4.09a - OCR Spec

AQA FP3 2016 June Q8

17 marks Challenging +1.2

8 The diagram shows the sketch of part of a curve, the pole $O$ and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is $r = 1 + \tan \theta$.
The point $A$ is the point on the curve at which $\theta = \frac { \pi } { 3 }$.
The perpendicular, $A N$, from $A$ to the initial line intersects the curve at the point $B$.

Find the exact length of $O A$.
1. Given that, at the point $B , \theta = \alpha$, show that $( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }$.
2. Hence, or otherwise, find $\alpha$ in terms of $\pi$.
Show that the area of triangle $O A B$ is $\frac { 3 + 2 \sqrt { 3 } } { 6 }$.
Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment $A B$.
[0pt] [7 marks]
\includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

AQA Further AS Paper 1 2023 June Q10

9 marks Standard +0.3

10

Write down the equation of the horizontal asymptote of $C$ 10
Find the value of $m$ and the value of $p$
10
10
Hence, or otherwise, write down the coordinates of the $y$-intercept of $C$
Without using calculus, show that the line $y = - 1$ does not intersect $C$

OCR Further Pure Core 1 2022 June Q5

11 marks Challenging +1.2

5 The diagram below shows the curve $C$ with polar equation $r = 3 ( 1 - \sin 2 \theta )$ for $0 \leqslant \theta \leqslant 2 \pi$. \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-5_728_963_303_239}

Show that a cartesian equation of $C$ is $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 9 ( x - y ) ^ { 4 }$.
Show that the line with equation $\mathrm { y } = \mathrm { x }$ is a line of symmetry of $C$.
In this question you must show detailed reasoning. Find the exact area of each of the loops of $C$.

OCR Further Pure Core 1 2024 June Q12

7 marks Challenging +1.8

12 For any positive parameter $k$, the curve $C _ { k }$ is defined by the polar equation $\mathrm { r } = \mathrm { k } ( \cos \theta + 1 ) + \frac { 10 } { \mathrm { k } } , 0 \leqslant \theta \leqslant 2 \pi$.
For each value of $k$ the curve is a single, closed loop with no self-intersections. The diagram shows $C _ { 10.5 }$ for the purpose of illustration. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-6_558_723_550_242} Each curve, $C _ { k }$, encloses a certain area, $A _ { k }$.
You are given that there is a single minimum value of $A _ { k }$.
Determine, in an exact form, the value of $k$ for which $C _ { k }$ encloses this minimum area.

OCR Further Pure Core 1 2020 November Q11

8 marks Standard +0.8

11 A curve has cartesian equation $x ^ { 3 } + y ^ { 3 } = 2 x y$. $C$ is the portion of the curve for which $x \geqslant 0$ and $y \geqslant 0$. The equation of $C$ in polar form is given by $r = \mathrm { f } ( \theta )$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

Find $f ( \theta )$.
Find an expression for $\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)$, giving your answer in terms of $\sin \theta$ and $\cos \theta$.
Hence find the line of symmetry of $C$.
Find the value of $r$ when $\theta = \frac { 1 } { 4 } \pi$.
By finding values of $\theta$ when $r = 0$, show that $C$ has a loop.

OCR Further Pure Core 2 2022 June Q2

5 marks Standard +0.3

2 Two polar curves, $C _ { 1 }$ and $C _ { 2 }$, are defined by $C _ { 1 } : r = 2 \theta$ and $C _ { 2 } : r = \theta + 1$ where $0 \leqslant \theta \leqslant 2 \pi$. $C _ { 1 }$ intersects the initial line at two points, the pole and the point $A$.

Write down the polar coordinates of $A$.
Determine the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$. The diagram below shows a sketch of $C _ { 1 }$. \includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
On the copy of this sketch in the Printed Answer Booklet, sketch $C _ { 2 }$.

OCR Further Pure Core 2 2020 November Q6

6 marks Challenging +1.8

6 The equation of a curve in polar coordinates is $r = \ln ( 1 + \sin \theta )$ for $\alpha \leqslant \theta \leqslant \beta$ where $\alpha$ and $\beta$ are non-negative angles. The curve consists of a single closed loop through the pole.

By solving the equation $r = 0$, determine the smallest possible values of $\alpha$ and $\beta$.
Find the area enclosed by the curve, giving your answer to 4 significant figures.
Hence, by considering the value of $r$ at $\theta = \frac { \alpha + \beta } { 2 }$, show that the loop is not circular.

OCR MEI Further Pure Core 2019 June Q7

8 marks Standard +0.3

7 A curve has cartesian equation $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y$, where $c$ is a positive constant.

Show that the polar equation of the curve is $r ^ { 2 } = c ^ { 2 } \sin 2 \theta$.
Sketch the curves $r = c \sqrt { \sin 2 \theta }$ and $r = - c \sqrt { \sin 2 \theta }$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
Answer all the questions.

OCR MEI Further Pure Core 2023 June Q7

6 marks Standard +0.8

7 The diagram below shows the curve with polar equation $r = a ( 1 - 2 \sin \theta )$ for $0 \leqslant \theta \leqslant 2 \pi$, where $a$ is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{76631941-3cd5-4b3e-a7e4-27b8f991975a-4_634_865_486_239} The curve crosses the initial line at A , and the points B and C are the lowest points on the two loops.

Find the values of $r$ and $\theta$ at the points A , B and C .
Find the set of values of $\theta$ for the points on the inner loop (shown in the diagram with a broken line).

OCR MEI Further Pure Core 2021 November Q14

14 marks Standard +0.8

14 A curve has polar equation $\mathrm { r } = \mathrm { a } ( \cos \theta + 2 \sin \theta )$, where $a$ is a positive constant and $0 \leqslant \theta \leqslant \pi$.

Determine the polar coordinates of the point on the curve which is furthest from the pole.
1. Show that the curve is a circle whose radius should be specified.
2. Write down the polar coordinates of the centre of the circle.

OCR MEI Further Pure with Technology 2019 June Q1

20 marks Challenging +1.8

1 A family of curves is given by the parametric equations $x ( t ) = \cos ( t ) - \frac { \cos ( ( m + 1 ) t ) } { m + 1 }$ and $y ( t ) = \sin ( t ) - \frac { \sin ( ( m + 1 ) t ) } { m + 1 }$ where $0 \leqslant t < 2 \pi$ and $m$ is a positive integer.

1. Sketch the curves in the cases $m = 3 , m = 4$ and $m = 5$ on separate axes in the Printed Answer Booklet.
2. State one common feature of these three curves.
3. State a feature for the case $m = 4$ which is absent in the cases $m = 3$ and $m = 5$.
1. Determine, in terms of $m$, the values of $t$ for which $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$ but $\frac { \mathrm { d } y } { \mathrm {~d} t } \neq 0$.
2. Describe the tangent to the curve at the points corresponding to such values of $t$.
1. Show that the curve lies between the circle centred at the origin with radius $$1 - \frac { 1 } { m + 1 }$$ and the circle centred at the origin with radius $$1 + \frac { 1 } { m + 1 }$$
2. Hence, or otherwise, show that the area $A$ bounded by the curve satisfies $$\frac { m ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } } < A < \frac { ( m + 2 ) ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } }$$
3. Find the limit of the area bounded by the curve as $m$ tends to infinity.
The arc length of a curve defined by parametric equations $x ( t )$ and $y ( t )$ between points corresponding to $t = c$ and $t = d$, where $c < d$, is $$\int _ { c } ^ { d } \sqrt { \left( \frac { \mathrm {~d} x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } } \mathrm {~d} t$$ Use this to show that the length of the curve is independent of $m$.

OCR MEI Further Pure with Technology 2022 June Q1

20 marks Challenging +1.8

1

A family of curves is given by the equation $$x ^ { 2 } + y ^ { 2 } + 2 a x y = 1 ( * )$$ where the parameter $a$ is a real number.
You may find it helpful to use a slider (for $a$ ) to investigate this family of curves.
1. On the axes in the Printed Answer Booklet, sketch the curve in each of the cases
  - $a = 0$
  - $a = 0.5$
  - $a = 2$
  - State a feature of the curve for the cases $a = 0 , a = 0.5$ that is not a feature of the curve in the case $a = 2$.
  - In the case $a = 1$, the curve consists of two straight lines. Determine the equations of these lines.
  - 1. Find an equation of the curve (*) in polar form.
    2. Hence, or otherwise, find, in exact form, the area bounded by the curve, the positive part of the $x$-axis and the positive part of the $y$-axis, in the case $a = 2$.
In this part of the question $m$ is any real number.

OCR MEI Further Pure with Technology Specimen Q1

19 marks Challenging +1.8

1 A family of curves has polar equation $r = \cos n \left( \frac { \theta } { n } \right) , 0 \leq \theta < n \pi$, where $n$ is a positive even integer.

(A) Sketch the curve for the cases $n = 2$ and $n = 4$.
(B) State two points which lie on every curve in the family.
(C) State one other feature common to all the curves.
(A) Write down an integral for the length of the curve for the case $n = 4$.
(B) Evaluate the integral.
(A) Using $t = \theta$ as the parameter, find a parametric form of the equation of the family of curves.
(B) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sin t \sin \left( \frac { t } { n } \right) - \cos t \cos \left( \frac { t } { n } \right) } { \sin t \cos \left( \frac { t } { n } \right) + \cos t \sin \left( \frac { t } { n } \right) }$.
Hence show that there are $n + 1$ points where the tangent to the curve is parallel to the $y$-axis.
By referring to appropriate sketches, show that the result in part (iv) is true in the case $n = 4$.
(A) Create a program to find all the solutions to $x ^ { 2 } \equiv - 1 ( \bmod p )$ where $0 \leq x < p$. Write out your program in full in the Printed Answer Booklet.
(B) Use the program to find the solutions to $x ^ { 2 } \equiv - 1 ( \bmod p )$ for the primes
$$\begin{aligned} ( 4 k ) ! & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( 2 k + 1 ) \times ( 2 k + 2 ) \times \ldots \times ( 4 k - 1 ) \times 4 k ( \bmod p ) \\ & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( - 2 k ) \times ( - ( 2 k - 1 ) ) \times \ldots \times ( - 2 ) \times ( - 1 ) ( \bmod p ) \\ & \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p ) \end{aligned}$$ (A) Explain why ( $2 k + 2$ ) can be written as ( $- ( 2 k - 1 )$ ) in line ( 2 ).
(B) Explain how line (3) has been obtained.
(C) Explain why, if $p$ is a prime of the form $p = 4 k + 1$, then $x ^ { 2 } \equiv - 1 ( \bmod p )$ will have at least one solution.
(D) Hence find a solution of $x ^ { 2 } \equiv - 1 ( \bmod 29 )$.
(A) Create a program that will find all the positive integers $n$, where $n < 1000$, such that $( n - 1 ) ! \equiv - 1 \left( \bmod n ^ { 2 } \right)$. Write out your program in full.
(B) State the values of $n$ obtained.
(C) A Wilson prime is a prime $p$ such that $( p - 1 ) ! \equiv - 1 \left( \bmod p ^ { 2 } \right)$. Write down all the Wilson primes $p$ where $p < 1000$.

Edexcel CP2 2021 June Q6

14 marks Challenging +1.2

The curve $C$ has equation

$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where $a$ and $p$ are positive constants and $p > 2$ There are exactly four points on $C$ where the tangent is perpendicular to the initial line.

Show that the range of possible values for $p$ is $$2 < p < 4$$
Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where $r$ is measured in centimetres. The depth of the pond is 90 centimetres.
Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
Given that the pond is initially empty,
determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
State a limitation of the model.

Edexcel CP2 2022 June Q7

10 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{33292670-3ad0-4125-a3bb-e4b7b21ed5f4-22_678_776_248_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $$r = 1 + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 3 }$$ Figure 1 also shows the tangent to $C$ at the point $A$.
This tangent is perpendicular to the initial line.

Use differentiation to prove that the polar coordinates of $A$ are $\left( 2 , \frac { \pi } { 4 } \right)$ The finite region $R$, shown shaded in Figure 1, is bounded by $C$, the tangent at $A$ and the initial line.
Use calculus to show that the exact area of $R$ is $\frac { 1 } { 2 } ( 1 - \ln 2 )$

Edexcel CP2 2023 June Q4

7 marks Challenging +1.2

(a) Sketch the polar curve $C$, with equation

$$r = 3 + \sqrt { 5 } \cos \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ On your sketch clearly label the pole, the initial line and the value of $r$ at the point where the curve intersects the initial line. The tangent to $C$ at the point $A$, where $0 < \theta < \frac { \pi } { 2 }$, is parallel to the initial line.
(b) Use calculus to show that at $A$ $$\cos \theta = \frac { 1 } { \sqrt { 5 } }$$ (c) Hence determine the value of $r$ at $A$.

Edexcel CP2 Specimen Q6

13 marks Challenging +1.8

(a) (i) Show on an Argand diagram the locus of points given by the values of $z$ satisfying

$$| z - 4 - 3 \mathbf { i } | = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that $\theta \in [ \alpha , \alpha + \pi ]$, where $\alpha = - \arctan \left( \frac { 4 } { 3 } \right)$,
(ii) show that this locus of points can be represented by the polar curve with equation $$r = 8 \cos \theta + 6 \sin \theta$$ The set of points $A$ is defined by $$A = \left\{ z : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\} \cap \{ z : | z - 4 - 3 \mathbf { i } | \leqslant 5 \}$$ (b) (i) Show, by shading on your Argand diagram, the set of points $A$.
(ii) Find the exact area of the region defined by $A$, giving your answer in simplest form.

Edexcel FP2 2019 June Q8

13 marks Challenging +1.8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4ba4a815-f53d-4de2-810b-b06e145f457b-24_547_629_242_717} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the vertical cross section of a child's spinning top. The point $A$ is vertically above the point $B$ and the height of the spinning top is 5 cm . The line $C D$ is perpendicular to $A B$ such that $C D$ is the maximum width of the spinning top.
The spinning top is modelled as the solid of revolution created when part of the curve with polar equation $$r ^ { 2 } = 25 \cos 2 \theta$$ is rotated through $2 \pi$ radians about the initial line.

Show that, according to the model, the surface area of the spinning top is $$k \pi ( 2 - \sqrt { 2 } ) \mathrm { cm } ^ { 2 }$$ where $k$ is a constant to be determined.
Show that, according to the model, the length $C D$ is $\frac { 5 \sqrt { 2 } } { 2 } \mathrm {~cm}$.

OCR Further Pure Core 2 2019 June Q9

11 marks Challenging +1.2

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 1 2018 March Q10

14 marks Challenging +1.8

10

(a) A curve has polar equation $r = 2 - \sec \theta$. Show that the cartesian equation of the curve can be written in the form $$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$ The figure shows a sketch of part of the curve with equation $y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$. \includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
(b) Explain why the curve is symmetrical in the $x$-axis.
(c) The line $x = a$ is an asymptote of the curve. State the value of $a$.
The enclosed loop shown in the figure is rotated through $180 ^ { \circ }$ about the $x$-axis. Find the exact volume of the solid formed. \section*{END OF QUESTION PAPER}

OCR FP2 Q8

13 marks Challenging +1.2

8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

State the greatest value of $r$ and the corresponding values of $\theta$.
Find the equations of the tangents at the pole.
Find the exact area enclosed by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 2 } \pi$.
Find, in simplified form, the cartesian equation of the curve.

AQA FP3 2006 January Q6

16 marks Challenging +1.2

6

A circle $C _ { 1 }$ has cartesian equation $x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36$. Show that the polar equation of $C _ { 1 }$ is $r = 12 \sin \theta$.
A curve $C _ { 2 }$ with polar equation $r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi$ is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651} Calculate the area bounded by $C _ { 2 }$.
The circle $C _ { 1 }$ intersects the curve $C _ { 2 }$ at the points $P$ and $Q$. Find, in surd form, the area of the quadrilateral $O P M Q$, where $M$ is the centre of the circle and $O$ is the pole.
(6 marks)

AQA FP3 2007 January Q2