Questions — OCR C3 (285 questions)

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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 \(2 \tan ^ { 2 } \beta = 3 + 9 \sec \beta\).
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 \(\int _ { 0 } ^ { 2 } \left( 1 + 10 \mathrm { e } ^ { \sqrt { x } } \right) \mathrm { d } x\).