Questions - OCR FP2 - A-Level Maths

The iteration $x _ { n + 1 } = \sqrt { \sin x _ { n } }$, with $x _ { 1 } = 0.875$, is to be used to find a real root, $\alpha$, of the equation $\mathrm { f } ( x ) = 0$. Find $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving the answers correct to 6 decimal places.
The error $e _ { n }$ is defined by $e _ { n } = \alpha - x _ { n }$. Given that $\alpha = 0.876726$, correct to 6 decimal places, find $e _ { 3 }$ and $e _ { 4 }$. Given that $\mathrm { g } ( x ) = \sqrt { \sin x }$, use $e _ { 3 }$ and $e _ { 4 }$ to estimate $\mathrm { g } ^ { \prime } ( \alpha )$.

OCR FP2 2010 January Q2

2 It is given that $\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )$.

Find $\mathrm { f } ( 0 )$ and $\mathrm { f } ^ { \prime } ( 0 )$, and show that $\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }$.
Hence find the Maclaurin series for $\mathrm { f } ( x )$ up to and including the term in $x ^ { 2 }$.

OCR FP2 2010 January Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-2_597_1006_973_571} A curve with no stationary points has equation $y = \mathrm { f } ( x )$. The equation $\mathrm { f } ( x ) = 0$ has one real root $\alpha$, and the Newton-Raphson method is to be used to find $\alpha$. The tangent to the curve at the point $\left( x _ { 1 } , \mathrm { f } \left( x _ { 1 } \right) \right)$ meets the $x$-axis where $x = x _ { 2 }$ (see diagram).

Show that $x _ { 2 } = x _ { 1 } - \frac { \mathrm { f } \left( x _ { 1 } \right) } { \mathrm { f } ^ { \prime } \left( x _ { 1 } \right) }$.
Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation $x = x _ { 1 }$, gives a sequence of approximations approaching $\alpha$.
Use the Newton-Raphson method, with a first approximation of 1 , to find a second approximation to the root of $x ^ { 2 } - 2 \sinh x + 2 = 0$.

OCR FP2 2010 January Q4

4 The equation of a curve, in polar coordinates, is $$r = \mathrm { e } ^ { - 2 \theta } , \quad \text { for } 0 \leqslant \theta \leqslant \pi .$$

Sketch the curve, stating the polar coordinates of the point at which $r$ takes its greatest value.
The pole is $O$ and points $P$ and $Q$, with polar coordinates ( $r _ { 1 } , \theta _ { 1 }$ ) and ( $r _ { 2 } , \theta _ { 2 }$ ) respectively, lie on the curve. Given that $\theta _ { 2 } > \theta _ { 1 }$, show that the area of the region enclosed by the curve and the lines $O P$ and $O Q$ can be expressed as $k \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right)$, where $k$ is a constant to be found.

OCR FP2 2010 January Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-3_591_1131_986_507} The diagram shows the curve with equation $y = \sqrt [ 3 ] { x }$, together with a set of $n$ rectangles of unit width.

By considering the areas of these rectangles, explain why $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } > \int _ { 0 } ^ { n } \sqrt [ 3 ] { x } \mathrm {~d} x$$
By drawing another set of rectangles and considering their areas, show that $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } < \int _ { 1 } ^ { n + 1 } \sqrt [ 3 ] { x } \mathrm {~d} x$$
Hence find an approximation to $\sum _ { n = 1 } ^ { 100 } \sqrt [ 3 ] { n }$, giving your answer correct to 2 significant figures.

OCR FP2 2010 January Q8

8 The equation of a curve is $$y = \frac { k x } { ( x - 1 ) ^ { 2 } } ,$$ where $k$ is a positive constant.

Write down the equations of the asymptotes of the curve.
Show that $y \geqslant - \frac { 1 } { 4 } k$.
Show that the $x$-coordinate of the stationary point of the curve is independent of $k$, and sketch the curve.

OCR FP2 2010 January Q9

Given that $y = \tanh ^ { - 1 } x$, for $- 1 < x < 1$, prove that $y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$.
It is given that $\mathrm { f } ( x ) = a \cosh x - b \sinh x$, where $a$ and $b$ are positive constants.
(a) Given that $b \geqslant a$, show that the curve with equation $y = \mathrm { f } ( x )$ has no stationary points.
(b) In the case where $a > 1$ and $b = 1$, show that $\mathrm { f } ( x )$ has a minimum value of $\sqrt { a ^ { 2 } - 1 }$.

OCR FP2 2011 January Q1

1 Use the substitution $t = \tan \frac { 1 } { 2 } x$ to find $\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x$.

OCR FP2 2011 January Q2

2 It is given that $\mathrm { f } ( x ) = \tanh ^ { - 1 } x$.

Show that $\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }$.
Hence find the Maclaurin series for $\mathrm { f } ( x )$, up to and including the term in $x ^ { 3 }$.

OCR FP2 2011 January Q3

3 The function f is defined by $\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }$, for $x \in \mathbb { R }$ and $a > 0$.

For the curve with equation $y = \mathrm { f } ( x )$,
(a) write down the equation of the asymptote,
(b) find the range of values that $y$ can take.
For the curve with equation $y ^ { 2 } = \mathrm { f } ( x )$, write down
(a) the equation of the line of symmetry,
(b) the maximum and minimum values of $y$,
(c) the set of values of $x$ for which the curve is defined.

OCR FP2 2011 January Q4

Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \sinh ^ { 4 } x \equiv \cosh 4 x - 4 \cosh 2 x + 3$$
Solve the equation $$\cosh 4 x - 3 \cosh 2 x + 1 = 0$$ giving your answer(s) in logarithmic form.

OCR FP2 2011 January Q5

5 The equation $$x ^ { 3 } - 5 x + 3 = 0$$ may be solved by the Newton-Raphson method. Successive approximations to a root are denoted by $x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots$.

Show that the Newton-Raphson formula can be written in the form $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$, where $$\mathrm { F } ( x ) = \frac { 2 x ^ { 3 } - 3 } { 3 x ^ { 2 } - 5 }$$
Find $\mathrm { F } ^ { \prime } ( x )$ and hence verify that $\mathrm { F } ^ { \prime } ( \alpha ) = 0$, where $\alpha$ is any one of the roots of equation (A).
Use the Newton-Raphson method to find the root of equation (A) which is close to 2 . Write down sufficient approximations to find the root correct to 4 decimal places.

OCR FP2 2011 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{debf6581-25ff-4692-bdfb-154675a3cdb0-3_608_1134_258_504} The diagram shows the curve $y = \mathrm { f } ( x )$, defined by $$f ( x ) = \begin{cases} x ^ { x } & \text { for } 0 < x \leqslant 1 ,
1 & \text { for } x = 0 . \end{cases}$$

By first taking logarithms, show that the curve has a stationary point at $x = \mathrm { e } ^ { - 1 }$. The area under the curve from $x = 0.5$ to $x = 1$ is denoted by $A$.
By considering the set of three rectangles shown in the diagram, show that a lower bound for $A$ is 0.388 .
By considering another set of three rectangles, find an upper bound for $A$, giving 3 decimal places in your answer. The area under the curve from $x = 0$ to $x = 0.5$ is denoted by $B$.
Draw a diagram to show rectangles which could be used to find lower and upper bounds for $B$, using not more than three rectangles for each bound. (You are not required to find the bounds.)

OCR FP2 2011 January Q7

7 A curve has polar equation $r = 1 + \cos 3 \theta$, for $- \pi < \theta \leqslant \pi$.

Show that the line $\theta = 0$ is a line of symmetry.
Find the equations of the tangents at the pole.
Find the exact value of the area of the region enclosed by the curve between $\theta = - \frac { 1 } { 3 } \pi$ and $\theta = \frac { 1 } { 3 } \pi$.

OCR FP2 2011 January Q8

Without using a calculator, show that $\sinh \left( \cosh ^ { - 1 } 2 \right) = \sqrt { 3 }$.
It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { \beta } \cosh ^ { n } x \mathrm {~d} x , \quad \text { where } \beta = \cosh ^ { - 1 } 2$$ Show that $n I _ { n } = 2 ^ { n - 1 } \sqrt { 3 } + ( n - 1 ) I _ { n - 2 }$, for $n \geqslant 2$.
Evaluate $I _ { 5 }$, giving your answer in the form $k \sqrt { 3 }$.

OCR FP2 2012 January Q1

1 Given that $\mathrm { f } ( x ) = \ln ( \cos 3 x )$, find $\mathrm { f } ^ { \prime } ( 0 )$ and $\mathrm { f } ^ { \prime \prime } ( 0 )$. Hence show that the first term in the Maclaurin series for $\mathrm { f } ( x )$ is $a x ^ { 2 }$, where the value of $a$ is to be found.

OCR FP2 2012 January Q2

2 By first completing the square in the denominator, find the exact value of $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { 4 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x$$

OCR FP2 2012 January Q3

3 Express $\frac { 2 x ^ { 3 } + x + 12 } { ( 2 x - 1 ) \left( x ^ { 2 } + 4 \right) }$ in partial fractions.

OCR FP2 2012 January Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-2_662_1063_986_484} The diagram shows the curve $y = \mathrm { e } ^ { - \frac { 1 } { x } }$ for $0 < x \leqslant 1$. A set of ( $n - 1$ ) rectangles is drawn under the curve as shown.

Explain why a lower bound for $\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x$ can be expressed as $$\frac { 1 } { n } \left( \mathrm { e } ^ { - n } + \mathrm { e } ^ { - \frac { n } { 2 } } + \mathrm { e } ^ { - \frac { n } { 3 } } + \ldots + \mathrm { e } ^ { - \frac { n } { n - 1 } } \right)$$
Using a set of $n$ rectangles, write down a similar expression for an upper bound for $\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x$.
Evaluate these bounds in the case $n = 4$, giving your answers correct to 3 significant figures.
When $n \geqslant N$, the difference between the upper and lower bounds is less than 0.001 . By expressing this difference in terms of $n$, find the least possible value of $N$.

OCR FP2 2012 January Q5

5 It is given that $\mathrm { f } ( x ) = x ^ { 3 } - k$, where $k > 0$, and that $\alpha$ is the real root of the equation $\mathrm { f } ( x ) = 0$. Successive approximations to $\alpha$, using the Newton-Raphson method, are denoted by $x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots$.

Show that $x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + k } { 3 x _ { n } ^ { 2 } }$.
Sketch the graph of $y = \mathrm { f } ( x )$, giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for $\left| \alpha - x _ { 2 } \right|$ to be greater than $\left| \alpha - x _ { 1 } \right|$. It is now given that $k = 100$ and $x _ { 1 } = 5$.
Write down the exact value of $\alpha$ and find $x _ { 2 }$ and $x _ { 3 }$ correct to 5 decimal places.
The error $e _ { n }$ is defined by $e _ { n } = \alpha - x _ { n }$. By finding $e _ { 1 } , e _ { 2 }$ and $e _ { 3 }$, verify that $e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }$.

OCR FP2 2012 January Q6

Prove that the derivative of $\cos ^ { - 1 } x$ is $- \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$. A curve has equation $y = \cos ^ { - 1 } \left( 1 - x ^ { 2 } \right)$, for $0 < x < \sqrt { 2 }$.
Find and simplify $\frac { \mathrm { d } y } { \mathrm {~d} x }$, and hence show that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = x \frac { \mathrm {~d} y } { \mathrm {~d} x }$$
Given that $y = \sinh ^ { - 1 } x$, prove that $y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)$.
It is given that $x$ satisfies the equation $\sinh ^ { - 1 } x - \cosh ^ { - 1 } x = \ln 2$. Use the logarithmic forms for $\sinh ^ { - 1 } x$ and $\cosh ^ { - 1 } x$ to show that $$\sqrt { x ^ { 2 } + 1 } - 2 \sqrt { x ^ { 2 } - 1 } = x$$ Hence, by squaring this equation, find the exact value of $x$.

OCR FP2 2012 January Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-4_606_915_219_557} The diagram shows two curves, $C _ { 1 }$ and $C _ { 2 }$, which intersect at the pole $O$ and at the point $P$. The polar equation of $C _ { 1 }$ is $r = \sqrt { 2 } \cos \theta$ and the polar equation of $C _ { 2 }$ is $r = \sqrt { 2 \sin 2 \theta }$. For both curves, $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. The value of $\theta$ at $P$ is $\alpha$.

Show that $\tan \alpha = \frac { 1 } { 2 }$.
Show that the area of the region common to $C _ { 1 }$ and $C _ { 2 }$, shaded in the diagram, is $\frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \alpha$.

OCR FP2 2012 January Q9

Show that $\tanh ( \ln n ) = \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 1 }$. It is given that, for non-negative integers $n , I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { n } u \mathrm {~d} u$.
Show that $I _ { n } - I _ { n - 2 } = - \frac { 1 } { n - 1 } \left( \frac { 3 } { 5 } \right) ^ { n - 1 }$, for $n \geqslant 2$.
Find the value of $I _ { 3 }$, giving your answer in the form $a + \ln b$, where $a$ and $b$ are constants.
Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac { 1 } { 2 } \left( \frac { 3 } { 5 } \right) ^ { 2 } + \frac { 1 } { 4 } \left( \frac { 3 } { 5 } \right) ^ { 4 } + \frac { 1 } { 6 } \left( \frac { 3 } { 5 } \right) ^ { 6 } + \ldots .$$

OCR FP2 2013 January Q1

1 Express $\frac { 5 x } { ( x - 1 ) \left( x ^ { 2 } + 4 \right) }$ in partial fractions.

OCR FP2 2013 January Q2

2 The equation of a curve is $y = \frac { x ^ { 2 } - 3 } { x - 1 }$.

Find the equations of the asymptotes of the curve.
Write down the coordinates of the points where the curve cuts the axes.
Show that the curve has no stationary points.
Sketch the curve and the asymptotes.

Questions — OCR FP2 (168 questions)

Browse by module

Browse by board

OCR FP2 2010 January Q1

OCR FP2 2010 January Q2

OCR FP2 2010 January Q3

OCR FP2 2010 January Q4

OCR FP2 2010 January Q7

OCR FP2 2010 January Q8

OCR FP2 2010 January Q9

OCR FP2 2011 January Q1

OCR FP2 2011 January Q2

OCR FP2 2011 January Q3

OCR FP2 2011 January Q4

OCR FP2 2011 January Q5

OCR FP2 2011 January Q6

OCR FP2 2011 January Q7

OCR FP2 2011 January Q8

OCR FP2 2012 January Q1

OCR FP2 2012 January Q2

OCR FP2 2012 January Q3

OCR FP2 2012 January Q4

OCR FP2 2012 January Q5

OCR FP2 2012 January Q6

OCR FP2 2012 January Q8

OCR FP2 2012 January Q9

OCR FP2 2013 January Q1

OCR FP2 2013 January Q2