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OCR FP2 2011 January Q1
5 marks Challenging +1.2
1 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to find \(\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x\).
OCR FP2 2011 January Q2
8 marks Standard +0.8
2 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } x\).
  1. Show that \(\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }\).
  2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 3 }\).
OCR FP2 2011 January Q3
9 marks Standard +0.3
3 The function f is defined by \(\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }\), for \(x \in \mathbb { R }\) and \(a > 0\).
  1. For the curve with equation \(y = \mathrm { f } ( x )\),
    (a) write down the equation of the asymptote,
    (b) find the range of values that \(y\) can take.
  2. For the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\), write down
    (a) the equation of the line of symmetry,
    (b) the maximum and minimum values of \(y\),
    (c) the set of values of \(x\) for which the curve is defined.
OCR FP2 2011 January Q4
9 marks Standard +0.8
4
  1. Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \sinh ^ { 4 } x \equiv \cosh 4 x - 4 \cosh 2 x + 3$$
  2. Solve the equation $$\cosh 4 x - 3 \cosh 2 x + 1 = 0$$ giving your answer(s) in logarithmic form.
OCR FP2 2011 January Q5
9 marks Standard +0.3
5 The equation $$x ^ { 3 } - 5 x + 3 = 0$$ may be solved by the Newton-Raphson method. Successive approximations to a root are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots\).
  1. Show that the Newton-Raphson formula can be written in the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where $$\mathrm { F } ( x ) = \frac { 2 x ^ { 3 } - 3 } { 3 x ^ { 2 } - 5 }$$
  2. Find \(\mathrm { F } ^ { \prime } ( x )\) and hence verify that \(\mathrm { F } ^ { \prime } ( \alpha ) = 0\), where \(\alpha\) is any one of the roots of equation (A).
  3. Use the Newton-Raphson method to find the root of equation (A) which is close to 2 . Write down sufficient approximations to find the root correct to 4 decimal places.
OCR FP2 2011 January Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{debf6581-25ff-4692-bdfb-154675a3cdb0-3_608_1134_258_504} The diagram shows the curve \(y = \mathrm { f } ( x )\), defined by $$f ( x ) = \begin{cases} x ^ { x } & \text { for } 0 < x \leqslant 1 , \\ 1 & \text { for } x = 0 . \end{cases}$$
  1. By first taking logarithms, show that the curve has a stationary point at \(x = \mathrm { e } ^ { - 1 }\). The area under the curve from \(x = 0.5\) to \(x = 1\) is denoted by \(A\).
  2. By considering the set of three rectangles shown in the diagram, show that a lower bound for \(A\) is 0.388 .
  3. By considering another set of three rectangles, find an upper bound for \(A\), giving 3 decimal places in your answer. The area under the curve from \(x = 0\) to \(x = 0.5\) is denoted by \(B\).
  4. Draw a diagram to show rectangles which could be used to find lower and upper bounds for \(B\), using not more than three rectangles for each bound. (You are not required to find the bounds.)
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 2012 January Q1
4 marks Standard +0.8
1 Given that \(\mathrm { f } ( x ) = \ln ( \cos 3 x )\), find \(\mathrm { f } ^ { \prime } ( 0 )\) and \(\mathrm { f } ^ { \prime \prime } ( 0 )\). Hence show that the first term in the Maclaurin series for \(\mathrm { f } ( x )\) is \(a x ^ { 2 }\), where the value of \(a\) is to be found.
OCR FP2 2012 January Q2
5 marks Standard +0.3
2 By first completing the square in the denominator, find the exact value of $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { 4 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x$$
OCR FP2 2012 January Q3
7 marks Standard +0.3
3 Express \(\frac { 2 x ^ { 3 } + x + 12 } { ( 2 x - 1 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.
OCR FP2 2012 January Q4
9 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-2_662_1063_986_484} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { x } }\) for \(0 < x \leqslant 1\). A set of ( \(n - 1\) ) rectangles is drawn under the curve as shown.
  1. Explain why a lower bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\) can be expressed as $$\frac { 1 } { n } \left( \mathrm { e } ^ { - n } + \mathrm { e } ^ { - \frac { n } { 2 } } + \mathrm { e } ^ { - \frac { n } { 3 } } + \ldots + \mathrm { e } ^ { - \frac { n } { n - 1 } } \right)$$
  2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\).
  3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures.
  4. When \(n \geqslant N\), the difference between the upper and lower bounds is less than 0.001 . By expressing this difference in terms of \(n\), find the least possible value of \(N\).
OCR FP2 2012 January Q5
11 marks Challenging +1.2
5 It is given that \(\mathrm { f } ( x ) = x ^ { 3 } - k\), where \(k > 0\), and that \(\alpha\) is the real root of the equation \(\mathrm { f } ( x ) = 0\). Successive approximations to \(\alpha\), using the Newton-Raphson method, are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots\).
  1. Show that \(x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + k } { 3 x _ { n } ^ { 2 } }\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for \(\left| \alpha - x _ { 2 } \right|\) to be greater than \(\left| \alpha - x _ { 1 } \right|\). It is now given that \(k = 100\) and \(x _ { 1 } = 5\).
  3. Write down the exact value of \(\alpha\) and find \(x _ { 2 }\) and \(x _ { 3 }\) correct to 5 decimal places.
  4. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). By finding \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\), verify that \(e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }\).
OCR FP2 2012 January Q6
8 marks Challenging +1.2
6
  1. Prove that the derivative of \(\cos ^ { - 1 } x\) is \(- \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\). A curve has equation \(y = \cos ^ { - 1 } \left( 1 - x ^ { 2 } \right)\), for \(0 < x < \sqrt { 2 }\).
  2. Find and simplify \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and hence show that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = x \frac { \mathrm {~d} y } { \mathrm {~d} x }$$
  3. Given that \(y = \sinh ^ { - 1 } x\), prove that \(y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).
  4. It is given that \(x\) satisfies the equation \(\sinh ^ { - 1 } x - \cosh ^ { - 1 } x = \ln 2\). Use the logarithmic forms for \(\sinh ^ { - 1 } x\) and \(\cosh ^ { - 1 } x\) to show that $$\sqrt { x ^ { 2 } + 1 } - 2 \sqrt { x ^ { 2 } - 1 } = x$$ Hence, by squaring this equation, find the exact value of \(x\).
OCR FP2 2012 January Q8
9 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-4_606_915_219_557} The diagram shows two curves, \(C _ { 1 }\) and \(C _ { 2 }\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C _ { 1 }\) is \(r = \sqrt { 2 } \cos \theta\) and the polar equation of \(C _ { 2 }\) is \(r = \sqrt { 2 \sin 2 \theta }\). For both curves, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The value of \(\theta\) at \(P\) is \(\alpha\).
  1. Show that \(\tan \alpha = \frac { 1 } { 2 }\).
  2. Show that the area of the region common to \(C _ { 1 }\) and \(C _ { 2 }\), shaded in the diagram, is \(\frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \alpha\).
OCR FP2 2012 January Q9
11 marks Challenging +1.8
9
  1. Show that \(\tanh ( \ln n ) = \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 1 }\). It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { n } u \mathrm {~d} u\).
  2. Show that \(I _ { n } - I _ { n - 2 } = - \frac { 1 } { n - 1 } \left( \frac { 3 } { 5 } \right) ^ { n - 1 }\), for \(n \geqslant 2\).
  3. Find the value of \(I _ { 3 }\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants.
  4. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac { 1 } { 2 } \left( \frac { 3 } { 5 } \right) ^ { 2 } + \frac { 1 } { 4 } \left( \frac { 3 } { 5 } \right) ^ { 4 } + \frac { 1 } { 6 } \left( \frac { 3 } { 5 } \right) ^ { 6 } + \ldots .$$
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 $$x _ { n + 1 } = \ln \left( x _ { n } - 1 \right) + 3$$ (a) Using this iterative formula with \(x _ { 1 } = 4.15\), find \(\beta\) correct to 3 decimal places. Show all your working.
    (b) Explain with the aid of a sketch why this iterative formula will not converge to \(\alpha\) whatever initial value is taken.
  2. (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 \(A\) ?