Questions — OCR FP2 (168 questions)

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OCR FP2 2007 January Q1
1 It is given that \(\mathrm { f } ( x ) = \ln ( 3 + x )\).
  1. Find the exact values of \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\), and show that \(f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 9 }\).
  2. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\), given that \(- 3 < x \leqslant 3\).
OCR FP2 2007 January Q2
2 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \tan ^ { - 1 } x\).
  1. Show by calculation that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval \(0.8 < x < 0.9\).
  2. Use the Newton-Raphson method, with a first approximation 0.8, to find the next approximation to this root. Give your answer correct to 3 decimal places.
OCR FP2 2007 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{268b605f-eb86-40df-946a-210da1355e83-2_686_967_998_589} The diagram shows the curve with equation \(y = \mathrm { e } ^ { x ^ { 2 } }\), for \(0 \leqslant x \leqslant 1\). The region under the curve between these limits is divided into four strips of equal width. The area of this region under the curve is \(A\).
  1. By considering the set of rectangles indicated in the diagram, show that an upper bound for \(A\) is 1.71 .
  2. By considering an appropriate set of four rectangles, find a lower bound for \(A\).
OCR FP2 2007 January Q4
4
  1. On separate diagrams, sketch the graphs of \(y = \sinh x\) and \(y = \operatorname { cosech } x\).
  2. Show that \(\operatorname { cosech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }\), and hence, using the substitution \(u = \mathrm { e } ^ { x }\), find \(\int \operatorname { cosech } x \mathrm {~d} x\).
OCR FP2 2007 January Q5
5 It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \cos x \mathrm {~d} x$$
  1. Prove that, for \(n \geqslant 2\), $$I _ { n } = \left( \frac { 1 } { 2 } \pi \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } .$$
  2. Find \(I _ { 4 }\) in terms of \(\pi\).
OCR FP2 2007 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{268b605f-eb86-40df-946a-210da1355e83-3_716_1431_852_356} The diagram shows the curve with equation \(y = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } }\), where \(a\) is a positive constant.
  1. Find the equations of the asymptotes of the curve.
  2. Sketch the curve with equation $$y ^ { 2 } = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } } .$$ State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.
OCR FP2 2007 January Q7
7
  1. Express \(\frac { 1 - t ^ { 2 } } { t ^ { 2 } \left( 1 + t ^ { 2 } \right) }\) in partial fractions.
  2. Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 - \cos x } \mathrm {~d} x = \sqrt { 3 } - 1 - \frac { 1 } { 6 } \pi$$
OCR FP2 2007 January Q8
8
  1. Define tanh \(y\) in terms of \(\mathrm { e } ^ { y }\) and \(\mathrm { e } ^ { - y }\).
  2. Given that \(y = \tanh ^ { - 1 } x\), where \(- 1 < x < 1\), prove that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
  3. Find the exact solution of the equation \(3 \cosh x = 4 \sinh x\), giving the answer in terms of a logarithm.
  4. Solve the equation $$\tanh ^ { - 1 } x + \ln ( 1 - x ) = \ln \left( \frac { 4 } { 5 } \right)$$
OCR FP2 2007 January Q9
9 The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$
  1. Sketch the curve.
  2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
  3. Find a cartesian equation of the curve. \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 January Q1
1 It is given that \(\mathrm { f } ( x ) = \ln ( 1 + \cos x )\).
  1. Find the exact values of \(f ( 0 ) , f ^ { \prime } ( 0 )\) and \(f ^ { \prime \prime } ( 0 )\).
  2. Hence find the first two non-zero terms of the Maclaurin series for \(\mathrm { f } ( x )\).
OCR FP2 2008 January Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-2_577_700_577_721} The diagram shows parts of the curves with equations \(y = \cos ^ { - 1 } x\) and \(y = \frac { 1 } { 2 } \sin ^ { - 1 } x\), and their point of intersection \(P\).
  1. Verify that the coordinates of \(P\) are \(\left( \frac { 1 } { 2 } \sqrt { 3 } , \frac { 1 } { 6 } \pi \right)\).
  2. Find the gradient of each curve at \(P\).
OCR FP2 2008 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-2_643_787_1621_680} The diagram shows the curve with equation \(y = \sqrt { 1 + x ^ { 3 } }\), for \(2 \leqslant x \leqslant 3\). The region under the curve between these limits has area \(A\).
  1. Explain why \(3 < A < \sqrt { 28 }\).
  2. The region is divided into 5 strips, each of width 0.2 . By using suitable rectangles, find improved lower and upper bounds between which \(A\) lies. Give your answers correct to 3 significant figures.
OCR FP2 2008 January Q4
4 The equation of a curve, in polar coordinates, is $$r = 1 + 2 \sec \theta , \quad \text { for } - \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$$
  1. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 6 } \pi\). [The result \(\int \sec \theta \mathrm { d } \theta = \ln | \sec \theta + \tan \theta |\) may be assumed.]
  2. Show that a cartesian equation of the curve is \(( x - 2 ) \sqrt { x ^ { 2 } + y ^ { 2 } } = x\).
OCR FP2 2008 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-3_606_890_815_630} The diagram shows the curve with equation \(y = x \mathrm { e } ^ { - x } + 1\). The curve crosses the \(x\)-axis at \(x = \alpha\).
  1. Use differentiation to show that the \(x\)-coordinate of the stationary point is 1 .
    \(\alpha\) is to be found using the Newton-Raphson method, with \(\mathrm { f } ( x ) = x \mathrm { e } ^ { - x } + 1\).
  2. Explain why this method will not converge to \(\alpha\) if an initial approximation \(x _ { 1 }\) is chosen such that \(x _ { 1 } > 1\).
  3. Use this method, with a first approximation \(x _ { 1 } = 0\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Find \(\alpha\), correct to 3 decimal places.
OCR FP2 2008 January Q6
6 The equation of a curve is \(y = \frac { 2 x ^ { 2 } - 11 x - 6 } { x - 1 }\).
  1. Find the equations of the asymptotes of the curve.
  2. Show that \(y\) takes all real values.
OCR FP2 2008 January Q7
7 It is given that, for integers \(n \geqslant 1\), $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x$$
  1. Use integration by parts to show that \(I _ { n } = 2 ^ { - n } + 2 n \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) ^ { n + 1 } } \mathrm {~d} x\).
  2. Show that \(2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }\).
  3. Find \(I _ { 2 }\) in terms of \(\pi\).
OCR FP2 2008 January Q8
8
  1. By using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$\sinh ^ { 3 } x = \frac { 1 } { 4 } \sinh 3 x - \frac { 3 } { 4 } \sinh x$$
  2. Find the range of values of the constant \(k\) for which the equation $$\sinh 3 x = k \sinh x$$ has real solutions other than \(x = 0\).
  3. Given that \(k = 4\), solve the equation in part (ii), giving the non-zero answers in logarithmic form.
OCR FP2 2008 January Q9
9
  1. Prove that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cosh ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
  2. Hence, or otherwise, find \(\int \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\).
  3. By means of a suitable substitution, find \(\int \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x\). \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 (OCR) 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 2006 June Q1
1 Find the first three non-zero terms of the Maclaurin series for $$( 1 + x ) \sin x$$ simplifying the coefficients.
OCR FP2 2006 June Q2
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
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
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
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
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
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 \(\theta = \frac { 1 } { 4 } \pi\).