Questions — OCR FP2 (173 questions)

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OCR FP2 2014 June Q7
11 marks Challenging +1.8
7 It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } x \mathrm {~d} x\).
  1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
  2. Explain why \(I _ { 2 n + 1 } < I _ { 2 n - 1 }\).
  3. It is given that \(I _ { 2 n + 1 } < I _ { 2 n } < I _ { 2 n - 1 }\). Take \(n = 5\) to find an interval within which the value of \(\pi\) lies.
OCR FP2 2014 June Q8
10 marks Standard +0.8
8 A curve has polar equation \(r = a ( 1 + \cos \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).
  1. Find the equation of the tangent at the pole.
  2. Sketch the curve.
  3. Find the area enclosed by the curve.
OCR FP2 2014 June Q9
12 marks Standard +0.3
9 The equation \(10 x - 8 \ln x = 28\) has a root \(\alpha\) in the interval [3,4]. The iteration \(x _ { n + 1 } = \mathrm { g } \left( x _ { n } \right)\), where \(\mathrm { g } ( x ) = 2.8 + 0.8 \ln x\) and \(x _ { 1 } = 3.8\), is to be used to find \(\alpha\).
  1. Find the value of \(\alpha\) correct to 5 decimal places. You should show the result of each step of the iteration to 6 decimal places.
  2. Illustrate this iteration by means of a sketch.
  3. The difference, \(\delta _ { r }\), between successive approximations is given by \(\delta _ { r } = x _ { r + 1 } - x _ { r }\). Find \(\delta _ { 3 }\).
  4. Given that \(\delta _ { n + 1 } \approx \mathrm {~g} ^ { \prime } ( \alpha ) \delta _ { n }\), for all positive integers \(n\), estimate the smallest value of \(n\) such that \(\delta _ { n } < 10 ^ { - 6 } \delta _ { 1 }\). \section*{OCR}
OCR FP2 2015 June Q1
3 marks Standard +0.8
1 By first expressing \(\tanh y\) in terms of exponentials, prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
OCR FP2 2015 June Q2
4 marks Standard +0.3
2 It is given that \(\mathrm { f } ( x ) = \ln ( 1 + \sin x )\). Using standard series, find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2015 June Q3
5 marks Standard +0.3
3 By first completing the square, find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } } \mathrm {~d} x\).
OCR FP2 2015 June Q4
9 marks Challenging +1.2
4 It is given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = n I _ { n - 1 } + k\) for \(n \geqslant 1\), where \(k\) is a constant to be determined.
  2. Find the exact value of \(I _ { 3 }\).
  3. Find the exact value of \(990 I _ { 8 } - I _ { 11 }\).
OCR FP2 2015 June Q5
9 marks Standard +0.8
5 It is given that \(y = \sin ^ { - 1 } 2 x\).
  1. Using the derivative of \(\sin ^ { - 1 } x\) given in the List of Formulae (MF1), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x }\).
  3. Hence show that \(\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } - 12 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0\).
  4. Using your results from parts (i), (ii) and (iii), find the Maclaurin series for \(\sin ^ { - 1 } 2 x\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2015 June Q6
12 marks Challenging +1.8
6 It is given that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - x - 1 = 0\) has three roots, one of which is positive.
  1. Show that the Newton-Raphson iterative formula for finding this root can be written $$x _ { n + 1 } = \frac { 6 x _ { n } ^ { 3 } + 5 x _ { n } ^ { 2 } + 1 } { 9 x _ { n } ^ { 2 } + 10 x _ { n } - 1 } .$$
  2. A sequence of iterates \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) which will find the positive root is such that the magnitude of the error in \(x _ { 2 }\) is greater than the magnitude of the error in \(x _ { 1 }\). On the graph given in the Printed Answer Book, mark a possible position for \(x _ { 1 }\).
  3. Apply the iterative formula in part (i) when the initial value is \(x _ { 1 } = - 1\). Describe the behaviour of the iterative sequence, illustrating your answer on the graph given in the Printed Answer Book.
  4. A sequence of approximations to the positive root is given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\). Successive differences \(x _ { r } - x _ { r - 1 } = d _ { r }\), where \(r \geqslant 2\), are such that \(d _ { r } \approx k \left( d _ { r - 1 } \right) ^ { 2 }\) where \(k\) is a constant. Show that \(d _ { 4 } \approx \frac { d _ { 3 } ^ { 3 } } { d _ { 2 } ^ { 2 } }\) and demonstrate this numerically when \(x _ { 1 } = 1\).
  5. Find the value of the positive root correct to 5 decimal places.
OCR FP2 2015 June Q7
10 marks Challenging +1.2
7 It is given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } - 25 } { ( x - 1 ) ( x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Write down the equations of the asymptotes of the curve \(y = \mathrm { f } ( x )\).
  3. Find the value of \(x\) where the graph of \(y = \mathrm { f } ( x )\) cuts the horizontal asymptote.
  4. Sketch the graph of \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR FP2 2015 June Q8
9 marks Standard +0.3
8 It is given that \(\mathrm { f } ( x ) = 2 \sinh x + 3 \cosh x\).
  1. Show that the curve \(y = \mathrm { f } ( x )\) has a stationary point at \(x = - \frac { 1 } { 2 } \ln 5\) and find the value of \(y\) at this point.
  2. Solve the equation \(\mathrm { f } ( x ) = 5\), giving your answers exactly. \section*{Question 9 begins on page 4.}
OCR FP2 2015 June Q9
11 marks Standard +0.8
9 The equation of a curve in polar coordinates is \(r = 2 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
  1. Sketch the curve.
  2. Find the area of the region enclosed by this curve.
  3. By expressing \(\sin 3 \theta\) in terms of \(\sin \theta\), show that a cartesian equation for the curve is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } y - 2 y ^ { 3 } .$$ \section*{END OF QUESTION PAPER}
OCR FP2 Q1
6 marks Standard +0.3
1
  1. Write down and simplify the first three non-zero terms of the Maclaurin series for \(\ln ( 1 + 3 x )\).
  2. Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.
OCR FP2 Q2
5 marks Standard +0.3
2 Use the Newton-Raphson method to find the root of the equation \(\mathrm { e } ^ { - x } = x\) which is close to \(x = 0.5\). Give the root correct to 3 decimal places.
OCR FP2 Q3
5 marks Moderate -0.3
3 Express \(\frac { x + 6 } { x \left( x ^ { 2 } + 2 \right) }\) in partial fractions.
OCR FP2 Q4
6 marks Standard +0.3
4 Answer the whole of this question on the insert provided.
\includegraphics[max width=\textwidth, alt={}]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-02_887_1273_1137_438}
The sketch shows the curve with equation \(y = \mathrm { F } ( x )\) and the line \(y = x\). The equation \(x = \mathrm { F } ( x )\) has roots \(x = \alpha\) and \(x = \beta\) as shown.
  1. Use the copy of the sketch on the insert to show how an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), with starting value \(x _ { 1 }\) such that \(0 < x _ { 1 } < \alpha\) as shown, converges to the root \(x = \alpha\).
  2. State what happens in the iteration in the following two cases.
    1. \(x _ { 1 }\) is chosen such that \(\alpha < x _ { 1 } < \beta\).
    2. \(x _ { 1 }\) is chosen such that \(x _ { 1 } > \beta\). \section*{Jan 2006} 4
    3. \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
    4. (a) \(\_\_\_\_\) (b) \(\_\_\_\_\) \section*{Jan 2006}
OCR FP2 Q5
8 marks Challenging +1.2
5
  1. Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
  2. Show that \(y\) cannot take values between - 3 and 1 .
OCR FP2 Q6
8 marks Standard +0.8
6
  1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
  2. Evaluate \(I _ { 3 }\), giving the answer in terms of e.
OCR FP2 Q7
9 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation \(y = \sqrt { x }\). A set of \(N\) rectangles of unit width is drawn, starting at \(x = 1\) and ending at \(x = N + 1\), where \(N\) is an integer (see diagram).
  1. By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
  2. By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
  3. Hence find, in terms of \(N\), limits between which \(\sum _ { r = 1 } ^ { N } \sqrt { r }\) lies. \section*{Jan 2006}
OCR FP2 Q8
13 marks Challenging +1.2
8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
  1. State the greatest value of \(r\) and the corresponding values of \(\theta\).
  2. Find the equations of the tangents at the pole.
  3. Find the exact area enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\).
  4. Find, in simplified form, the cartesian equation of the curve.
OCR FP2 Q9
12 marks Standard +0.3
9
  1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), prove that $$\sinh 2 x = 2 \sinh x \cosh x$$
  2. Show that the curve with equation $$y = \cosh 2 x - 6 \sinh x$$ has just one stationary point, and find its \(x\)-coordinate in logarithmic form. Determine the nature of the stationary point.
OCR FP2 2009 January Q1
6 marks Standard +0.3
  1. Write down and simplify the first three terms of the Maclaurin series for \(e^{2x}\). [2]
  2. Hence show that the Maclaurin series for $$\ln(e^{2x} + e^{-2x})$$ begins \(\ln a + bx^2\), where \(a\) and \(b\) are constants to be found. [4]
OCR FP2 2009 January Q2
12 marks Standard +0.8
It is given that \(\alpha\) is the only real root of the equation \(x^3 + 2x - 28 = 0\) and that \(1.8 < \alpha < 2\).
  1. The iteration \(x_{n+1} = \sqrt[3]{28 - 2x_n}\), with \(x_1 = 1.9\), is to be used to find \(\alpha\). Find the values of \(x_2\), \(x_3\) and \(x_4\), giving the answers correct to 7 decimal places. [3]
  2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). Given that \(\alpha = 1.891 574 9\), correct to 7 decimal places, evaluate \(\frac{e_3}{e_2}\) and \(\frac{e_4}{e_3}\). Comment on these values in relation to the gradient of the curve with equation \(y = \sqrt[3]{28 - 2x}\) at \(x = \alpha\). [3]
OCR FP2 2009 January Q3
7 marks Standard +0.3
  1. Prove that the derivative of \(\sin^{-1} x\) is \(\frac{1}{\sqrt{1-x^2}}\). [3]
  2. Given that $$\sin^{-1} 2x + \sin^{-1} y = \frac{1}{2}\pi,$$ find the exact value of \(\frac{dy}{dx}\) when \(x = \frac{1}{4}\). [4]
OCR FP2 2009 January Q4
6 marks Standard +0.8
  1. By means of a suitable substitution, show that $$\int \frac{x^2}{\sqrt{x^2-1}} dx$$ can be transformed to \(\int \cosh^2 \theta \, d\theta\). [2]
  2. Hence show that \(\int \frac{x^2}{\sqrt{x^2-1}} dx = \frac{1}{2}\sqrt{x^2-1} + \frac{1}{2}\cosh^{-1} x + c\). [4]