OCR C3 2009 June Q9
9
  1. Show that, for all non-zero values of the constant \(k\), the curve $$y = \frac { k x ^ { 2 } - 1 } { k x ^ { 2 } + 1 }$$ has exactly one stationary point.
  2. Show that, for all non-zero values of the constant \(m\), the curve $$y = \mathrm { e } ^ { m x } \left( x ^ { 2 } + m x \right)$$ has exactly two stationary points.
OCR C3 2011 June Q1
1 Find
  1. \(\int 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\),
  2. \(\int 10 ( 2 x + 1 ) ^ { - 1 } \mathrm {~d} x\).
OCR C3 2011 June Q2
2 The curve \(y = \ln x\) is transformed by:
a reflection in the \(x\)-axis, followed by a stretch with scale factor 3 parallel to the \(y\)-axis, followed by a translation in the positive \(y\)-direction by \(\ln 4\).
Find the equation of the resulting curve, giving your answer in the form \(y = \ln ( \mathrm { f } ( x ) )\).
OCR C3 2011 June Q3
3
  1. Given that \(7 \sin 2 \alpha = 3 \sin \alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), find the exact value of \(\cos \alpha\).
  2. Given that \(3 \cos 2 \beta + 19 \cos \beta + 13 = 0\), where \(90 ^ { \circ } < \beta < 180 ^ { \circ }\), find the exact value of \(\sec \beta\).
OCR C3 2011 June Q4
4
  1. Show by means of suitable sketch graphs that the equation $$( x - 2 ) ^ { 4 } = x + 16$$ has exactly 2 real roots.
  2. State the value of the smaller root.
  3. Use the iterative formula $$x _ { n + 1 } = 2 + \sqrt [ 4 ] { x _ { n } + 16 }$$ with a suitable starting value, to find the larger root correct to 3 decimal places.
OCR C3 2011 June Q5
5 The equation of a curve is \(y = x ^ { 2 } \ln ( 4 x - 3 )\). Find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point on the curve for which \(x = 2\).
OCR C3 2011 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{fc7679bf-a9a1-493d-bf89-35206382787f-3_576_821_258_662} The diagram shows the curve with equation \(y = \sqrt { 3 x - 5 }\). The tangent to the curve at the point \(P\) passes through the origin. The shaded region is bounded by the curve, the \(x\)-axis and the line \(O P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 10 } { 3 }\) and hence find the exact area of the shaded region.
OCR C3 2011 June Q7
7 The functions \(\mathrm { f } , \mathrm { g }\) and h are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = | x | , \quad \mathrm { g } ( x ) = 3 x + 5 \quad \text { and } \quad \mathrm { h } ( x ) = \mathrm { gg } ( x ) .$$
  1. Solve the equation \(\mathrm { g } ( x + 2 ) = \mathrm { f } ( - 12 )\).
  2. Find \(\mathrm { h } ^ { - 1 } ( x )\).
  3. Determine the values of \(x\) for which $$x + \mathrm { f } ( x ) = 0 .$$
OCR C3 2011 June Q8
8 An experiment involves two substances, Substance 1 and Substance 2, whose masses are changing. The mass, \(M _ { 1 }\) grams, of Substance 1 at time \(t\) hours is given by $$M _ { 1 } = 400 \mathrm { e } ^ { - 0.014 t } .$$ The mass, \(M _ { 2 }\) grams, of Substance 2 is increasing exponentially and the mass at certain times is shown in the following table.
\(t\) (hours)01020
\(M _ { 2 }\) (grams)75120192
A critical stage in the experiment is reached at time \(T\) hours when the masses of the two substances are equal.
  1. Find the rate at which the mass of Substance 1 is decreasing when \(t = 10\), giving your answer in grams per hour correct to 2 significant figures.
  2. Show that \(T\) is the root of an equation of the form \(\mathrm { e } ^ { k t } = c\), where the values of the constants \(k\) and \(c\) are to be stated.
  3. Hence find the value of \(T\) correct to 3 significant figures.
OCR C3 2011 June Q9
9
  1. Prove that \(\frac { \sin ( \theta - \alpha ) + 3 \sin \theta + \sin ( \theta + \alpha ) } { \cos ( \theta - \alpha ) + 3 \cos \theta + \cos ( \theta + \alpha ) } \equiv \tan \theta\) for all values of \(\alpha\).
  2. Find the exact value of \(\frac { 4 \sin 149 ^ { \circ } + 12 \sin 150 ^ { \circ } + 4 \sin 151 ^ { \circ } } { 3 \cos 149 ^ { \circ } + 9 \cos 150 ^ { \circ } + 3 \cos 151 ^ { \circ } }\).
  3. It is given that \(k\) is a positive constant. Solve, for \(0 ^ { \circ } < \theta < 60 ^ { \circ }\) and in terms of \(k\), the equation $$\frac { \sin \left( 6 \theta - 15 ^ { \circ } \right) + 3 \sin 6 \theta + \sin \left( 6 \theta + 15 ^ { \circ } \right) } { \cos \left( 6 \theta - 15 ^ { \circ } \right) + 3 \cos 6 \theta + \cos \left( 6 \theta + 15 ^ { \circ } \right) } = k .$$
OCR C3 2012 June Q1
1 Solve the inequality \(| 2 x - 5 | > | x + 1 |\).
OCR C3 2012 June Q2
2 It is given that \(p = \mathrm { e } ^ { 280 }\) and \(q = \mathrm { e } ^ { 300 }\).
  1. Use logarithm properties to show that \(\ln \left( \frac { \mathrm { e } \mathrm { p } ^ { 2 } } { q } \right) = 261\).
  2. Find the smallest integer \(n\) which satisfies the inequality \(5 ^ { n } > p q\).
OCR C3 2012 June Q3
3 It is given that \(\theta\) is the acute angle such that \(\sec \theta \sin \theta = 36 \cot \theta\).
  1. Show that \(\tan \theta = 6\).
  2. Hence, using an appropriate formula in each case, find the exact value of
    (a) \(\tan \left( \theta - 45 ^ { \circ } \right)\),
    (b) \(\quad \tan 2 \theta\).
OCR C3 2012 June Q4
4
  1. Show that \(\int _ { 0 } ^ { 4 } \frac { 18 } { \sqrt { 6 x + 1 } } \mathrm {~d} x = 24\).
  2. Find \(\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 2 \right) ^ { 2 } \mathrm {~d} x\), giving your answer in terms of e .
OCR C3 2012 June Q5
5
  1. It is given that \(k\) is a positive constant. By sketching the graphs of $$y = 14 - x ^ { 2 } \text { and } y = k \ln x$$ on a single diagram, show that the equation $$14 - x ^ { 2 } = k \ln x$$ has exactly one real root.
  2. The real root of the equation \(14 - x ^ { 2 } = 3 \ln x\) is denoted by \(\alpha\).
    (a) Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
    (b) Use the iterative formula \(x _ { n + 1 } = \sqrt { 14 - 3 \ln x _ { n } }\), with a suitable starting value, to find \(\alpha\). Show the result of each iteration, and give \(\alpha\) correct to 2 decimal places.
OCR C3 2012 June Q6
6 The volume, \(V \mathrm {~m} ^ { 3 }\), of liquid in a container is given by $$V = \left( 3 h ^ { 2 } + 4 \right) ^ { \frac { 3 } { 2 } } - 8 ,$$ where \(h \mathrm {~m}\) is the depth of the liquid.
  1. Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 0.6\), giving your answer correct to 2 decimal places.
  2. Liquid is leaking from the container. It is observed that, when the depth of the liquid is 0.6 m , the depth is decreasing at a rate of 0.015 m per hour. Find the rate at which the volume of liquid in the container is decreasing at the instant when the depth is 0.6 m .
OCR C3 2012 June Q7
7 The function f is defined for all real values of \(x\) by \(\mathrm { f } ( x ) = 2 x + 5\). The function g is defined for all real values of \(x\) and is such that \(\mathrm { g } ^ { - 1 } ( x ) = \sqrt [ 3 ] { x - a }\), where \(a\) is a constant. It is given that \(\mathrm { fg } ^ { - 1 } ( 12 ) = 9\). Find the value of \(a\) and hence solve the equation \(\operatorname { gf } ( x ) = 68\).
OCR C3 2012 June Q8
8
  1. Express \(3 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence
    (a) solve the equation \(3 \sin \theta + 4 \cos \theta + 1 = 0\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\),
    (b) find the values of the positive constants \(k\) and \(c\) such that $$- 37 \leqslant k ( 3 \sin \theta + 4 \cos \theta ) + c \leqslant 43$$ for all values of \(\theta\).
OCR C3 2012 June Q9
9
  1. Show that the derivative with respect to \(y\) of $$y \ln ( 2 y ) - y$$ is \(\ln ( 2 y )\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{390105da-0cba-4f82-8c8f-1f36090b1564-3_465_631_1859_717} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \mathrm { e } ^ { x ^ { 2 } }\). The point \(P \left( 2 , \frac { 1 } { 2 } \mathrm { e } ^ { 4 } \right)\) lies on the curve. The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = \frac { 1 } { 2 } e ^ { 4 }\). Find the exact volume of the solid produced when the shaded region is rotated completely about the \(y\)-axis.
  3. Hence find the volume of the solid produced when the region bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\) is rotated completely about the \(y\)-axis. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR C3 2013 June Q1
1 Find
  1. \(\quad \int ( 4 - 3 x ) ^ { 7 } \mathrm {~d} x\),
  2. \(\quad \int ( 4 - 3 x ) ^ { - 1 } \mathrm {~d} x\).
OCR C3 2013 June Q2
2 Using an appropriate identity in each case, find the possible values of
  1. \(\sin \alpha\) given that \(4 \cos 2 \alpha = \sin ^ { 2 } \alpha\),
  2. \(\sec \beta\) given that \(2 \tan ^ { 2 } \beta = 3 + 9 \sec \beta\).