Questions C3 (1301 questions)

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OCR C3 2009 January Q1
5 marks Moderate -0.8
1 Find
  1. \(\int 8 \mathrm { e } ^ { - 2 x } \mathrm {~d} x\),
  2. \(\int ( 4 x + 5 ) ^ { 6 } \mathrm {~d} x\).
OCR C3 2009 January Q2
5 marks Moderate -0.5
2
  1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Deduce an approximation to \(\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x\).
OCR C3 2009 January Q3
7 marks Moderate -0.3
3
  1. Express \(2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta }\) in terms of \(\sec \theta\).
  2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$
OCR C3 2009 January Q4
7 marks Moderate -0.3
4 For each of the following curves, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the exact \(x\)-coordinate of the stationary point:
  1. \(y = \left( 4 x ^ { 2 } + 1 \right) ^ { 5 }\),
  2. \(y = \frac { x ^ { 2 } } { \ln x }\).
OCR C3 2009 January Q5
8 marks Moderate -0.3
5 The mass, \(M\) grams, of a certain substance is increasing exponentially so that, at time \(t\) hours, the mass is given by $$M = 40 \mathrm { e } ^ { k t }$$ where \(k\) is a constant. The following table shows certain values of \(t\) and \(M\).
\(t\)02163
\(M\)80
  1. In either order,
    1. find the values missing from the table,
    2. determine the value of \(k\).
    3. Find the rate at which the mass is increasing when \(t = 21\).
OCR C3 2009 January Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_627_689_264_726} The function f is defined for all real values of \(x\) by $$f ( x ) = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$ The graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) meet at the point \(P\), and the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) meets the \(x\)-axis at \(Q\) (see diagram).
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and determine the \(x\)-coordinate of the point \(Q\).
  2. State how the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related geometrically, and hence show that the \(x\)-coordinate of the point \(P\) is the root of the equation $$x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$
  3. Use an iterative process, based on the equation \(x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }\), to find the \(x\)-coordinate of \(P\), giving your answer correct to 2 decimal places.
OCR C3 2009 January Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_419_700_1809_721} The diagram shows the curve \(y = \mathrm { e } ^ { k x } - a\), where \(k\) and \(a\) are constants.
  1. Give details of the pair of transformations which transforms the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { k x } - a\).
  2. Sketch the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\).
  3. Given that the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\) passes through the points \(( 0,13 )\) and \(( \ln 3,13 )\), find the values of \(k\) and \(a\).
OCR C3 2009 January Q8
10 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-4_538_702_264_719} The diagram shows the curve with equation $$y = \frac { 6 } { \sqrt { x } } - 3$$ The point \(P\) has coordinates \(( 0 , p )\). The shaded region is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = p\). The shaded region is rotated completely about the \(y\)-axis to form a solid of volume \(V\).
  1. Show that \(V = 16 \pi \left( 1 - \frac { 27 } { ( p + 3 ) ^ { 3 } } \right)\).
  2. It is given that \(P\) is moving along the \(y\)-axis in such a way that, at time \(t\), the variables \(p\) and \(t\) are related by $$\frac { \mathrm { d } p } { \mathrm {~d} t } = \frac { 1 } { 3 } p + 1 .$$ Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} t }\) at the instant when \(p = 9\).
OCR C3 2009 January Q9
12 marks Standard +0.8
9
  1. By first expanding \(\cos ( 2 \theta + \theta )\), prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
  2. Hence prove that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
  3. Show that the only solutions of the equation $$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$ are odd multiples of \(90 ^ { \circ }\).
OCR C3 2011 January Q1
3 marks Moderate -0.8
1 Solve the equation \(| 3 x + 4 a | = 5 a\), where \(a\) is a positive constant.
OCR C3 2011 January Q2
4 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_538_1061_388_541} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 7 ) = 0\) and that there are stationary points at \(( - 2 , - 6 )\) and \(( 0,0 )\). Sketch the curve with equation \(y = - 4 \mathrm { f } ( x + 3 )\), indicating the coordinates of the stationary points.
OCR C3 2011 January Q3
3 marks Standard +0.3
3 A giant spherical balloon is being inflated in a theme park. The radius of the balloon is increasing at a rate of 12 cm per hour. Find the rate at which the surface area of the balloon is increasing at the instant when the radius is 150 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per hour correct to 2 significant figures.
[0pt] [Surface area of sphere \(= 4 \pi r ^ { 2 }\).]
OCR C3 2011 January Q4
7 marks Standard +0.3
4
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR C3 2011 January Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_559_1191_1749_479} The diagram shows the curve with equation \(y = \frac { 6 } { \sqrt { 3 x - 2 } }\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 1 , x = a\) and \(y = 0\), where \(a\) is a constant greater than 1 . It is given that the area of \(R\) is 16 square units. Find the value of \(a\) and hence find the exact volume of the solid formed when \(R\) is rotated completely about the \(x\)-axis.
[0pt] [9]
OCR C3 2011 January Q6
9 marks Standard +0.8
6 The curve with equation \(y = \frac { 3 x + 4 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }\) has a stationary point at \(P\). It is given that \(P\) is close to the point with coordinates \(( 2.4 , - 1.6 )\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \frac { 16 } { 3 } x + 1 }$$
  2. By first using an iterative process based on the equation in part (i), find the coordinates of \(P\), giving each coordinate correct to 3 decimal places.
OCR C3 2011 January Q7
12 marks Standard +0.3
7 The function f is defined for \(x > 0\) by \(\mathrm { f } ( x ) = \ln x\) and the function g is defined for all real values of \(x\) by \(\mathrm { g } ( x ) = x ^ { 2 } + 8\).
  1. Find the exact, positive value of \(x\) which satisfies the equation \(\operatorname { fg } ( x ) = 8\).
  2. State which one of f and g has an inverse and define that inverse function.
  3. Find the exact value of the gradient of the curve \(y = \operatorname { gf } ( x )\) at the point with \(x\)-coordinate \(\mathrm { e } ^ { 3 }\).
  4. Use Simpson's rule with four strips to find an approximate value of $$\int _ { - 4 } ^ { 4 } \mathrm { fg } ( x ) \mathrm { d } x$$ giving your answer correct to 3 significant figures.
OCR C3 2011 January Q8
12 marks Standard +0.3
8
    1. Sketch the graph of \(y = \operatorname { cosec } x\) for \(0 < x < 4 \pi\).
    2. It is given that \(\operatorname { cosec } \alpha = \operatorname { cosec } \beta\), where \(\frac { 1 } { 2 } \pi < \alpha < \pi\) and \(2 \pi < \beta < \frac { 5 } { 2 } \pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\).
    1. Write down the identity giving \(\tan 2 \theta\) in terms of \(\tan \theta\).
    2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2 \phi \tan 4 \phi\), showing all your working.
OCR C3 2011 January Q9
13 marks Standard +0.3
9
  1. The function f is defined for all real values of \(x\) by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$
    1. Show that \(\mathrm { f } ^ { \prime } ( x ) > 0\) for all \(x\).
    2. Show that the set of values of \(x\) for which \(\mathrm { f } ^ { \prime \prime } ( x ) > 0\) is the same as the set of values of \(x\) for which \(\mathrm { f } ( x ) > 0\), and state what this set of values is.
    3. \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-04_634_830_641_699} The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x } ,$$ where \(k\) is a constant greater than 1 . The graph of \(y = \mathrm { g } ( x )\) is shown above. Find the range of g , giving your answer in simplified form.
OCR C3 2011 January Q10
Moderate -0.8
10
8 (b) (i)
8 (b) (ii)
\multirow{29}{*}{}
{}
9
  1. (a)
\multirow{12}{*}{}D
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-
\multirow[t]{25}{*}{9
  • (b)}
    • □
    9
  • \section*{RECOGNISING ACHIEVEMENT} RECOGNISING ACHIEVEMENT
    OCR C3 2012 January Q1
    3 marks Moderate -0.8
    1 Show that \(\int _ { \sqrt { 2 } } ^ { \sqrt { 6 } } \frac { 2 } { x } \mathrm {~d} x = \ln 3\).
    OCR C3 2012 January Q2
    5 marks Moderate -0.3
    2 \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-2_490_713_447_660} The diagram shows part of the curve \(y = \frac { 6 } { ( 2 x + 1 ) ^ { 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 1\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis.
    OCR C3 2012 January Q3
    7 marks Moderate -0.3
    3 Find the equation of the normal to the curve \(y = \frac { x ^ { 2 } + 4 } { x + 2 }\) at the point \(\left( 1 , \frac { 5 } { 3 } \right)\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    OCR C3 2012 January Q4
    7 marks Standard +0.3
    4 The acute angles \(\alpha\) and \(\beta\) are such that $$2 \cot \alpha = 1 \text { and } 24 + \sec ^ { 2 } \beta = 10 \tan \beta \text {. }$$
    1. State the value of \(\tan \alpha\) and determine the value of \(\tan \beta\).
    2. Hence find the exact value of \(\tan ( \alpha + \beta )\).
    OCR C3 2012 January Q5
    8 marks Moderate -0.8
    5 \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-3_844_837_242_621} It is given that f is a one-one function defined for all real values. The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). The coordinates of certain points on the curve are shown in the following table.
    \(x\)2468101214
    \(y\)181419232526
    1. State the value of \(\mathrm { ff } ( 6 )\) and the value of \(\mathrm { f } ^ { - 1 } ( 8 )\).
    2. On the copy of the diagram, sketch the curve \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating how the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related.
    3. Use Simpson's rule with 6 strips to find an approximation to \(\int _ { 2 } ^ { 14 } \mathrm { f } ( x ) \mathrm { d } x\).
    OCR C3 2012 January Q6
    10 marks Standard +0.3
    6 \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-4_476_709_251_683} The diagram shows the curve with equation \(x = \ln \left( y ^ { 3 } + 2 y \right)\). At the point \(P\) on the curve, the gradient is 4 and it is given that \(P\) is close to the point with coordinates (7.5,12).
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
    2. Show that the \(y\)-coordinate of \(P\) satisfies the equation $$y = \frac { 12 y ^ { 2 } + 8 } { y ^ { 2 } + 2 }$$
    3. By first using an iterative process based on the equation in part (ii), find the coordinates of \(P\), giving each coordinate correct to 3 decimal places.