Questions FP2 (1279 questions)

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OCR MEI FP2 2010 June Q4
18 marks Challenging +1.2
4
  1. Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for \(\cosh 2 x\).
  2. Sketch the curve with equation \(y = \cosh x - 1\). The region bounded by this curve, the \(x\)-axis, and the line \(x = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
  3. Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
    Determine, in exact logarithmic form, the \(x\)-coordinate of the stationary point.
OCR MEI FP2 2010 June Q5
18 marks Challenging +1.2
5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).
  1. Firstly consider cases in which \(k\) is a positive even integer.
    (A) State the shape of the curve when \(k = 2\).
    (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
    (C) Describe the shape that the curve tends to as \(k\) becomes very large.
    (D) State the range of possible values of \(x\) and \(y\).
  2. Now consider cases in which \(k\) is a positive odd integer.
    (A) Explain why \(x\) and \(y\) may take any value.
    (B) State the shape of the curve when \(k = 1\).
    (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
    (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
  3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
  4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
    (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
    (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.
OCR FP2 2007 January Q1
5 marks Standard +0.3
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
6 marks Standard +0.3
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
6 marks Standard +0.3
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
9 marks Standard +0.3
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
9 marks Challenging +1.2
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
8 marks Challenging +1.2
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
9 marks Challenging +1.2
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
9 marks Standard +0.8
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
11 marks Challenging +1.2
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.
OCR FP2 2008 January Q1
6 marks Standard +0.3
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
8 marks Standard +0.8
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
9 marks Standard +0.8
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
8 marks Standard +0.8
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
9 marks Challenging +1.3
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
10 marks Standard +0.8
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
11 marks Standard +0.8
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 \(\int \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x\).
OCR FP2 2006 June Q1
3 marks Moderate -0.5
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
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 $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$