Questions Further Pure Core 2

Find the acute angle between the line $\mathbf { r } = \left( \begin{array} { c } 3
2
- 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5
3
2 \end{array} \right)$ and $\Pi$.
Determine the geometrical relationship between the line $\mathbf { r } = \left( \begin{array} { l } 1
4
2 \end{array} \right) + \mu \left( \begin{array} { r } 3
- 1
5 \end{array} \right)$ and $\Pi$.

OCR Further Pure Core 2 Specimen Q8

8 The equation of a curve is $y = \cosh ^ { 2 } x - 3 \sinh x$. Show that $\left( \ln \left( \frac { 3 + \sqrt { 13 } } { 2 } \right) , - \frac { 5 } { 4 } \right)$ is the only stationary point on the curve.

OCR Further Pure Core 2 Specimen Q9

9 A curve has equation $x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }$, where $x$ and $y$ are not both zero.

Show that the equation of the curve in polar coordinates is $r ^ { 2 } = \frac { 2 } { 2 - \sin ^ { 2 } 2 \theta }$.
Deduce that no point on the curve $x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }$ is further than $\sqrt { 2 }$ from the origin.

OCR Further Pure Core 2 Specimen Q10

10 Let $C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos r \theta$. Show that $C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos 10 \theta$.

OCR Further Pure Core 2 Specimen Q11

11 During an industrial process substance $X$ is converted into substance $Z$. Some of the substance $X$ goes through an intermediate phase, and is converted to substance $Y$, before being converted to substance $Z$. The situation is modelled by $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 0.3 x - 0.2 y \text { and } \frac { \mathrm { d } z } { \mathrm {~d} t } = 0.2 y + 0.1 x$$ where $x , y$ and $z$ are the amounts in kg of $X , Y$ and $Z$ at time $t$ hours after the process starts. Initially there is 10 kg of substance $X$ and nothing of substances $Y$ and $Z$. The amount of substance $X$ decreases exponentially. The initial rate of decrease is 4 kg per hour.

Show that $x = A \mathrm { e } ^ { - 0.4 t }$, stating the value of $A$.
(a) Show that $\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } z } { \mathrm {~d} t } = 0$.
(b) Comment on this result in the context of the industrial process.
Express $y$ in terms of $t$.
Determine the maximum amount of substance $Y$ present during the process.
How long does it take to produce 9 kg of substance $Z$ ? \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR Further Pure Core 2 2019 June Q6

Show that the motion of the particle can be modelled by the differential equation \end{itemize} $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when $t = 0$.
Find $v$ in terms of $t$.
Find the velocity of the particle when $t = 2$. When $t = 2$ the force acting in the positive $x$-direction is replaced by a constant force of magnitude $\frac { 1 } { 2 } \mathrm {~N}$ in the same direction.
Refine the differential equation given in part (a) to model the motion for $t \geqslant 2$.
Use the refined model from part (d) to find an exact expression for $v$ in terms of $t$ for $t \geqslant 2$.
$6 \quad A$ is a fixed point on a smooth horizontal surface. A particle $P$ is initially held at $A$ and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after 0.2 seconds at point $B$ whose displacement is 0.2 m from $A$. The point $M$ is halfway between $A$ and $B$. The displacement of $P$ from $M$ at time $t$ seconds after release is denoted by $x \mathrm {~m}$.
On the axes provided in the Printed Answer Booklet, sketch a graph of $x$ against $t$ for $0 \leqslant t \leqslant 0.4$.
Find the displacement of $P$ from $M$ at 0.75 seconds after release.

OCR Further Pure Core 2 2019 June Q9

Find the exact area enclosed by the curve.
Show that the greatest value of $r$ on the curve is $\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }$.

OCR Further Pure Core 2 2022 June Q8

Show that $\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta$, where $a$ is an integer to be determined.
Hence show that $\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }$, where $b$ and $c$ are integers to be determined.

OCR Further Pure Core 2 Specimen Q7

Use the Maclaurin series for $\sin x$ to work out the series expansion of $\sin x \sin 2 x \sin 4 x$ up to and including the term in $x ^ { 3 }$.
Hence find, in exact surd form, an approximation to the least positive root of the equation $2 \sin x \sin 2 x \sin 4 x = x$.

OCR Further Pure Core 2 2021 November Q2

2 In this question you must show detailed reasoning. The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $z _ { 1 } = 3 - 7 \mathrm { i }$ and $z _ { 2 } = 2 + 4 \mathrm { i }$.

Express each of the following as exact numbers in the form $a + b \mathrm { i }$.
1. $3 z _ { 1 } + 4 z _ { 2 }$
2. $z _ { 1 } z _ { 2 }$
3. $\frac { Z _ { 1 } } { Z _ { 2 } }$
Write $z _ { 1 }$ in modulus-argument form giving the modulus in exact form and the argument correct to $\mathbf { 3 }$ significant figures.

OCR Further Pure Core 2 2019 June Q3

3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

OCR Further Pure Core 2 2022 June Q3

3 In this question you must show detailed reasoning. The roots of the equation $4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0$ are $\alpha , \beta$ and $\gamma$. Find a cubic equation with integer coefficients whose roots are $\alpha + \beta , \beta + \gamma$ and $\gamma + \alpha$.

OCR Further Pure Core 2 2022 June Q9

9 In this question you must show detailed reasoning.

Show that $\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta$, where $a$ is an integer to be determined.
Hence show that $\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }$, where $b$ and $c$ are integers to be determined.

OCR Further Pure Core 2 2023 June Q4

4 In this question you must show detailed reasoning. The region $R$ is bounded by the curve with equation $\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }$, the $x$-axis and the lines with equations $x = \frac { 1 } { 2 }$ and $x = 1$ (see diagram). The units of the axes are cm .
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when $R$ is rotated by $2 \pi$ radians about the $x$-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

OCR Further Pure Core 2 2023 June Q5

5 In this question you must show detailed reasoning.

Using the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, show that $\sinh 2 x \equiv 2 \sinh x \cosh x$.
Solve the equation $15 \sinh x + 16 \cosh x - 6 \sinh 2 x = 20$, giving all your answers in logarithmic form.

OCR Further Pure Core 2 2023 June Q10

10 In this question you must show detailed reasoning. A region, $R$, of the floor of an art gallery is to be painted for the purposes of an art installation. A suitable polar coordinate system is set up on the floor of the gallery with units in metres and radians. $R$ is modelled as being the region enclosed by two curves, $C _ { 1 }$ and $C _ { 2 }$. The polar equations of $C _ { 1 }$ and $C _ { 2 }$ are $$\begin{array} { l l } C _ { 1 } : r = 5 , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi
C _ { 2 } : r = 3 \cosh \theta , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \end{array}$$ Both curves are shown in the diagram, with $R$ indicated.
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-6_1481_821_836_251} The gallery must buy tins of paint to paint $R$. Each tin of paint can cover an area of $0.5 \mathrm {~m} ^ { 2 }$.
Determine the smallest number of tins of paint that the gallery must buy in order to be able to paint $R$ completely.

OCR Further Pure Core 2 2020 November Q8

8 In this question you must show detailed reasoning. The complex number $- 4 + i \sqrt { 48 }$ is denoted by $z$.

Determine the cube roots of $z$, giving the roots in exponential form. The points which represent the cube roots of $z$ are denoted by $A , B$ and $C$ and these form a triangle in an Argand diagram.
Write down the angles that any lines of symmetry of triangle $A B C$ make with the positive real axis, justifying your answer.

OCR Further Pure Core 2 2018 March Q1

1 Plane $\Pi$ has equation $3 x - y + 2 z = 33$. Line $l$ has the following vector equation. $$l : \quad \mathbf { r } = \left( \begin{array} { l } 1
0
5 \end{array} \right) + \lambda \left( \begin{array} { l }

OCR Further Pure Core 2 2018 March Q3

3 \end{array} \right)$$

Find the acute angle between $\Pi$ and $l$.
Find the coordinates of the point of intersection of $\Pi$ and $l$.
$S$ is the point $( 4,5 , - 5 )$. Find the shortest distance from $S$ to $\Pi$. 2 The complex number $2 + \mathrm { i }$ is denoted by $z$.
Show that $z ^ { 2 } = 3 + 4 \mathrm { i }$.
Plot the following on the Argand diagram in the Printed Answer Booklet.
- $z$
- $z ^ { 2 }$
- State the relationship between $\left| z ^ { 2 } \right|$ and $| z |$.
- State the relationship between $\arg \left( z ^ { 2 } \right)$ and $\arg ( z )$.
3 In this question you must show detailed reasoning. Use the formula $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$ to evaluate $121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }$.

OCR Further Pure Core 2 2018 March Q5

5 \end{array} \right) + \lambda \left( \begin{array} { l } 2
2
3 \end{array} \right)$$

Find the acute angle between $\Pi$ and $l$.
Find the coordinates of the point of intersection of $\Pi$ and $l$.
$S$ is the point $( 4,5 , - 5 )$. Find the shortest distance from $S$ to $\Pi$. 2 The complex number $2 + \mathrm { i }$ is denoted by $z$.
Show that $z ^ { 2 } = 3 + 4 \mathrm { i }$.
Plot the following on the Argand diagram in the Printed Answer Booklet.
- $z$
- $z ^ { 2 }$
- State the relationship between $\left| z ^ { 2 } \right|$ and $| z |$.
- State the relationship between $\arg \left( z ^ { 2 } \right)$ and $\arg ( z )$.
3 In this question you must show detailed reasoning. Use the formula $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$ to evaluate $121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }$. 4 You are given that the cubic equation $2 x ^ { 3 } - 3 x ^ { 2 } + x + 4 = 0$ has three roots, $\alpha , \beta$ and $\gamma$.
By making a suitable substitution to obtain a related cubic equation, determine the value of $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$. 5 In this question you must show detailed reasoning.
An ant starts from a fixed point $O$ and walks in a straight line for 1.5 s . Its velocity, $v \mathrm {~cm} \mathrm {~s} ^ { - 1 }$, can be modelled by $v = \frac { 1 } { \sqrt { 9 - t ^ { 2 } } }$. By finding the mean value of $v$ in $0 \leqslant t \leqslant 1.5$, deduce the average velocity of the ant.

OCR Further Pure Core 2 2018 March Q6

6 In this question you must show detailed reasoning.

Find the coordinates of all stationary points on the graph of $y = 6 \sinh ^ { 2 } x - 13 \cosh x$, giving your answers in an exact, simplified form.
By finding the second derivative, classify the stationary points found in part (i).

OCR Further Pure Core 2 2018 March Q7

7 In the following set of simultaneous equations, $a$ and $b$ are constants. $$\begin{aligned} 3 x + 2 y - z & = 5
2 x - 4 y + 7 z & = 60
a x + 20 y - 25 z & = b \end{aligned}$$

In the case where $a = 10$, solve the simultaneous equations, giving your solution in terms of $b$.
Determine the value of $a$ for which there is no unique solution for $x , y$ and $z$.
(a) Find the values of $\alpha$ and $\beta$ for which $\alpha ( 2 y - z ) + \beta ( - 4 y + 7 z ) = 20 y - 25 z$ for any $y$ and $z$.
(b) Hence, for the case where there is no unique solution for $x , y$ and $z$, determine the value of $b$ for which there is an infinite number of solutions.
(c) When $a$ takes the value in part (ii) and $b$ takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations.

OCR Further Pure Core 2 2018 March Q8

8 In this question you must show detailed reasoning.
Show that $\int _ { 0 } ^ { 2 } \frac { 2 x ^ { 2 } + 3 x - 1 } { x ^ { 3 } - 3 x ^ { 2 } + 4 x - 12 } \mathrm {~d} x = \frac { 3 } { 8 } \pi - \ln 9$.

OCR Further Pure Core 2 2018 March Q9

9 In this question you must show detailed reasoning.

Show that $\mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } = 2 \mathrm { i } \sin \theta$.
Hence, show that $\frac { 2 } { \mathrm { e } ^ { 2 \mathrm { i } \theta } - 1 } = - ( 1 + \mathrm { i } \cot \theta )$.
Two series, $C$ and $S$, are defined as follows. $$\begin{aligned} & C = 2 + 2 \cos \frac { \pi } { 10 } + 2 \cos \frac { \pi } { 5 } + 2 \cos \frac { 3 \pi } { 10 } + 2 \cos \frac { 2 \pi } { 5 }
& S = 2 \sin \frac { \pi } { 10 } + 2 \sin \frac { \pi } { 5 } + 2 \sin \frac { 3 \pi } { 10 } + 2 \sin \frac { 2 \pi } { 5 } \end{aligned}$$ By considering $C + \mathrm { i } S$, find a simplified expression for $C$ in terms of only integers and $\cot \frac { \pi } { 20 }$.
Verify that $S = C - 2$ and, by considering the series in their original form, explain why this is so. \section*{END OF QUESTION PAPER}

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