Questions — OCR (4619 questions)

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OCR FP2 2006 June Q2
6 marks Moderate -0.5
2
  1. Given that \(y = \tan ^ { - 1 } x\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
  2. Verify that \(y = \tan ^ { - 1 } x\) satisfies the equation $$\left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0$$
OCR FP2 2006 June Q3
6 marks Standard +0.8
3 The equation of a curve is \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. State the equation of the asymptote of the curve.
  2. Show that \(- \frac { 1 } { 6 } \leqslant y \leqslant \frac { 1 } { 2 }\).
OCR FP2 2006 June Q4
7 marks Standard +0.3
4
  1. Using the definition of \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), prove that $$\cosh 2 x = 2 \cosh ^ { 2 } x - 1$$
  2. Hence solve the equation $$\cosh 2 x - 7 \cosh x = 3$$ giving your answer in logarithmic form.
OCR FP2 2006 June Q5
7 marks Challenging +1.2
5
  1. Express \(t ^ { 2 } + t + 1\) in the form \(( t + a ) ^ { 2 } + b\).
  2. By using the substitution \(\tan \frac { 1 } { 2 } x = t\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$
OCR FP2 2006 June Q6
8 marks Challenging +1.8
6
\includegraphics[max width=\textwidth, alt={}, center]{52b43f20-e0e6-4ddd-9518-bea9782982bf-3_623_1354_262_392} The diagram shows the curve with equation \(y = 3 ^ { x }\) for \(0 \leqslant x \leqslant 1\). The area \(A\) under the curve between these limits is divided into \(n\) strips, each of width \(h\) where \(n h = 1\).
  1. By using the set of rectangles indicated on the diagram, show that \(A > \frac { 2 h } { 3 ^ { h } - 1 }\).
  2. By considering another set of rectangles, show that \(A < \frac { ( 2 h ) 3 ^ { h } } { 3 ^ { h } - 1 }\).
  3. Given that \(h = 0.001\), use these inequalities to find values between which \(A\) lies.
OCR FP2 2006 June Q7
11 marks Standard +0.8
7 The equation of a curve, in polar coordinates, is $$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$
  1. Find the equation of the tangent at the pole.
  2. State the greatest value of \(r\) and the corresponding value of \(\theta\).
  3. Sketch the curve.
  4. Find the exact area of the region enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 4 } \pi\).
OCR FP2 2006 June Q8
11 marks Standard +0.8
8 The curve with equation \(y = \frac { \sinh x } { x ^ { 2 } }\), for \(x > 0\), has one turning point.
  1. Show that the \(x\)-coordinate of the turning point satisfies the equation \(x - 2 \tanh x = 0\).
  2. Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next two approximations, \(x _ { 2 }\) and \(x _ { 3 }\), to the positive root of \(x - 2 \tanh x = 0\).
  3. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\). (You are not expected to evaluate \(x _ { 4 }\).)
OCR FP2 2006 June Q9
13 marks Challenging +1.8
9
  1. Given that \(y = \sinh ^ { - 1 } x\), prove that \(y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).
  2. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \alpha } \sinh ^ { n } \theta \mathrm {~d} \theta$$ where \(\alpha = \sinh ^ { - 1 } 1\). Show that $$n I _ { n } = \sqrt { 2 } - ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2 .$$
  3. Evaluate \(I _ { 4 }\), giving your answer in terms of \(\sqrt { 2 }\) and logarithms.
OCR FP2 2007 June Q1
4 marks Standard +0.3
1 The equation of a curve, in polar coordinates, is $$r = 2 \sin 3 \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi .$$ Find the exact area of the region enclosed by the curve between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
OCR FP2 2007 June Q2
5 marks Moderate -0.5
2
  1. Given that \(\mathrm { f } ( x ) = \sin \left( 2 x + \frac { 1 } { 4 } \pi \right)\), show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \sqrt { 2 } ( \sin 2 x + \cos 2 x )\).
  2. Hence find the first four terms of the Maclaurin series for \(\mathrm { f } ( x )\). [You may use appropriate results given in the List of Formulae.]
OCR FP2 2007 June Q3
6 marks Standard +0.8
3 It is given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( x - 1 ) \left( x ^ { 2 } + 9 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int f ( x ) \mathrm { d } x\).
OCR FP2 2007 June Q4
7 marks Standard +0.8
4
  1. Given that $$y = x \sqrt { 1 - x ^ { 2 } } - \cos ^ { - 1 } x$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a simplified form.
  2. Hence, or otherwise, find the exact value of \(\int _ { 0 } ^ { 1 } 2 \sqrt { 1 - x ^ { 2 } } \mathrm {~d} x\).
OCR FP2 2007 June Q5
8 marks Challenging +1.2
5 It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x$$
  1. Show that, for \(n \geqslant 1\), $$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
  2. Find \(I _ { 3 }\) in terms of e.
OCR FP2 2007 June Q6
11 marks Challenging +1.2
6
\includegraphics[max width=\textwidth, alt={}, center]{dd0e327e-6125-4970-8cfa-cefcedfec06f-3_822_1373_264_386} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) for \(x > 0\), together with a set of \(n\) rectangles of unit width, starting at \(x = 1\).
  1. By considering the areas of these rectangles, explain why $$\frac { 1 } { 1 ^ { 2 } } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } > \int _ { 1 } ^ { n + 1 } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
  2. By considering the areas of another set of rectangles, explain why $$\frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 4 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } < \int _ { 1 } ^ { n } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
  3. Hence show that $$1 - \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < 2 - \frac { 1 } { n }$$
  4. Hence give bounds between which \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\) lies.
OCR FP2 2007 June Q7
10 marks Standard +0.3
7
  1. Using the definitions of hyperbolic functions in terms of exponentials, prove that $$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
  2. Given that \(\cosh x \cosh y = 9\) and \(\sinh x \sinh y = 8\), show that \(x = y\).
  3. Hence find the values of \(x\) and \(y\) which satisfy the equations given in part (ii), giving the answers in logarithmic form.
OCR FP2 2007 June Q8
10 marks Standard +0.8
8 The iteration \(x _ { n + 1 } = \frac { 1 } { \left( x _ { n } + 2 \right) ^ { 2 } }\), with \(x _ { 1 } = 0.3\), is to be used to find the real root, \(\alpha\), of the equation \(x ( x + 2 ) ^ { 2 } = 1\).
  1. Find the value of \(\alpha\), correct to 4 decimal places. You should show the result of each step of the iteration process.
  2. Given that \(\mathrm { f } ( x ) = \frac { 1 } { ( x + 2 ) ^ { 2 } }\), show that \(\mathrm { f } ^ { \prime } ( \alpha ) \neq 0\).
  3. The difference, \(\delta _ { r }\), between successive approximations is given by \(\delta _ { r } = x _ { r + 1 } - x _ { r }\). Find \(\delta _ { 3 }\).
  4. Given that \(\delta _ { r + 1 } \approx \mathrm { f } ^ { \prime } ( \alpha ) \delta _ { r }\), find an estimate for \(\delta _ { 10 }\).
OCR FP2 2007 June Q9
11 marks Standard +0.8
9 It is given that the equation of a curve is $$y = \frac { x ^ { 2 } - 2 a x } { x - a }$$ where \(a\) is a positive constant.
  1. Find the equations of the asymptotes of the curve.
  2. Show that \(y\) takes all real values.
  3. Sketch the curve \(y = \frac { x ^ { 2 } - 2 a x } { x - a }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
OCR FP2 2008 June Q1
5 marks Standard +0.3
1 It is given that \(\mathrm { f } ( x ) = \frac { 2 a x } { ( x - 2 a ) \left( x ^ { 2 } + a ^ { 2 } \right) }\), where \(a\) is a non-zero constant. Express \(\mathrm { f } ( x )\) in partial fractions.
OCR FP2 2008 June Q2
5 marks Standard +0.8
2
\includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-2_341_1043_466_552} The diagram shows the curve \(y = \mathrm { f } ( x )\). The curve has a maximum point at ( 0,5 ) and crosses the \(x\)-axis at \(( - 2,0 ) , ( 3,0 )\) and \(( 4,0 )\). Sketch the curve \(y ^ { 2 } = \mathrm { f } ( x )\), showing clearly the coordinates of any turning points and of any points where this curve crosses the axes.
OCR FP2 2008 June Q3
6 marks
3 By using the substitution \(t = \tan \frac { 1 } { 2 } x\), find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 - \cos x } \mathrm {~d} x$$ giving the answer in terms of \(\pi\).
OCR FP2 2008 June Q4
8 marks Standard +0.3
4
  1. Sketch, on the same diagram, the curves with equations \(y = \operatorname { sech } x\) and \(y = x ^ { 2 }\).
  2. By using the definition of \(\operatorname { sech } x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that the \(x\)-coordinates of the points at which these curves meet are solutions of the equation $$x ^ { 2 } = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 } .$$
  3. The iteration $$x _ { n + 1 } = \sqrt { \frac { 2 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { 2 x _ { n } } + 1 } }$$ can be used to find the positive root of the equation in part (ii). With initial value \(x _ { 1 } = 1\), the approximations \(x _ { 2 } = 0.8050 , x _ { 3 } = 0.8633 , x _ { 4 } = 0.8463\) and \(x _ { 5 } = 0.8513\) are obtained, correct to 4 decimal places. State with a reason whether, in this case, the iteration produces a 'staircase' or a ‘cobweb’ diagram.
OCR FP2 2008 June Q5
8 marks Challenging +1.2
5 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$
  1. By considering \(I _ { n } + I _ { n - 2 }\), or otherwise, show that, for \(n \geqslant 2\), $$( n - 1 ) \left( I _ { n } + I _ { n - 2 } \right) = 1 .$$
  2. Find \(I _ { 4 }\) in terms of \(\pi\).
OCR FP2 2008 June Q6
7 marks Standard +0.3
6 It is given that \(\mathrm { f } ( x ) = 1 - \frac { 7 } { x ^ { 2 } }\).
  1. Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2.5\), to find the next approximations \(x _ { 2 }\) and \(x _ { 3 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 6 decimal places. [3]
  2. The root of \(\mathrm { f } ( x ) = 0\) for which \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) are approximations is denoted by \(\alpha\). Write down the exact value of \(\alpha\).
  3. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Find \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\), giving your answers correct to 5 decimal places. Verify that \(e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }\).
OCR FP2 2008 June Q7
10 marks Challenging +1.2
7 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\), for \(x > - \frac { 1 } { 2 }\).
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { 1 + 2 x }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\).
  2. Show that the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\) can be written as \(\ln a + b x + c x ^ { 2 }\), for constants \(a , b\) and \(c\) to be found.
OCR FP2 2008 June Q8
11 marks Challenging +1.2
8 The equation of a curve, in polar coordinates, is $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

  1. \includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-3_268_796_1567_717} The diagram shows the part of the curve for which \(0 \leqslant \theta \leqslant \alpha\), where \(\theta = \alpha\) is the equation of the tangent to the curve at \(O\). Find \(\alpha\) in terms of \(\pi\).
  2. (a) If \(\mathrm { f } ( \theta ) = 1 - \sin 2 \theta\), show that \(\mathrm { f } \left( \frac { 1 } { 2 } ( 2 k + 1 ) \pi - \theta \right) = \mathrm { f } ( \theta )\) for all \(\theta\), where \(k\) is an integer.
    (b) Hence state the equations of the lines of symmetry of the curve $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
  3. Sketch the curve with equation $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$ State the maximum value of \(r\) and the corresponding values of \(\theta\).