Questions — OCR (4907 questions)

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OCR C3 2007 January Q3
7 marks Standard +0.3
3
  1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$ on the same diagram, show that the equation $$x ^ { 5 } + b x - a = 0$$ has exactly one real root.
  2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }\), with a suitable starting value, to find the real root of the equation \(x ^ { 5 } + 2 x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places.
OCR C3 2007 January Q4
7 marks Moderate -0.3
4
  1. Given that \(x = ( 4 t + 9 ) ^ { \frac { 1 } { 2 } }\) and \(y = 6 \mathrm { e } ^ { \frac { 1 } { 2 } x + 1 }\), find expressions for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) when \(t = 4\), giving your answer correct to 3 significant figures.
OCR C3 2007 January Q5
8 marks Moderate -0.3
5
  1. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).
OCR C3 2007 January Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-3_483_956_264_593} The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).
  1. Find the exact area of the shaded region.
  2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer.
OCR C3 2007 January Q7
8 marks Standard +0.8
7 The curve \(y = \ln x\) is transformed to the curve \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.
  1. Give full details of the translation and stretch involved.
  2. Sketch the graph of \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\).
  3. Sketch, on another diagram, the graph of \(y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|\).
  4. State, in terms of \(a\), the set of values of \(x\) for which \(\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)\).
OCR C3 2007 January Q8
11 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-4_538_1443_262_351} The diagram shows the curve with equation \(y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve and the line \(P Q\).
  1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2 .
  2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places.
  3. Deduce an approximation to the area of region \(B\).
OCR C3 2007 January Q9
12 marks Standard +0.3
9 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi \\ \mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$
  1. State the range of f and the range of g .
  2. Show that \(\operatorname { gf } ( 0.5 ) = 2.16\), correct to 3 significant figures, and explain why \(\mathrm { fg } ( 0.5 )\) is not defined.
  3. Find the set of values of \(x\) for which \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\) is not defined.
OCR C3 2008 January Q1
5 marks Easy -1.2
1 Functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = 2 x - 5$$ Evaluate
  1. \(f g ( 1 )\),
  2. \(\mathrm { f } ^ { - 1 } ( 12 )\).
OCR C3 2008 January Q2
6 marks Moderate -0.3
2 The sequence defined by $$x _ { 1 } = 3 , \quad x _ { n + 1 } = \sqrt [ 3 ] { 31 - \frac { 5 } { 2 } x _ { n } }$$ converges to the number \(\alpha\).
  1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
  2. Find an equation of the form \(a x ^ { 3 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root.
OCR C3 2008 January Q3
7 marks Moderate -0.3
3
  1. Solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation \(\sec \frac { 1 } { 2 } \alpha = 4\).
  2. Solve, for \(0 ^ { \circ } < \beta < 180 ^ { \circ }\), the equation \(\tan \beta = 7 \cot \beta\).
OCR C3 2008 January Q4
6 marks Standard +0.3
4 Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = \left( h ^ { 6 } + 16 \right) ^ { \frac { 1 } { 2 } } - 4$$
  1. Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\).
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures.
OCR C3 2008 January Q5
8 marks Moderate -0.3
5
  1. Find \(\int ( 3 x + 7 ) ^ { 9 } \mathrm {~d} x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_537_881_402_671} The diagram shows the curve \(y = \frac { 1 } { 2 \sqrt { x } }\). The shaded region is bounded by the curve and the lines \(x = 3 , x = 6\) and \(y = 0\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced, simplifying your answer.
OCR C3 2008 January Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_641_837_1306_657} The diagram shows the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).
  1. Give details of the pair of geometrical transformations which transforms the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\) to the graph of \(y = \sin ^ { - 1 } x\).
  2. Sketch the graph of \(y = \left| - \sin ^ { - 1 } ( x - 1 ) \right|\).
  3. Find the exact solutions of the equation \(\left| - \sin ^ { - 1 } ( x - 1 ) \right| = \frac { 1 } { 3 } \pi\).
OCR C3 2008 January Q7
10 marks Standard +0.3
7 A curve has equation \(y = \frac { x \mathrm { e } ^ { 2 x } } { x + k }\), where \(k\) is a non-zero constant.
  1. Differentiate \(x \mathrm { e } ^ { 2 x }\), and show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } + 2 k x + k \right) } { ( x + k ) ^ { 2 } }\).
  2. Given that the curve has exactly one stationary point, find the value of \(k\), and determine the exact coordinates of the stationary point.
OCR C3 2008 January Q8
10 marks Standard +0.8
8 The definite integral \(I\) is defined by $$I = \int _ { 0 } ^ { 6 } 2 ^ { x } \mathrm {~d} x$$
  1. Use Simpson's rule with 6 strips to find an approximate value of \(I\).
  2. By first writing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where the constant \(k\) is to be determined, find the exact value of \(I\).
  3. Use the answers to parts (i) and (ii) to deduce that \(\ln 2 \approx \frac { 9 } { 13 }\).
OCR C3 2008 January Q9
12 marks Standard +0.8
9
  1. Use the identity for \(\cos ( A + B )\) to prove that $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) \equiv \sqrt { 3 } - 2 \sin 2 \theta .$$
  2. Hence find the exact value of \(4 \cos 82.5 ^ { \circ } \cos 52.5 ^ { \circ }\).
  3. Solve, for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation \(4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = 1\).
  4. Given that there are no values of \(\theta\) which satisfy the equation $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = k ,$$ determine the set of values of the constant \(k\).
OCR C3 2005 June Q1
4 marks Moderate -0.8
1 The function f is defined for all real values of \(x\) by $$f ( x ) = 10 - ( x + 3 ) ^ { 2 } .$$
  1. State the range of f .
  2. Find the value of \(\mathrm { ff } ( - 1 )\).
OCR C3 2005 June Q2
4 marks Moderate -0.8
2 Find the exact solutions of the equation \(| 6 x - 1 | = | x - 1 |\).
OCR C3 2005 June Q3
6 marks Moderate -0.3
3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t } .$$
  1. Find the value of \(t\) for which the mass is 25 grams.
  2. Find the rate at which the mass is decreasing when \(t = 55\).
OCR C3 2005 June Q4
8 marks Moderate -0.3
4
  1. \includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-2_579_785_1279_721} The diagram shows the curve \(y = \frac { 2 } { \sqrt { } x }\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1 , x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed.
  2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int _ { 1 } ^ { 5 } \sqrt { } \left( x ^ { 2 } + 1 \right) \mathrm { d } x ,$$ giving your answer correct to 3 decimal places.
OCR C3 2005 June Q5
8 marks Standard +0.3
5
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac { 7 } { 2 }\), giving all solutions for which \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR C3 2005 June Q6
7 marks Moderate -0.3
6
  1. Find the exact value of the \(x\)-coordinate of the stationary point of the curve \(y = x \ln x\).
  2. The equation of a curve is \(y = \frac { 4 x + c } { 4 x - c }\), where \(c\) is a non-zero constant. Show by differentiation that this curve has no stationary points.
OCR C3 2005 June Q7
9 marks Standard +0.3
7
  1. Write down the formula for \(\cos 2 x\) in terms of \(\cos x\).
  2. Prove the identity \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } \equiv 4 - 2 \sec ^ { 2 } x\).
  3. Solve, for \(0 < x < 2 \pi\), the equation \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } = 3 \tan x - 7\).
OCR C3 2005 June Q8
13 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-3_588_915_954_614} The diagram shows part of each of the curves \(y = e ^ { \frac { 1 } { 5 } x }\) and \(y = \sqrt [ 3 ] { } ( 3 x + 8 )\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.
  1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3.
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 5 } { 3 } \ln ( 3 x + 8 )\).
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places.
  4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). \includegraphics[max width=\textwidth, alt={}, center]{e0e2a26b-d4d6-46ea-ac12-a882f3465e5e-4_625_647_264_749} The function f is defined by \(\mathrm { f } ( x ) = \sqrt { } ( m x + 7 ) - 4\), where \(x \geqslant - \frac { 7 } { m }\) and \(m\) is a positive constant. The diagram shows the curve \(y = \mathrm { f } ( x )\).
  1. A sequence of transformations maps the curve \(y = \sqrt { } x\) to the curve \(y = \mathrm { f } ( x )\). Give details of these transformations.
  2. Explain how you can tell that f is a one-one function and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. It is given that the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5]
OCR C3 2006 June Q1
5 marks Moderate -0.8
1 Find the equation of the tangent to the curve \(y = \sqrt { 4 x + 1 }\) at the point ( 2,3 ).