Questions C3 (1200 questions)

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OCR C3 2013 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{71e01d8f-d0ed-4f17-b7cd-6f5a93bbe329-2_435_472_932_794} The diagram shows a container in the form of a right circular cone. The angle between the axis and the slant height is \(\alpha\), where \(\alpha = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\). Initially the container is empty, and then liquid is added at the rate of \(14 \mathrm {~cm} ^ { 3 }\) per minute. The depth of liquid in the container at time \(t\) minutes is \(x \mathrm {~cm}\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid in the container when the depth is \(x \mathrm {~cm}\) is given by $$V = \frac { 1 } { 12 } \pi x ^ { 3 } .$$ [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  2. Find the rate at which the depth of the liquid is increasing at the instant when the depth is 8 cm . Give your answer in cm per minute correct to 2 decimal places.
OCR C3 2013 June Q4
4 Find the exact value of the gradient of the curve $$y = \sqrt { 4 x - 7 } + \frac { 4 x } { 2 x + 1 }$$ at the point for which \(x = 4\).
OCR C3 2013 June Q5
5
  1. Give full details of a sequence of two transformations needed to transform the graph of \(y = | x |\) to the graph of \(y = | 2 ( x + 3 ) |\).
  2. Solve the inequality \(| x | > | 2 ( x + 3 ) |\), showing all your working.
OCR C3 2013 June Q6
6 The value of \(\int _ { 0 } ^ { 8 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x\) obtained by using Simpson's rule with four strips is denoted by \(A\).
  1. Find the value of \(A\) correct to 3 significant figures.
  2. Explain why an approximate value of \(\int _ { 0 } ^ { 8 } \ln \left( 9 + 6 x ^ { 2 } + x ^ { 4 } \right) \mathrm { d } x\) is \(2 A\).
  3. Explain why an approximate value of \(\int _ { 0 } ^ { 8 } \ln \left( 3 \mathrm { e } + \mathrm { e } x ^ { 2 } \right) \mathrm { d } x\) is \(A + 8\).
OCR C3 2013 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{71e01d8f-d0ed-4f17-b7cd-6f5a93bbe329-3_428_751_703_641} The diagram shows the curve \(y = \mathrm { f } ( x )\), where f is the function defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 3 + 4 \mathrm { e } ^ { - x }$$
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).
  3. The straight line \(y = x\) meets the curve \(y = \mathrm { f } ( x )\) at the point \(P\). By using an iterative process based on the equation \(x = \mathrm { f } ( x )\), with a starting value of 3 , find the coordinates of the point \(P\). Show all your working and give each coordinate correct to 3 decimal places.
  4. How is the point \(P\) related to the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) ?
OCR C3 2013 June Q8
8
  1. Express \(4 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence
    (a) solve the equation \(4 \cos \theta - 2 \sin \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
    (b) determine the greatest and least values of $$25 - ( 4 \cos \theta - 2 \sin \theta ) ^ { 2 }$$ as \(\theta\) varies, and, in each case, find the smallest positive value of \(\theta\) for which that value occurs.
OCR C3 2013 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{71e01d8f-d0ed-4f17-b7cd-6f5a93bbe329-4_661_915_269_557} The diagram shows the curve $$y = \mathrm { e } ^ { 2 x } - 18 x + 15 .$$ The curve crosses the \(y\)-axis at \(P\) and the minimum point is \(Q\). The shaded region is bounded by the curve and the line \(P Q\).
  1. Show that the \(x\)-coordinate of \(Q\) is \(\ln 3\).
  2. Find the exact area of the shaded region.
OCR C3 2014 June Q1
1 Given that \(y = 4 x ^ { 2 } \ln x\), find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = \mathrm { e } ^ { 2 }\).
OCR C3 2014 June Q2
2 By first using appropriate identities, solve the equation $$5 \cos 2 \theta \operatorname { cosec } \theta = 2$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
OCR C3 2014 June Q3
3
  1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sqrt { x } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
  2. Deduce an approximation to \(\int _ { 0 } ^ { 2 } \left( 1 + 10 \mathrm { e } ^ { \sqrt { x } } \right) \mathrm { d } x\).
OCR C3 2014 June Q4
4 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = \sqrt [ 3 ] { x - 10 }$$
  1. Evaluate \(\mathrm { f } ^ { - 1 } ( - 50 )\).
  2. Show that \(\operatorname { fg } ( x ) = 2 x - 16\).
  3. Differentiate \(\operatorname { gf } ( x )\) with respect to \(x\).
OCR C3 2014 June Q5
5
  1. The mass, \(M\) grams, of a substance at time \(t\) years is given by $$M = 58 \mathrm { e } ^ { - 0.33 t }$$ Find the rate at which the mass is decreasing at the instant when \(t = 4\). Give your answer correct to 2 significant figures.
  2. The mass of a second substance is increasing exponentially. The initial mass is 42.0 grams and, 6 years later, the mass is 51.8 grams. Find the mass at a time 24 years after the initial value.
OCR C3 2014 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_524_720_246_676} The diagram shows the curve \(y = x ^ { 4 } - 8 x\).
  1. By sketching a second curve on the copy of the diagram, show that the equation $$x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$$ has two real roots. State the equation of the second curve.
  2. The larger root of the equation \(x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0\) is denoted by \(\alpha\).
    (a) Show by calculation that \(2.1 < \alpha < 2.2\).
    (b) Use an iterative process based on the equation $$x = \sqrt [ 4 ] { 9 + 8 x - x ^ { 2 } } ,$$ with a suitable starting value, to find \(\alpha\) correct to 3 decimal places. Give the result of each step of the iterative process.
OCR C3 2014 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_547_851_1749_605} The diagram shows the curve \(y = \sqrt { \frac { 3 } { 4 x + 1 } }\) for \(0 \leqslant x \leqslant 20\). The point \(P\) on the curve has coordinates \(\left( 20 , \frac { 1 } { 9 } \sqrt { 3 } \right)\). The shaded region \(R\) is enclosed by the curve and the lines \(x = 0\) and \(y = \frac { 1 } { 9 } \sqrt { 3 }\).
  1. Find the exact area of \(R\).
  2. Find the exact volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis.
OCR C3 2014 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-4_616_1024_296_516} The diagram shows the curve \(y = \frac { 2 x + 4 } { x ^ { 2 } + 5 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the two stationary points.
  2. The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$ (a) Sketch the curve \(y = \mathrm { g } ( x )\) and state the range of g .
    (b) It is given that the equation \(\mathrm { g } ( x ) = k\), where \(k\) is a constant, has exactly two distinct real roots. Write down the set of possible values of \(k\).
OCR C3 2014 June Q9
9
  1. Express \(5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence
    (a) give details of the transformations needed to transform the curve \(y = 5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) to the curve \(y = \sin \theta\),
    (b) find the smallest positive value of \(\beta\) satisfying the equation $$5 \cos \left( \frac { 1 } { 3 } \beta - 40 ^ { \circ } \right) + 3 \cos \left( \frac { 1 } { 3 } \beta + 20 ^ { \circ } \right) = 3 .$$ \section*{END OF QUESTION PAPER}
OCR C3 2015 June Q1
1 Find the equation of the tangent to the curve \(y = \frac { 5 x + 4 } { 3 x - 8 }\) at the point \(( 2 , - 7 )\).
OCR C3 2015 June Q2
2 It is given that \(\theta\) is the acute angle such that \(\cot \theta = 4\). Without using a calculator, find the exact value of
  1. \(\tan \left( \theta + 45 ^ { \circ } \right)\),
  2. \(\operatorname { cosec } \theta\).
OCR C3 2015 June Q3
3 The volume, \(V\) cubic metres, of water in a reservoir is given by $$V = 3 ( 2 + \sqrt { h } ) ^ { 6 } - 192 ,$$ where \(h\) metres is the depth of the water. Water is flowing into the reservoir at a constant rate of 150 cubic metres per hour. Find the rate at which the depth of water is increasing at the instant when the depth is 1.4 metres.
OCR C3 2015 June Q4
4 It is given that \(| x + 3 a | = 5 a\), where \(a\) is a positive constant. Find, in terms of \(a\), the possible values of $$| x + 7 a | - | x - 7 a |$$
OCR C3 2015 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-2_455_643_1327_694} The diagram shows the curve \(y = \mathrm { e } ^ { 3 x } - 6 \mathrm { e } ^ { 2 x } + 32\).
  1. Find the exact \(x\)-coordinate of the minimum point and verify that the \(y\)-coordinate of the minimum point is 0 .
  2. Find the exact area of the region (shaded in the diagram) enclosed by the curve and the axes.
OCR C3 2015 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-3_553_579_274_726} The diagram shows the curve \(y = 8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right)\). The end-points \(A\) and \(B\) of the curve have coordinates ( \(a , - 4 \pi\) ) and ( \(b , 4 \pi\) ) respectively.
  1. State the values of \(a\) and \(b\).
  2. It is required to find the root of the equation \(8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right) = x\).
    (a) Show by calculation that the root lies between 1.7 and 1.8.
    (b) In order to find the root, the iterative formula $$x _ { n + 1 } = p + \sin \left( q x _ { n } \right) ,$$ with a suitable starting value, is to be used. Determine the values of the constants \(p\) and \(q\) and hence find the root correct to 4 significant figures. Show the result of each step of the iteration process.
OCR C3 2015 June Q7
7
  1. Find the exact value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
  2. Use Simpson's rule with two strips to show that an approximate value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\) can be expressed in the form \(m + n \sqrt [ 3 ] { 36 }\), where the values of the constants \(m\) and \(n\) are to be stated.
  3. Use the results from parts (i) and (ii) to find an approximate value of \(\sqrt [ 3 ] { 36 }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers. \section*{Question 8 begins on page 4.}
OCR C3 2015 June Q8
8 The functions \(f\) and \(g\) are defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = 2 + \ln ( x + 3 ) \text { for } x \geqslant 0
\mathrm {~g} ( x ) = a x ^ { 2 } \text { for all real values of } x , \text { where } a \text { is a positive constant. } \end{gathered}$$
  1. Given that \(\operatorname { gf } \left( \mathrm { e } ^ { 4 } - 3 \right) = 9\), find the value of \(a\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Given that \(\mathrm { ff } \left( \mathrm { e } ^ { N } - 3 \right) = \ln \left( 53 \mathrm { e } ^ { 2 } \right)\), find the value of \(N\).
OCR C3 2015 June Q9
9 It is given that \(\mathrm { f } ( \theta ) = \sin \left( \theta + 30 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right)\).
  1. Show that \(\mathrm { f } ( \theta ) = \cos \theta\). Hence show that $$f ( 4 \theta ) + 4 f ( 2 \theta ) \equiv 8 \cos ^ { 4 } \theta - 3 .$$
  2. Hence
    (a) determine the greatest and least values of \(\frac { 1 } { \mathrm { f } ( 4 \theta ) + 4 \mathrm { f } ( 2 \theta ) + 7 }\) as \(\theta\) varies,
    (b) solve the equation $$\sin \left( 12 \alpha + 30 ^ { \circ } \right) + \cos \left( 12 \alpha + 60 ^ { \circ } \right) + 4 \sin \left( 6 \alpha + 30 ^ { \circ } \right) + 4 \cos \left( 6 \alpha + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < \alpha < 60 ^ { \circ }\). \section*{END OF QUESTION PAPER}