Questions FP2 - A-Level Maths

Express the polar equation of $C$ in the form $r = \mathrm { f } ( \theta )$ and state the limits between which $\theta$ lies. The line $\theta = \alpha$ is a line of symmetry of $C$.
Find and simplify an expression for $\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)$ and hence explain why $\alpha = \frac { 1 } { 4 } \pi$.
Find the value of $r$ when $\theta = \frac { 1 } { 4 } \pi$.
Sketch the curve $C$.

OCR FP2 2011 June Q5

Prove that, if $y = \sin ^ { - 1 } x$, then $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$.
Find the Maclaurin series for $\sin ^ { - 1 } x$, up to and including the term in $x ^ { 3 }$.
Use the result of part (ii) and the Maclaurin series for $\ln ( 1 + x )$ to find the Maclaurin series for $\left( \sin ^ { - 1 } x \right) \ln ( 1 + x )$, up to and including the term in $x ^ { 4 }$.

OCR FP2 2011 June Q6

6 It is given that $I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } ( 1 - x ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x$, for $n \geqslant 0$.

Show that $I _ { n } = \frac { 2 n } { 2 n + 5 } I _ { n - 1 }$, for $n \geqslant 1$.
Hence find the exact value of $I _ { 3 }$.

OCR FP2 2011 June Q7

Sketch the graph of $y = \tanh x$ and state the value of the gradient when $x = 0$. On the same axes, sketch the graph of $y = \tanh ^ { - 1 } x$. Label each curve and give the equations of the asymptotes.
Find $\int _ { 0 } ^ { k } \tanh x \mathrm {~d} x$, where $k > 0$.
Deduce, or show otherwise, that $\int _ { 0 } ^ { \tanh k } \tanh ^ { - 1 } x \mathrm {~d} x = k \tanh k - \ln ( \cosh k )$.

OCR FP2 2011 June Q8

Use the substitution $x = \cosh ^ { 2 } u$ to find $\int \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x$, giving your answer in the form $\mathrm { f } ( x ) + \ln ( \mathrm { g } ( x ) )$.
\includegraphics[max width=\textwidth, alt={}, center]{d25d17c4-a87c-4dcf-900c-400086af6610-3_693_1041_927_593}
Hence calculate the exact area of the region between the curve $y = \sqrt { \frac { x } { x - 1 } }$, the $x$-axis and the lines $x = 1$ and $x = 4$ (see diagram).
What can you say about the volume of the solid of revolution obtained when the region defined in part (ii) is rotated completely about the $x$-axis? Justify your answer.

OCR FP2 2016 June Q1

By first expanding $\left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 3 }$, or otherwise, show that $\cosh 3 x \equiv 4 \cosh ^ { 3 } x - 3 \cosh x$.
Solve the equation $\cosh 3 x = 6 \cosh x$, giving your answers in exact logarithmic form.

OCR FP2 2016 June Q2

2 It is given that $\mathrm { f } ( x ) = \frac { x ( x - 1 ) } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) }$. Express $\mathrm { f } ( x )$ in partial fractions and hence find the exact value of $\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$.

OCR FP2 2016 June Q3

3 The diagram shows the curve $y = \mathrm { f } ( x )$. Points $A , B , C$ and $D$ on the curve have coordinates ( $- 1,0 ) , ( 2,0 )$, $( 5,0 )$ and $( 0,2 )$ respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a31997f4-7890-42c1-9725-1b7058e8741f-2_593_1221_1041_406} On the copy of this diagram in the Printed Answer Book, sketch the curve $y ^ { 2 } = \mathrm { f } ( x )$, giving the coordinates of the points where the curve crosses the axes.

OCR FP2 2016 June Q4

4 You are given the equation $( 2 x - 1 ) ^ { 2 } - \mathrm { e } ^ { x } = 0$.

Verify that 0 is a root of the equation. There are also two other roots, $\alpha$ and $\beta$, where $0 < \alpha < \beta$.
The iterative formula $x _ { r + 1 } = \ln \left( 2 x _ { r } - 1 \right) ^ { 2 }$ is to be used to find a root of the equation.
(a) Sketch the line $y = x$ and the curve $y = \ln ( 2 x - 1 ) ^ { 2 }$ on the same axes, showing the roots $0 , \alpha$ and $\beta$.
(b) By drawing a 'staircase' diagram on your sketch, starting with a value of $x$ that is between $\alpha$ and $\beta$, show that this iteration does not converge to $\alpha$.
(c) Using this iterative formula with $x _ { 1 } = 3.75$, find the value of $\beta$ correct to 3 decimal places.
Using the Newton-Raphson method with $x _ { 1 } = 1.6$, find the root $\alpha$ of the equation $( 2 x - 1 ) ^ { 2 } - \mathrm { e } ^ { x } = 0$ correct to 5 significant figures. Show the result of each iteration.

OCR FP2 2016 June Q5

5 It is given that $y = \tan ^ { - 1 } 2 x$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 0$.
Find the Maclaurin series for $y$ up to and including the term in $x ^ { 3 }$. Show all your working.
The result in part (ii), together with the value $x = \frac { 1 } { 2 }$, is used to find an estimate for $\pi$. Show that this estimate is only correct to 1 significant figure.

OCR FP2 2016 June Q6

6 The equation of a curve in polar coordinates is $r = \sin 5 \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 5 } \pi$.

Sketch the curve and write down the equations of the tangents at the pole.
The line of symmetry meets the curve at the pole and at one other point $A$. Find the equation of the line of symmetry and the cartesian coordinates of $A$.
Find the area of the region enclosed by this curve.

OCR FP2 2016 June Q7

By using a set of rectangles of unit width to approximate an area under the curve $y = \frac { 1 } { x }$, show that $\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x }$ is infinite.
By using a set of rectangles of unit width to approximate an area under the curve $y = \frac { 1 } { x ^ { 2 } }$, find an upper limit for the series $\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } }$.

OCR FP2 2016 June Q8

8 It is given that $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$ where $n$ is a positive integer.

By writing $\sec ^ { n } x = \sec ^ { n - 2 } x \sec ^ { 2 } x$, or otherwise, show that $$( n - 1 ) I _ { n } = ( \sqrt { 2 } ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 } \text { for } n > 1 .$$
Show that $I _ { 8 } = \frac { 96 } { 35 }$.
Prove by induction that $I _ { 2 n }$ is rational for all values of $n > 1$. \section*{END OF QUESTION PAPER}

OCR FP2 Specimen Q1

Starting from the definition of $\cosh x$ in terms of $\mathrm { e } ^ { x }$, show that $\cosh 2 x = 2 \cosh ^ { 2 } x - 1$.
Given that $\cosh 2 x = k$, where $k > 1$, express each of $\cosh x$ and $\sinh x$ in terms of $k$.

OCR FP2 Specimen Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-2_728_951_486_534} The diagram shows the graph of $$y = \frac { 2 x ^ { 2 } + 3 x + 3 } { x + 1 }$$

Find the equations of the asymptotes of the curve.
Prove that the values of $y$ between which there are no points on the curve are - 5 and 3 .

OCR FP2 Specimen Q3

Find the first three terms of the Maclaurin series for $\ln ( 2 + x )$.
Write down the first three terms of the series for $\ln ( 2 - x )$, and hence show that, if $x$ is small, then $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$

OCR FP2 Specimen Q4

4 The equation of a curve, in polar coordinates, is $$r = 2 \cos 2 \theta \quad ( - \pi < \theta \leqslant \pi ) .$$

Find the values of $\theta$ which give the directions of the tangents at the pole. One loop of the curve is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_362_720_653_708}
Find the exact value of the area of the region enclosed by the loop.

OCR FP2 Specimen Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_444_999_1258_539} The diagram shows the curve $y = \frac { 1 } { x + 1 }$ together with four rectangles of unit width.

Explain how the diagram shows that $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } < \int _ { 0 } ^ { 4 } \frac { 1 } { x + 1 } \mathrm {~d} x$$ The curve $y = \frac { 1 } { x + 2 }$ passes through the top left-hand corner of each of the four rectangles shown.
By considering the rectangles in relation to this curve, write down a second inequality involving $\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 }$ and a definite integral.
By considering a suitable range of integration and corresponding rectangles, show that $$\ln ( 500.5 ) < \sum _ { r = 2 } ^ { 1000 } \frac { 1 } { r } < \ln ( 1000 ) .$$

OCR FP2 Specimen Q6

Given that $I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \sqrt { } ( 1 - x ) \mathrm { d } x$, prove that, for $n \geqslant 1$, $$( 2 n + 3 ) I _ { n } = 2 n I _ { n - 1 } .$$
Hence find the exact value of $I _ { 2 }$.

OCR FP2 Specimen Q7

7 The curve with equation $$y = \frac { x } { \cosh x }$$ has one stationary point for $x > 0$.

Show that the $x$-coordinate of this stationary point satisfies the equation $x \tanh x - 1 = 0$. The positive root of the equation $x \tanh x - 1 = 0$ is denoted by $\alpha$.
Draw a sketch showing (for positive values of $x$ ) the graph of $y = \tanh x$ and its asymptote, and the graph of $y = \frac { 1 } { x }$. Explain how you can deduce from your sketch that $\alpha > 1$.
Use the Newton-Raphson method, taking first approximation $x _ { 1 } = 1$, to find further approximations $x _ { 2 }$ and $x _ { 3 }$ for $\alpha$.
By considering the approximate errors in $x _ { 1 }$ and $x _ { 2 }$, estimate the error in $x _ { 3 }$.

OCR FP2 Specimen Q8

Use the substitution $t = \tan \frac { 1 } { 2 } x$ to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x = 2 \sqrt { } 2 \int _ { 0 } ^ { 1 } \frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t$$
Express $\frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) }$ in partial fractions.
Hence find $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x$, expressing your answer in an exact form.

OCR MEI FP2 Q5

5 A curve has equation $y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }$, where $k$ is a positive constant and $k \neq 2$.

Find the equations of the three asymptotes.
Use your graphical calculator to obtain rough sketches of the curve in the two separate cases $k < 2$ and $k > 2$.
In the case $k < 2$, your sketch may not show clearly the shape of the curve near $x = 0$. Use calculus to show that the curve has a minimum point when $x = 0$.
In the case $k > 2$, your sketch may not show clearly how the curve approaches its asymptote as $x \rightarrow + \infty$. Show algebraically that the curve crosses this asymptote.
Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases $k < 2$ and $k > 2$. These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{MEI STRUCTURED MATHEMATICS} Further Methods for Advanced Mathematics (FP2)
Tuesday

OCR MEI FP2 2006 January Q1

A curve has polar equation $r = a \cos 3 \theta$ for $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, where $a$ is a positive constant.
1. Sketch the curve, using a continuous line for sections where $r > 0$ and a broken line for sections where $r < 0$.
2. Find the area enclosed by one of the loops.
Find the exact value of $\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$.
Use a trigonometric substitution to find $\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + 3 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$.

OCR MEI FP2 2006 January Q2

2 In this question, $\theta$ is a real number with $0 < \theta < \frac { 1 } { 6 } \pi$, and $w = \frac { 1 } { 2 } \mathrm { e } ^ { 3 \mathrm { j } \theta }$.

State the modulus and argument of each of the complex numbers $$w , \quad w ^ { * } \quad \text { and } \quad \mathrm { j } w .$$ Illustrate these three complex numbers on an Argand diagram.
Show that $( 1 + w ) \left( 1 + w ^ { * } \right) = \frac { 5 } { 4 } + \cos 3 \theta$. Infinite series $C$ and $S$ are defined by $$\begin{aligned} & C = \cos 2 \theta - \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 8 \theta - \frac { 1 } { 8 } \cos 11 \theta + \ldots
& S = \sin 2 \theta - \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 8 \theta - \frac { 1 } { 8 } \sin 11 \theta + \ldots \end{aligned}$$
Show that $C = \frac { 4 \cos 2 \theta + 2 \cos \theta } { 5 + 4 \cos 3 \theta }$, and find a similar expression for $S$.

OCR MEI FP2 2006 January Q3

3 The matrix $\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3
- 2 & - 3 & 6
2 & 2 & - 4 \end{array} \right)$.

Show that the characteristic equation for $\mathbf { M }$ is $\lambda ^ { 3 } + 6 \lambda ^ { 2 } - 9 \lambda - 14 = 0$.
Show that - 1 is an eigenvalue of $\mathbf { M }$, and find the other two eigenvalues.
Find an eigenvector corresponding to the eigenvalue - 1 .
Verify that $\left( \begin{array} { l } 3
0
1 \end{array} \right)$ and $\left( \begin{array} { r } 0
3
- 2 \end{array} \right)$ are eigenvectors of $\mathbf { M }$.
Write down a matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { M } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.
Use the Cayley-Hamilton theorem to express $\mathbf { M } ^ { - 1 }$ in the form $a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }$. Section B (18 marks)

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