Questions — OCR C3 (285 questions)

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OCR C3 2012 January Q5
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
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.
OCR C3 2012 January Q7
7
  1. Substance \(A\) is decaying exponentially and its mass is recorded at regular intervals. At time \(t\) years, the mass, \(M\) grams, of substance \(A\) is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$ (a) Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
    (b) Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
  2. Substance \(B\) is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance \(B\) after a further year.
OCR C3 2012 January Q8
8
  1. Express \(\cos 4 \theta\) in terms of \(\sin 2 \theta\) and hence show that \(\cos 4 \theta\) can be expressed in the form \(1 - k \sin ^ { 2 } \theta \cos ^ { 2 } \theta\), where \(k\) is a constant to be determined.
  2. Hence find the exact value of \(\sin ^ { 2 } \left( \frac { 1 } { 24 } \pi \right) \cos ^ { 2 } \left( \frac { 1 } { 24 } \pi \right)\).
  3. By expressing \(2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta\) in terms of \(\cos 4 \theta\), find the greatest and least possible values of $$2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$$ as \(\theta\) varies.
    \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-5_606_926_267_552} The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = k \left( x ^ { 2 } + 4 x \right) ,$$ where \(k\) is a positive constant. The diagram shows the curve with equation \(y = \mathrm { f } ( x )\).
  4. The curve \(y = x ^ { 2 }\) can be transformed to the curve \(y = \mathrm { f } ( x )\) by the following sequence of transformations: a translation parallel to the \(x\)-axis,
    a translation parallel to the \(y\)-axis,
    a stretch. a translation parallel to the \(x\)-axis, a translation parallel to the \(y\)-axis, a stretch.
    Give details, in terms of \(k\) where appropriate, of these transformations.
  5. Find the range of f in terms of \(k\).
  6. It is given that there are three distinct values of \(x\) which satisfy the equation \(| \mathrm { f } ( x ) | = 20\). Find the value of \(k\) and determine exactly the three values of \(x\) which satisfy the equation in this case.
OCR C3 2013 January Q1
1 For each of the following curves, find the gradient at the point with \(x\)-coordinate 2 .
  1. \(y = \frac { 3 x } { 2 x + 1 }\)
  2. \(y = \sqrt { 4 x ^ { 2 } + 9 }\)
OCR C3 2013 January Q2
2 The acute angle \(A\) is such that \(\tan A = 2\).
  1. Find the exact value of \(\operatorname { cosec } A\).
  2. The angle \(B\) is such that \(\tan ( A + B ) = 3\). Using an appropriate identity, find the exact value of \(\tan B\).
OCR C3 2013 January Q3
3
  1. Given that \(| t | = 3\), find the possible values of \(| 2 t - 1 |\).
  2. Solve the inequality \(| x - \sqrt { 2 } | > | x + 3 \sqrt { 2 } |\).
OCR C3 2013 January Q4
4 The mass, \(m\) grams, of a substance is increasing exponentially so that the mass at time \(t\) hours is given by $$m = 250 \mathrm { e } ^ { 0.021 t } .$$
  1. Find the time taken for the mass to increase to twice its initial value, and deduce the time taken for the mass to increase to 8 times its initial value.
  2. Find the rate at which the mass is increasing at the instant when the mass is 400 grams.
OCR C3 2013 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-2_454_770_1628_635} The diagram shows the curve \(y = \frac { 6 } { \sqrt { 3 x + 1 } }\). The shaded region is bounded by the curve and the lines \(x = 2 , x = 9\) and \(y = 0\).
  1. Show that the area of the shaded region is \(4 \sqrt { 7 }\) square units.
  2. The shaded region is rotated completely about the \(x\)-axis. Show that the volume of the solid produced can be written in the form \(k \ln 2\), where the exact value of the constant \(k\) is to be determined.
OCR C3 2013 January Q7
7
  1. By sketching the curves \(y = \ln x\) and \(y = 8 - 2 x ^ { 2 }\) on a single diagram, show that the equation $$\ln x = 8 - 2 x ^ { 2 }$$ has exactly one real root.
  2. Explain how your diagram shows that the root is between 1 and 2 .
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt { 4 - \frac { 1 } { 2 } \ln x _ { n } } ,$$ with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places.
  4. The curves \(y = \ln x\) and \(y = 8 - 2 x ^ { 2 }\) are each translated by 2 units in the positive \(x\)-direction and then stretched by scale factor 4 in the \(y\)-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places.
    \includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-3_389_917_1117_557} The diagram shows the curve with equation $$x = ( y + 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  5. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  6. Find the gradient of the curve at each of the points \(A\) and \(B\), giving each answer correct to 2 decimal places.
OCR C3 2013 January Q8
8 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 2 } + 4 a x + a ^ { 2 } \text { and } \mathrm { g } ( x ) = 4 x - 2 a ,$$ where \(a\) is a positive constant.
  1. Find the range of f in terms of \(a\).
  2. Given that \(\mathrm { fg } ( 3 ) = 69\), find the value of \(a\) and hence find the value of \(x\) such that \(\mathrm { g } ^ { - 1 } ( x ) = x\).
OCR C3 2013 January Q9
9
  1. Prove that $$\cos ^ { 2 } \left( \theta + 45 ^ { \circ } \right) - \frac { 1 } { 2 } ( \cos 2 \theta - \sin 2 \theta ) \equiv \sin ^ { 2 } \theta .$$
  2. Hence solve the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta + 45 ^ { \circ } \right) - 3 ( \cos \theta - \sin \theta ) = 2$$ for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
  3. It is given that there are two values of \(\theta\), where \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), satisfying the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 3 } \theta + 45 ^ { \circ } \right) - 3 \left( \cos \frac { 2 } { 3 } \theta - \sin \frac { 2 } { 3 } \theta \right) = k ,$$ where \(k\) is a constant. Find the set of possible values of \(k\).
OCR C3 2009 June Q1
1 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a690aa5-63a7-4569-afa8-0746814ebab4-2_533_375_267_404} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a690aa5-63a7-4569-afa8-0746814ebab4-2_533_379_267_882} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a690aa5-63a7-4569-afa8-0746814ebab4-2_531_373_267_1366} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Each diagram above shows part of a curve, the equation of which is one of the following: $$y = \sin ^ { - 1 } x , \quad y = \cos ^ { - 1 } x , \quad y = \tan ^ { - 1 } x , \quad y = \sec x , \quad y = \operatorname { cosec } x , \quad y = \cot x .$$ State which equation corresponds to
  1. Fig. 1,
  2. Fig. 2,
  3. Fig. 3.
OCR C3 2009 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-2_477_833_1493_657} The diagram shows the curve with equation \(y = ( 2 x - 3 ) ^ { 2 }\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis.
OCR C3 2009 June Q3
3 The angles \(\alpha\) and \(\beta\) are such that $$\tan \alpha = m + 2 \quad \text { and } \quad \tan \beta = m$$ where \(m\) is a constant.
  1. Given that \(\sec ^ { 2 } \alpha - \sec ^ { 2 } \beta = 16\), find the value of \(m\).
  2. Hence find the exact value of \(\tan ( \alpha + \beta )\).
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 \(x = 2\).