Questions - OCR Further Pure Core 1

OCR Further Pure Core 1 2021 November Q1

1

Sketch on a single Argand diagram the loci given by
1. $\quad | z - 1 + 2 i | = 3$,
2. $\quad | z + 1 | = | z - 2 |$.
Indicate, by shading, the region of the Argand diagram for which $| z - 1 + 2 i | \leqslant 3$ and $| z + 1 | \leqslant | z - 2 |$.

OCR Further Pure Core 1 2021 November Q2

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

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

OCR Further Pure Core 1 2021 November Q3

3 A function $\mathrm { f } ( \mathrm { z } )$ is defined on all complex numbers z by $\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 3 \mathrm { z } ^ { 2 } + \mathrm { k } \mathrm { z } - 5$ where $k$ is a real constant. The roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ are $\alpha , \beta$ and $\gamma$. You are given that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5$.

Explain why $f ( z ) = 0$ has only one real root.
Find the value of $k$.
Find a cubic equation with integer coefficients that has roots $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$.

OCR Further Pure Core 1 2021 November Q4

4 Points $A , B$ and $C$ have coordinates ( $4,2,0$ ), ( $1,5,3$ ) and ( $1,4 , - 2$ ) respectively. The line $l$ passes through $A$ and $B$.

Find a cartesian equation for $l$.
$M$ is the point on $l$ that is closest to $C$.
Find the coordinates of $M$.
Find the exact area of the triangle $A B C$.

OCR Further Pure Core 1 2021 November Q5

5 Use de Moivre's theorem to find the constants $A , B$ and $C$ in the identity $\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$.
$6 O$ is the origin of a coordinate system whose units are cm .
The points $A , B , C$ and $D$ have coordinates ( 1,0 ), ( 1,4 ), ( 6,9 ) and ( 0,9 ) respectively.
The arc $B C$ is part of the curve with equation $x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37$.
The closed shape $O A B C D$ is formed, in turn, from the line segments $O A$ and $A B$, the arc $B C$ and the line segments $C D$ and $D O$ (see diagram).
A funnel can be modelled by rotating $O A B C D$ by $2 \pi$ radians about the $y$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_641_1131_808_242} Find the volume of the funnel according to the model.

OCR Further Pure Core 1 2021 November Q7

7 The diagram below shows the curve with polar equation $r = \sin 3 \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_385_807_1834_260}

Find the values of $\theta$ at the pole.
Find the polar coordinates of the point on the curve where $r$ takes its maximum value.
In this question you must show detailed reasoning. Find the exact area enclosed by the curve.
Given that $\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$, find a cartesian equation for the curve.

OCR Further Pure Core 1 2021 November Q8

8 You are given that $\mathrm { f } ( x ) = 4 \sinh x + 3 \cosh x$.

Show that the curve $y = f ( x )$ has no turning points.
Determine the exact solution of the equation $\mathrm { f } ( x ) = 5$.

OCR Further Pure Core 1 2021 November Q9

9 You are given that the matrix $\left( \begin{array} { c c } 2 & 1
- 1 & 0 \end{array} \right)$ represents a transformation T .

You are given that the line with equation $\mathrm { y } = \mathrm { kx }$ is invariant under T . Determine the value of $k$.
Determine whether the line with equation $\mathrm { y } = \mathrm { kx }$ in part (a) is a line of invariant points under T .

OCR Further Pure Core 1 2021 November Q10

10 Using an algebraic method, determine the least value of $n$ for which $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } \geqslant 0.49$.

OCR Further Pure Core 1 2021 November Q11

11 The displacement of a door from its equilibrium (closed) position is measured by the angle, $\theta$ radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that $\theta$ can take positive and negative values. The door is released from rest from an open position at time $t = 0$. A proposed differential equation to model the motion of the door for $t \geqslant 0$ is $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm { d } t } + 3 \theta = 0$ where $\lambda$ is a constant and $\lambda \geqslant 0$.

1. According to the model, for what value of $\lambda$ will the motion of the door be simple harmonic?
2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic.
Find the range of values of $\lambda$ for which the model predicts that the door will never pass through the equilibrium position.
Sketch a possible graph of $\theta$ against $t$ when $\lambda$ lies outside the range found in part (b) but the motion is not simple harmonic.

OCR Further Pure Core 1 Specimen Q1

1 Show that $\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }$.

OCR Further Pure Core 1 Specimen Q2

2 In this question you must show detailed reasoning. The equation $\mathrm { f } ( x ) = 0$, where $\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169$, has a root $x = 2 + 3 \mathrm { i }$.

Express $\mathrm { f } ( x )$ as a product of two quadratic factors.
Hence write down all the roots of the equation $\mathrm { f } ( x ) = 0$.

OCR Further Pure Core 1 Specimen Q3

3 In this question you must show detailed reasoning. The diagram below shows the curve $r = 2 \cos 4 \theta$ for $- k \pi \leq \theta \leq k \pi$ where $k$ is a constant to be determined. Calculate the exact area enclosed by the curve.

OCR Further Pure Core 1 Specimen Q4

4 Draw the region in an Argand diagram for which $| z | \leq 2$ and $| z | > | z - 3 i |$.

OCR Further Pure Core 1 Specimen Q5

5

Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }$.
Find $\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x$.

OCR Further Pure Core 1 Specimen Q6

6 The equation $x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.
The equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$ has roots $\alpha \beta , \beta \gamma$ and $\gamma \alpha$.
Find the values of $p , q$ and $r$.

OCR Further Pure Core 1 Specimen Q7

7 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\frac { x - 3 } { 1 } = \frac { y - 5 } { 2 } = \frac { z + 2 } { - 3 }$ and $\frac { x - 4 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z - 7 } { 4 }$.

Find the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.
Find a cartesian equation of the plane which contains $l _ { 1 }$ and is parallel to $l _ { 2 }$.

OCR Further Pure Core 1 Specimen Q8

8

Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3
2 x + 4 y + 5 z = & 9
7 x + 11 y + 12 z = & 20 \end{array}$$
Determine the values of $p$ and $k$ for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r l } x + & y + & z = & 3
2 x + & 4 y + & 5 z = & 9
7 x + & 11 y + & p z = & k \end{array}$$

OCR Further Pure Core 1 Specimen Q9

9 Prove by induction that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$

OCR Further Pure Core 1 Specimen Q10

10 The Argand diagram below shows the origin $O$ and pentagon $A B C D E$, where $A , B , C , D$ and $E$ are the points that represent the complex numbers $a , b , c , d$ and $e$, and where $a$ is a positive real number. You are given that these five complex numbers are the roots of the equation $z ^ { 5 } - a ^ { 5 } = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

Justify each of the following statements.
(a) $A , B , C , D$ and $E$ lie on a circle with centre $O$.
(b) $A B C D E$ is a regular pentagon.
(c) $b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c$
(d) $b ^ { * } = e$
(e) $a + b + c + d + e = 0$
The midpoints of sides $A B , B C , C D , D E$ and $E A$ represent the complex numbers $p , q , r , s$ and $t$. Determine a polynomial equation, with real coefficients, that has roots $p , q , r , s$ and $t$.

OCR Further Pure Core 1 Specimen Q11

11 A company is required to weigh any goods before exporting them overseas. When a crate is placed on a set of weighing scales, the mass displayed takes time to settle down to its final value. The company wishes to model the mass, $m \mathrm {~kg}$, which is displayed $t$ seconds after a crate X is placed on the scales.
For the displayed mass it is assumed that the rate of change of the quantity $\left( 0.5 \frac { \mathrm {~d} m } { \mathrm {~d} t } + m \right)$ with respect to time is proportional to $( 80 - m )$.

Show that $\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 2 \mathrm {~km} = 160 \mathrm { k }$, where $k$ is a real constant. It is given that the complementary function for the differential equation in part (i) is $\mathrm { e } ^ { \lambda t } ( A \cos 2 t + B \sin 2 t )$, where $A$ and $B$ are arbitrary constants.
Show that $k = \frac { 5 } { 2 }$ and state the value of the constant $\lambda$. When X is initially placed on the scales the displayed mass is zero and the rate of increase of the displayed mass is $160 \mathrm {~kg} \mathrm {~s} ^ { - 1 }$.
Find $m$ in terms of $t$.
Describe the long term behaviour of $m$.
With reference to your answer to part (iv), comment on a limitation of the model.
(a) Find the value of $m$ that corresponds to the stationary point on the curve $m = \mathrm { f } ( t )$ with the smallest positive value of $t$.
(b) Interpret this value of $m$ in the context of the model.
Adapt the differential equation $\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 5 m = 400$ to model the mass displayed $t$ seconds after a crate Y , of mass 100 kg , is placed on the scales. \section*{END OF QUESTION PAPER} \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{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 1 2023 June Q9

9 In this question you must show detailed reasoning.}

Use de Moivre's theorem to determine constants $A$, $B$ and $C$ such that $$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$ The function f is defined by
$\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1$.
Show that $\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$.
\includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260} The diagram shows the curve with equation $\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$ for $0 \leqslant x < 1$ and the asymptote $x = 1$. The region $R$ is the unbounded region between the curve, the $x$-axis, the line $x = 0$ and the line $x = 1$. You are given that the area of $R$ is finite.
Determine the exact area of $R$.

OCR Further Pure Core 1 2023 June Q6

6 In this question you must show detailed reasoning.} The power output, $p$ watts, of a machine at time $t$ hours after it is switched on can be modelled by the equation $\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )$ for $t \geqslant 0$. Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to $\mathbf { 2 }$ decimal places.

OCR Further Pure Core 1 2022 June Q1

1 In this question you must show detailed reasoning.

Show that $\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }$. The region $R$ is bounded by the curve with equation $\mathrm { y } = \sqrt { \operatorname { sinhx } }$, the $x$-axis and the line with equation $x = 2 \ln 3$ (see diagram). The units of the axes are centimetres.
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242} A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating $R$ completely about the $x$-axis.
Determine, according to the model, the exact volume of one chocolate piece.

OCR Further Pure Core 1 2018 March Q1

1 In this question you must show detailed reasoning.
Find the square roots of $24 + 10 \mathrm { i }$, giving your answers in the form $a + b \mathrm { i }$.

Questions — OCR Further Pure Core 1 (134 questions)

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