Questions — OCR C3 (339 questions)

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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.
OCR C3 2012 January Q7
9 marks Standard +0.3
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 $$M = 40 \mathrm { e } ^ { - 0.132 t }$$
    1. Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
    2. Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
    3. 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
10 marks Standard +0.3
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 2011 June Q1
5 marks Moderate -0.8
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
4 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.8
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
8 marks Moderate -0.8
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
10 marks Moderate -0.3
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
12 marks Standard +0.3
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
5 marks Standard +0.3
1 Solve the inequality \(| 2 x - 5 | > | x + 1 |\).
OCR C3 2012 June Q2
6 marks Moderate -0.5
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
7 marks Moderate -0.3
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
    1. \(\tan \left( \theta - 45 ^ { \circ } \right)\),
    2. \(\quad \tan 2 \theta\).
OCR C3 2012 June Q4
8 marks Moderate -0.3
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
10 marks Standard +0.3
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 \(\alpha\).
    1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
    2. 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
7 marks Standard +0.3
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 marks Standard +0.3
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
11 marks Standard +0.3
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
    1. solve the equation \(3 \sin \theta + 4 \cos \theta + 1 = 0\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\),
    2. 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
11 marks Standard +0.3
9
  1. Show that the derivative with respect to \(y\) of $$y \ln ( 2 y ) - y$$ is \(\ln ( 2 y )\).

  2. [diagram]
    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.