Questions Further Additional Pure AS

Using the vector product, calculate the area of triangle $A B C$.
Hence find, in simplest surd form, the perpendicular distance from $C$ to the line through $A$ and $B$.

OCR Further Additional Pure AS 2018 June Q2

2 The surface with equation $z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y$ has two stationary points.

Verify that one of these stationary points is at the origin.
Find the coordinates of the second stationary point.

OCR Further Additional Pure AS 2018 June Q3

3 Given that $n$ is a positive integer, show that the numbers ( $4 n + 1$ ) and ( $6 n + 1$ ) are co-prime.

OCR Further Additional Pure AS 2018 June Q4

4 The group $G$ consists of a set of six matrices under matrix multiplication. Two of the elements of $G$ are $\mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 1 & - 1
0 & - 1 \end{array} \right)$.

Determine each of the following:
- $\mathbf { A } ^ { 2 }$
- $\mathbf { B } ^ { 2 }$
- Determine all the elements of $G$.
- State the order of each non-identity element of $G$.
- State, with justification, whether $G$ is
- abelian
- cyclic.

OCR Further Additional Pure AS 2018 June Q5

5 For integers $a$ and $b$, with $a \geqslant 0$ and $0 \leqslant b \leqslant 99$, the numbers $M$ and $N$ are such that $$M = 100 a + b \text { and } N = a - 9 b .$$

By considering the number $M + 2 N$, show that $17 \mid M$ if and only if $17 \mid N$.
Demonstrate step-by-step how an algorithm based on the result of part (i) can be used to show that 2058376813901 is a multiple of 17 .

OCR Further Additional Pure AS 2018 June Q6

6 The Fibonacci sequence $\left\{ F _ { n } \right\}$ is defined by $F _ { 0 } = 0 , F _ { 1 } = 1$ and $F _ { n } = F _ { n - 1 } + F _ { n - 2 }$ for all $n \geqslant 2$.

Show that $F _ { n + 5 } = 5 F _ { n + 1 } + 3 F _ { n }$
Prove that $F _ { n }$ is a multiple of 5 when $n$ is a multiple of 5 .

OCR Further Additional Pure AS 2018 June Q7

7 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical $z$-axis. The model is represented by part of the surface with equation $z = \mathrm { f } ( x , y )$ and $O$ is on the surface. The point $P$ is on the rim of the dish and directly above the $x$-axis.
The object, $B$, modelled as a point on the $z$-axis is the receiving box which collects the TV signals reflected by the dish.
\includegraphics[max width=\textwidth, alt={}, center]{f2166e0a-cd4c-40af-b4b4-04ef4919d996-3_753_995_584_525}

The horizontal plane $\Pi _ { 1 }$, containing the point $P$, intersects the surface of the model in a contour of the surface.
(a) Sketch this contour in the Printed Answer Booklet.
(b) State a suitable equation for this contour.
A second plane, $\Pi _ { 2 }$, containing both $P$ and the $z$-axis, intersects the surface of the model in a section of the surface.
(a) Sketch this section in the Printed Answer Booklet.
(b) State a suitable equation for this section.
A proposed equation for the surface is $z = a x ^ { 2 } + b y ^ { 2 }$. What can you say about the constants $a$ and $b$ within this equation? Justify your answers.
The real TV satellite dish has the following measurements (in metres): the height of $P$ above $O$ is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of
- $a$ and $b$,
- any other constants stated within the answers to parts (i)(b) and (ii)(b).
- Incoming satellite signals arrive at the dish in linear "beams" travelling parallel to the $z$-axis. They are then 'bounced' off the dish to the receiving box at $B$.
- On the diagram for part (ii)(a) in the Printed Answer Booklet draw some of these beams and mark $B$.
- If the values of $a$ and $b$ were changed, what would happen?

OCR Further Additional Pure AS 2019 June Q1

1 In decimal (base 10) form, the number $N$ is 15260.

Express $N$ in binary (base 2) form.
Using the binary form of $N$, show that $N$ is divisible by 7 .

OCR Further Additional Pure AS 2019 June Q2

The convergent sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$ is defined by $a _ { 0 } = 1$ and $\mathrm { a } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { a } _ { \mathrm { n } } } + \frac { 4 } { \sqrt { \mathrm { a } _ { \mathrm { n } } } }$ for $n \geqslant 0$. Calculate the limit of the sequence.
The convergent sequence $\left\{ \mathrm { b } _ { \mathrm { n } } \right\}$ is defined by $\mathrm { b } _ { 0 } = 1$ and $\mathrm { b } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { b } _ { \mathrm { n } } } + \frac { \mathrm { k } } { \sqrt { \mathrm { b } _ { \mathrm { n } } } }$ for $n \geqslant 0$, where $k$ is a constant. Determine the value of $k$ for which the limit of the sequence is 9 .

OCR Further Additional Pure AS 2019 June Q3

3 The non-zero vectors $\mathbf { x }$ and $\mathbf { y }$ are such that $\mathbf { x } \times \mathbf { y } = \mathbf { 0 }$.

Explain the geometrical significance of this statement.
Use your answer to part (a) to explain how the line equation $\mathbf { r } = \mathbf { a } + t \mathbf { d }$ can be written in the form $( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }$.

OCR Further Additional Pure AS 2019 June Q4

4 The sequence $\left\{ \mathrm { u } _ { \mathrm { n } } \right\}$ is defined by $u _ { 1 } = 1$ and $\mathrm { u } _ { \mathrm { n } + 1 } = 2 \mathrm { u } _ { \mathrm { n } } + \mathrm { n } ^ { 2 }$ for $\mathrm { n } \geqslant 1$.
Determine $u _ { n }$ as a function of $n$.

OCR Further Additional Pure AS 2019 June Q5

5 The tetrahedron $T$, shown below, has vertices at $O ( 0,0,0 ) , A ( 1,2,2 ) , B ( 2,1,2 )$ and $C ( 2,2,1 )$.
\includegraphics[max width=\textwidth, alt={}, center]{59fa1650-a296-471e-93b9-0988177cd89d-3_360_464_319_555} Diagram not drawn to scale Show that the surface area of $T$ is $\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )$.

OCR Further Additional Pure AS 2019 June Q6

Determine all values of $x$ for which $16 x \equiv 5 ( \bmod 101 )$.
Solve
1. $95 x \equiv 6 ( \bmod 101 )$,
2. $95 x \equiv 5 ( \bmod 101 )$.

OCR Further Additional Pure AS 2019 June Q7

7 You are given the set $S = \{ 1,5,7,11,13,17 \}$ together with $\times _ { 18 }$, the operation of multiplication modulo 18.

Complete the Cayley table for $\left( S , \times _ { 18 } \right)$ given in the Printed Answer Booklet.
Prove that ( $S , \times _ { 18 }$ ) is a group. (You may assume that $\times _ { 18 }$ is associative.)
Write down the order of each element of the group.
Show that $\left( S , \times _ { 18 } \right)$ is a cyclic group.
1. Give an example of a non-cyclic group of order 6 .
2. Give one reason why your example is structurally different to $\left( S , { } _ { 18 } \right)$.

OCR Further Additional Pure AS 2019 June Q8

8 The motion of two remote controlled helicopters $P$ and $Q$ is modelled as two points moving along straight lines. Helicopter $P$ moves on the line $\mathbf { r } = \left( \begin{array} { r } 2 + 4 p
- 3 + p
1 + 3 p \end{array} \right)$ and helicopter $Q$ moves on the line $\mathbf { r } = \left( \begin{array} { l } 5 + 8 q
2 + q
5 + 4 q \end{array} \right)$.
The function $z$ denotes $( P Q ) ^ { 2 }$, the square of the distance between $P$ and $Q$.

Show that $z = 26 p ^ { 2 } + 81 q ^ { 2 } - 90 p q - 58 p + 90 q + 50$.
Use partial differentiation to find the values of $p$ and $q$ for which $z$ has a stationary point.
With the aid of a diagram, explain why this stationary point must be a minimum point, rather than a maximum point or a saddle point.
Hence find the shortest possible distance between the two helicopters. The model is now refined by modelling each helicopter as a sphere of radius 0.5 units.
Explain how this will change your answer to part (d). \section*{END OF QUESTION PAPER}

OCR Further Additional Pure AS 2022 June Q1

1 The points $A , B$ and $C$ have position vectors $\mathbf { a } = \left( \begin{array} { l } 1
1
3 \end{array} \right) , \mathbf { b } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right)$ and $\mathbf { c } = \left( \begin{array} { r } - 5
1
2 \end{array} \right)$ respectively, relative to the origin $O$.

Calculate, in its simplest exact form, the area of triangle $O A B$.
Show that $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) + \mathbf { b } \times ( \mathbf { c } \times \mathbf { a } ) + \mathbf { c } \times ( \mathbf { a } \times \mathbf { b } ) = \mathbf { 0 }$.

OCR Further Additional Pure AS 2022 June Q2

2 The surface $S$ has equation $z = x ^ { 3 } + y ^ { 3 } - 2 x ^ { 2 } - 5 y ^ { 2 } + 3 x y$.
It is given that $S$ has two stationary points; one at the origin, $O$, and the other at the point $A$.
Determine the coordinates of $A$.

OCR Further Additional Pure AS 2022 June Q3

3 The sequence $\left\{ U _ { n } \right\}$ is given by $U _ { 1 } = 0 , U _ { 2 } = - 1$ and $U _ { n + 2 } = U _ { n + 1 } + U _ { n } + n - 1$ for $n \geqslant 1$.

List the first seven terms of this sequence. The Fibonacci sequence $\left\{ \mathrm { F } _ { \mathrm { n } } \right\}$ is given by $\mathrm { F } _ { 1 } = 1 , \mathrm {~F} _ { 2 } = 1$ and $\mathrm { F } _ { \mathrm { n } + 2 } = \mathrm { F } _ { \mathrm { n } + 1 } + \mathrm { F } _ { \mathrm { n } }$ for $n \geqslant 1$.
1. By comparing the two sequences, give the relationship between $\mathrm { U } _ { \mathrm { n } }$ and $\mathrm { F } _ { \mathrm { n } }$.
2. Show that the relationship found in part (b)(i) holds for all $n \geqslant 1$.

OCR Further Additional Pure AS 2022 June Q4

4 Let $\mathrm { N } = 10 \mathrm { a } + \mathrm { b }$ and $\mathrm { M } = \mathrm { a } + 3 \mathrm {~b}$, where $a$ and $b$ are integers such that $a \geqslant 1$ and $0 \leqslant b \leqslant 9$.

Prove that $29 \mid N$ if and only if $29 \mid M$.
Use an iterative method based on the result of part (a) to show that 899364472 is a multiple of 29 .

OCR Further Additional Pure AS 2022 June Q5

5 A research student is using 3-D graph-plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea-level at regular intervals, (approximately) distributed along a straight line. Each island takes the form of a single peak; also, along the line of islands, the heights of these peaks decrease in size in an (approximately) regular fashion (see Fig. 1.1). \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{06496165-0b83-4050-ae26-fa5a0614bd46-3_476_812_495_246} \captionsetup{labelformat=empty} \caption{Fig. 1.1}

\end{figure} The student's model uses the surface with equation $\mathrm { z } = \sin \mathrm { x } + \sin \mathrm { y }$, a part of which is shown in Fig. 1.2 below. The surface of the sea is taken to be the plane $z = 0$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{06496165-0b83-4050-ae26-fa5a0614bd46-3_789_951_1270_242} \captionsetup{labelformat=empty} \caption{Fig. 1.2}

\end{figure}

- Describe two problems with this model.
- Suggest revisions to this model so that each of these problems is addressed.
- Still using their original model, the student examines the contour $z = 2$ for their surface only to find that the software shows what appears to be an empty graph.
Explain what has happened.

OCR Further Additional Pure AS 2022 June Q6

6 The sequence $\left\{ u _ { n } \right\}$ is such that $u _ { 1 } = 7 , u _ { 2 } = 37 , u _ { 3 } = 337 , u _ { 4 } = 3337 , \ldots$.

Write down a first-order recurrence system for $\left\{ \mathrm { u } _ { \mathrm { n } } \right\}$.
By solving the recurrence system of part (a), show that $\mathrm { u } _ { \mathrm { n } } = \frac { 1 } { 3 } \left( 10 ^ { \mathrm { n } } + 11 \right)$.
Prove that $\left\{ \mathrm { u } _ { \mathrm { n } } \right\}$ contains infinitely many terms which are multiples of 37 .

OCR Further Additional Pure AS 2022 June Q7

7 The diagram below shows an equilateral triangle $A B C$. The three lines of reflection symmetry of $A B C$ (the lines $a , b$ and $c$ ) are shown as broken lines. The point of intersection of these three lines, $O$, is the centre of rotational symmetry of the triangle.
\includegraphics[max width=\textwidth, alt={}, center]{06496165-0b83-4050-ae26-fa5a0614bd46-4_533_538_884_246} The group $D _ { 3 }$ is defined as the set of symmetries of $A B C$ under the composition of the following transformations.
$i$ : the identity transformation
$a$ : reflection in line $a$
$b$ : reflection in line $b$
$c$ : reflection in line $c$
$p$ : an anticlockwise rotation about $O$ through $120 ^ { \circ }$
$q$ : a clockwise rotation about $O$ through $120 ^ { \circ }$
Note that the lines $a , b$ and $c$ are unaffected by the transformations and remain fixed.

On the diagrams provided in the Printed Answer Booklet, show each of the six elements of $D _ { 3 }$ obtained when the above transformations are applied to triangle $A B C$.
Complete the Cayley table given in the Printed Answer Booklet.
List all the proper subgroups of $D _ { 3 }$.
State, with justification, whether $D _ { 3 }$ is
1. cyclic,
2. abelian.
The group $H$, also of order 6, is the set of rotational symmetries of the regular hexagon. Describe two structural differences between $D _ { 3 }$ and $H$. \section*{END OF QUESTION PAPER}

OCR Further Additional Pure AS 2023 June Q1

Express 205 in the form $7 q + r$ for positive integers $q$ and $r$, with $0 \leqslant r < 7$.
Given that $7 \mid ( 205 \times 8666 )$, use the result of part (a) to justify that $7 \mid 8666$.

OCR Further Additional Pure AS 2023 June Q2

2 For all positive integers $n$, the terms of the sequence $\left\{ u _ { n } \right\}$ are given by the formula $u _ { n } = 3 n ^ { 2 } + 3 n + 7 ( \bmod 10 )$.

Show that $u _ { n + 5 } = u _ { n }$ for all positive integers $n$.
Hence describe the behaviour of the sequence, justifying your answer.

OCR Further Additional Pure AS 2023 June Q3

3 A surface has equation $z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y$ for all real values of $x$ and $y$. Determine the coordinates of all stationary points of this surface.

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