Questions - OCR FP2 - A-Level Maths

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 FP2 2009 January Q1

Write down and simplify the first three terms of the Maclaurin series for $\mathrm { e } ^ { 2 x }$.
Hence show that the Maclaurin series for $$\ln \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)$$ begins $\ln a + b x ^ { 2 }$, where $a$ and $b$ are constants to be found.

OCR FP2 2009 January Q2

2 It is given that $\alpha$ is the only real root of the equation $x ^ { 5 } + 2 x - 28 = 0$ and that $1.8 < \alpha < 2$.

The iteration $x _ { n + 1 } = \sqrt [ 5 ] { 28 - 2 x _ { n } }$, 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.
The error $e _ { n }$ is defined by $e _ { n } = \alpha - x _ { n }$. Given that $\alpha = 1.8915749$, 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 $y = \sqrt [ 5 ] { 28 - 2 x }$ at $x = \alpha$.
Prove that the derivative of $\sin ^ { - 1 } x$ is $\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$.
Given that $$\sin ^ { - 1 } 2 x + \sin ^ { - 1 } y = \frac { 1 } { 2 } \pi$$ find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = \frac { 1 } { 4 }$.
By means of a suitable substitution, show that $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x$$ can be transformed to $\int \cosh ^ { 2 } \theta \mathrm {~d} \theta$.
Hence show that $\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } x \sqrt { x ^ { 2 } - 1 } + \frac { 1 } { 2 } \cosh ^ { - 1 } x + c$.

OCR FP2 2009 January Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-3_661_734_267_703} The diagram shows the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 4.36$$ The curve has turning points at $x = 1$ and $x = 2$ and crosses the $x$-axis at $x = \alpha , x = \beta$ and $x = \gamma$, where $0 < \alpha < \beta < \gamma$.

The Newton-Raphson method is to be used to find the roots of the equation $\mathrm { f } ( x ) = 0$, with $x _ { 1 } = k$.
(a) To which root, if any, would successive approximations converge in each of the cases $k < 0$ and $k = 1$ ?
(b) What happens if $1 < k < 2$ ?
Sketch the curve with equation $y ^ { 2 } = \mathrm { f } ( x )$. State the coordinates of the points where the curve crosses the $x$-axis and the coordinates of any turning points.
Using the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that $$1 + 2 \sinh ^ { 2 } x \equiv \cosh 2 x .$$
Solve the equation $$\cosh 2 x - 5 \sinh x = 4$$ giving your answers in logarithmic form.

OCR FP2 2009 January Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-4_511_609_264_769} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2 \cos \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi .$$ The points $P , Q , R$ and $S$ on the curve are such that the straight lines $P O R$ and $Q O S$ are perpendicular, where $O$ is the pole. The point $P$ has polar coordinates ( $r , \alpha$ ).

Show that $O P + O Q + O R + O S = k$, where $k$ is a constant to be found.
Given that $\alpha = \frac { 1 } { 4 } \pi$, find the exact area bounded by the curve and the lines $O P$ and $O Q$ (shaded in the diagram).

OCR FP2 2009 January Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-5_579_1363_267_390} The diagram shows the curve with equation $y = \frac { 1 } { x + 1 }$. A set of $n$ rectangles of unit width is drawn, starting at $x = 0$ and ending at $x = n$, where $n$ is an integer.

By considering the areas of these rectangles, explain why $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n + 1 } < \ln ( n + 1 ) .$$
By considering the areas of another set of rectangles, show that $$1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } > \ln ( n + 1 ) .$$
Hence show that $$\ln ( n + 1 ) + \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n + 1 } \frac { 1 } { r } < \ln ( n + 1 ) + 1$$
State, with a reason, whether $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r }$ is convergent.

OCR FP2 2009 January Q9

9 A curve has equation $$y = \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) }$$ where $a$ is a positive constant.

Explain why the curve has no asymptotes parallel to the $y$-axis.
Find, in terms of $a$, the set of values of $y$ for which there are no points on the curve.
Find the exact value of $\int _ { a } ^ { 2 a } \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) } \mathrm { d } x$, showing that it is independent of $a$.

Questions — OCR FP2 (168 questions)

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