Questions — OCR FP2 (173 questions)

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OCR FP2 2011 January Q7
10 marks Challenging +1.2
7 A curve has polar equation \(r = 1 + \cos 3 \theta\), for \(- \pi < \theta \leqslant \pi\).
  1. Show that the line \(\theta = 0\) is a line of symmetry.
  2. Find the equations of the tangents at the pole.
  3. Find the exact value of the area of the region enclosed by the curve between \(\theta = - \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\).
OCR FP2 2011 January Q8
12 marks Challenging +1.8
8
  1. Without using a calculator, show that \(\sinh \left( \cosh ^ { - 1 } 2 \right) = \sqrt { 3 }\).
  2. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \beta } \cosh ^ { n } x \mathrm {~d} x , \quad \text { where } \beta = \cosh ^ { - 1 } 2$$ Show that \(n I _ { n } = 2 ^ { n - 1 } \sqrt { 3 } + ( n - 1 ) I _ { n - 2 }\), for \(n \geqslant 2\).
  3. Evaluate \(I _ { 5 }\), giving your answer in the form \(k \sqrt { 3 }\).
OCR FP2 2013 January Q1
5 marks Moderate -0.5
1 Express \(\frac { 5 x } { ( x - 1 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.
OCR FP2 2013 January Q2
10 marks Standard +0.8
2 The equation of a curve is \(y = \frac { x ^ { 2 } - 3 } { x - 1 }\).
  1. Find the equations of the asymptotes of the curve.
  2. Write down the coordinates of the points where the curve cuts the axes.
  3. Show that the curve has no stationary points.
  4. Sketch the curve and the asymptotes.
OCR FP2 2013 January Q3
6 marks Standard +0.3
3 By first expressing \(\cosh x\) and \(\sinh x\) in terms of exponentials, solve the equation $$3 \cosh x - 4 \sinh x = 7$$ giving your answer in an exact logarithmic form.
OCR FP2 2013 January Q4
8 marks Standard +0.8
4 You are given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { 2 x } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } n I _ { n - 1 }\) for \(n \geqslant 1\).
  2. Find \(I _ { 3 }\) in terms of e.
OCR FP2 2013 January Q5
11 marks Standard +0.8
5 You are given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\).
  1. Find \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\).
  2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = - 2 \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
  3. Find a similar expression for \(\mathrm { f } ^ { \prime \prime \prime } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime \prime } ( 0 )\).
  4. Find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2013 January Q6
6 marks Standard +0.3
6 By first completing the square, find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x ^ { 2 } + 4 x + 8 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.
OCR FP2 2013 January Q7
13 marks Challenging +1.2
7 A curve has polar equation \(r = 5 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Sketch the curve, indicating the line of symmetry and stating the polar coordinates of the point \(P\) on the curve which is furthest away from the pole.
  2. Calculate the area enclosed by the curve.
  3. Find the cartesian equation of the tangent to the curve at \(P\).
  4. Show that a cartesian equation of the curve is \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( 10 x y ) ^ { 2 }\).
OCR FP2 2013 January Q8
13 marks Standard +0.3
8 It is required to solve the equation \(\ln ( x - 1 ) - x + 3 = 0\).
You are given that there are two roots, \(\alpha\) and \(\beta\), where \(1.1 < \alpha < 1.2\) and \(4.1 < \beta < 4.2\).
  1. The root \(\beta\) can be found using the iterative formula $$x _ { n + 1 } = \ln \left( x _ { n } - 1 \right) + 3$$
    1. Using this iterative formula with \(x _ { 1 } = 4.15\), find \(\beta\) correct to 3 decimal places. Show all your working.
    2. Explain with the aid of a sketch why this iterative formula will not converge to \(\alpha\) whatever initial value is taken.
    3. (a) Show that the Newton-Raphson iterative formula for this equation can be written in the form $$x _ { n + 1 } = \frac { 3 - 2 x _ { n } - \left( x _ { n } - 1 \right) \ln \left( x _ { n } - 1 \right) } { 2 - x _ { n } }$$ (b) Use this formula with \(x _ { 1 } = 1.2\) to find \(\alpha\) correct to 3 decimal places.
OCR FP2 2009 June Q1
5 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{cf77e51a-1d3f-423a-be59-96ec60fbeb67-2_568_959_269_593} The diagram shows the curve with equation \(y = \ln ( \cos x )\), for \(0 \leqslant x \leqslant 1.5\). The region bounded by the curve, the \(x\)-axis and the line \(x = 1.5\) has area \(A\). The region is divided into five strips, each of width 0.3 .
  1. By considering the set of rectangles indicated in the diagram, find an upper bound for \(A\). Give the answer correct to 3 decimal places.
  2. By considering another set of five suitable rectangles, find a lower bound for \(A\). Give the answer correct to 3 decimal places.
  3. How could you reduce the difference between the upper and lower bounds for \(A\) ?
OCR FP2 2009 June Q2
4 marks Standard +0.8
2 Given that \(y = \frac { x ^ { 2 } + x + 1 } { ( x - 1 ) ^ { 2 } }\), prove that \(y \geqslant \frac { 1 } { 4 }\) for all \(x \neq 1\).
OCR FP2 2009 June Q3
6 marks Standard +0.3
3
  1. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }\), find \(\mathrm { f } ^ { \prime } ( 0 )\) and \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
  2. Hence find the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).
OCR FP2 2009 June Q4
6 marks Standard +0.8
4 Express \(\frac { x ^ { 3 } } { ( x - 2 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.
OCR FP2 2009 June Q5
7 marks Standard +0.8
5 It is given that \(I = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \cos \theta } { 1 + \cos \theta } \mathrm { d } \theta\).
  1. By using the substitution \(t = \tan \frac { 1 } { 2 } \theta\), show that \(I = \int _ { 0 } ^ { 1 } \left( \frac { 2 } { 1 + t ^ { 2 } } - 1 \right) \mathrm { d } t\).
  2. Hence find \(I\) in terms of \(\pi\).
OCR FP2 2009 June Q6
6 marks Standard +0.8
6 Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of \(a\).
OCR FP2 2009 June Q7
10 marks Standard +0.3
7
  1. Sketch the graph of \(y = \operatorname { coth } x\), and give the equations of any asymptotes.
  2. It is given that \(\mathrm { f } ( x ) = x \tanh x - 2\). Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 4 decimal places.
  3. If \(\mathrm { f } ( x ) = 0\), show that \(\operatorname { coth } x = \frac { 1 } { 2 } x\). Hence write down the roots of \(\mathrm { f } ( x ) = 0\), correct to 4 decimal places.
OCR FP2 2009 June Q8
14 marks Standard +0.3
8
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
    1. \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
    2. \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
    3. Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
    4. Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that $$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
    5. Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).
OCR FP2 2009 June Q9
14 marks Challenging +1.2
9
  1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ Show that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
  2. The equation of a curve, in polar coordinates, is $$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$
    1. Find the equations of the tangents at the pole and sketch the curve.
    2. Find the exact area of the region enclosed by the curve. RECOGNISING ACHIEVEMENT
OCR FP2 2010 June Q1
4 marks Moderate -0.5
1 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } 2 x\) and \(\mathrm { g } ( x ) = p \tan ^ { - 1 } x\), where \(p\) is a constant. Find the value of \(p\) for which \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 2 } \right) = \mathrm { g } ^ { \prime } \left( \frac { 1 } { 2 } \right)\).
OCR FP2 2010 June Q2
6 marks Standard +0.8
2 Given that the first three terms of the Maclaurin series for \(( 1 + \sin x ) \mathrm { e } ^ { 2 x }\) are identical to the first three terms of the binomial series for \(( 1 + a x ) ^ { n }\), find the values of the constants \(a\) and \(n\). (You may use appropriate results given in the List of Formulae (MF1).)
OCR FP2 2010 June Q3
6 marks Challenging +1.2
3 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x = 1 + \sqrt { 3 }$$
OCR FP2 2010 June Q4
7 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-2_947_1305_986_420} The diagram shows the curve with equation $$y = \frac { a x + b } { x + c }$$ where \(a , b\) and \(c\) are constants.
  1. Given that the asymptotes of the curve are \(x = - 1\) and \(y = - 2\) and that the curve passes through \(( 3,0 )\), find the values of \(a , b\) and \(c\).
  2. Sketch the curve with equation $$y ^ { 2 } = \frac { a x + b } { x + c }$$ for the values of \(a , b\) and \(c\) found in part (i). State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.
OCR FP2 2010 June Q5
8 marks Challenging +1.2
5 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } } ( 1 - 2 x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x$$
  1. Prove that, for \(n \geqslant 1\), $$I _ { n } = 2 n I _ { n - 1 } - 1$$
  2. Find the exact value of \(I _ { 3 }\).
OCR FP2 2010 June Q6
7 marks Standard +0.8
6
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 2 } + 1 } }\).
  2. Given that \(y = \cosh \left( a \sinh ^ { - 1 } x \right)\), where \(a\) is a constant, show that $$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \frac { \mathrm {~d} y } { \mathrm {~d} x } - a ^ { 2 } y = 0$$