Questions - OCR - A-Level Maths

OCR Further Pure Core AS 2023 June Q5

4 marks Moderate -0.3

5 In this question you must show detailed reasoning. The roots of the equation $5 x ^ { 2 } - 3 x + 12 = 0$ are $\alpha$ and $\beta$. By considering the symmetric functions of the roots, $\alpha + \beta$ and $\alpha \beta$, determine the exact value of $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$.

OCR Further Pure Core AS 2023 June Q6

6 marks Moderate -0.3

6 Prove by induction that $4 \times 8 ^ { n } + 66$ is divisible by 14 for all integers $n \geqslant 0$.

OCR Further Pure Core AS 2023 June Q7

6 marks Standard +0.8

7 In this question you must show detailed reasoning. Matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c c } a & - 6 & a - 3 \\ a + 9 & a & 4 \\ 0 & - 13 & a - 1 \end{array} \right)$ where $a$ is a constant.
Find all possible values of $a$ for which $\operatorname { det } \mathbf { A }$ has the same value as it has when $a = 2$.

OCR Further Pure Core AS 2023 June Q8

9 marks Moderate -0.3

8

Solve the equation $\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }$.
Prove algebraically that, for non-zero $z , z = - z ^ { * }$ if and only if $z$ is purely imaginary.
The complex numbers $z$ and $z ^ { * }$ are represented on an Argand diagram by the points $A$ and $B$ respectively.
1. State, for any $z$, the single transformation which transforms $A$ to $B$.
2. Use a geometric argument to prove that $z = z ^ { * }$ if and only if $z$ is purely real.

OCR Further Pure Core AS 2023 June Q9

10 marks Standard +0.8

9 Matrix $\mathbf { R }$ is given by $\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b \\ 0 & 1 & 0 \\ b & 0 & a \end{array} \right)$ where $a$ and $b$ are constants.

Find $\mathbf { R } ^ { 2 }$ in terms of $a$ and $b$. The constants $a$ and $b$ are given by $a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )$ and $b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )$.
By determining exact expressions for $a b$ and $a ^ { 2 } - b ^ { 2 }$ and using the result from part (a), show that $\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1 \\ 0 & 2 & 0 \\ 1 & 0 & \sqrt { 3 } \end{array} \right)$ where $k$ is a real number whose value is to be determined.
Find $\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }$ and $\mathbf { R } ^ { 24 }$.
Describe fully the transformation represented by $\mathbf { R }$. \section*{END OF QUESTION PAPER}

OCR Further Pure Core AS 2021 November Q1

5 marks Standard +0.3

1 The lines $l _ { 1 }$ and $l _ { 2 }$ have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 8 \\ - 11 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 5 \\ 3 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6 \\ 11 \\ 8 \end{array} \right) + \mu \left( \begin{array} { r } - 3 \\ 1 \\ - 1 \end{array} \right) \end{aligned}$$

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect.
Write down the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core AS 2021 November Q2

4 marks Standard +0.3

2 The equation $2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 5 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Use a substitution to find a cubic equation with integer coefficients whose roots are $\alpha + 1 , \beta + 1$ and $\gamma + 1$.

OCR Further Pure Core AS 2021 November Q3

6 marks Standard +0.3

3 In this question you must show detailed reasoning.
The equation $x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0$ has a root $3 - 5 i$.
Find the other three roots of this equation.

OCR Further Pure Core AS 2021 November Q4

7 marks Standard +0.3

4

A locus $C _ { 1 }$ is defined by $C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}$.
1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 1 }$.
2. Find the cartesian equation of the boundary line of the region representing $C _ { 1 }$, giving your answer in the form $a x + b y + c = 0$.
A locus $C _ { 2 }$ is defined by $C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}$. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 2 }$.

OCR Further Pure Core AS 2021 November Q5

8 marks Moderate -0.3

5 Matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r l } - 1 & 0 \\ 0 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$.

Use $\mathbf { A }$ and $\mathbf { B }$ to disprove the proposition: "Matrix multiplication is commutative". Matrix $\mathbf { B }$ represents the transformation $\mathrm { T } _ { \mathrm { B } }$.
Describe the transformation $\mathrm { T } _ { \mathrm { B } }$.
By considering the inverse transformation of $\mathrm { T } _ { \mathrm { B } }$, determine $\mathbf { B } ^ { - 1 }$. Matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 3 \end{array} \right)$ and represents the transformation $\mathrm { T } _ { \mathrm { C } }$.
The transformation $\mathrm { T } _ { \mathrm { BC } }$ is transformation $\mathrm { T } _ { \mathrm { C } }$ followed by transformation $\mathrm { T } _ { \mathrm { B } }$.
An object shape of area 5 is transformed by $\mathrm { T } _ { \mathrm { BC } }$ to an image shape $N$.
Determine the area of $N$.

OCR Further Pure Core AS 2021 November Q6

6 marks Moderate -0.8

6 In this question you must show detailed reasoning.

Solve the equation $2 z ^ { 2 } - 10 z + 25 = 0$ giving your answers in the form $\mathrm { a } + \mathrm { bi }$.
Solve the equation $3 \omega - 2 = \mathrm { i } ( 5 + 2 \omega )$ giving your answer in the form $\mathrm { a } + \mathrm { bi }$.

OCR Further Pure Core AS 2021 November Q7

5 marks Standard +0.3

7 Prove that $2 ^ { 3 n } - 3 ^ { n }$ is divisible by 5 for all integers $n \geqslant 1$.

OCR Further Pure Core AS 2021 November Q8

6 marks Standard +0.8

8 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1 \\ 1 - t & 6 & t \\ 2 - 2 t & 2 - 2 t & 1 \end{array} \right)$.

Find, in fully factorised form, an expression for $\operatorname { det } \mathbf { A }$ in terms of $t$.
State the values of $t$ for which $\mathbf { A }$ is singular. You are given the following system of equations in $x , y$ and $z$, where $b$ is a real number. $$\begin{aligned} \left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5 \\ \left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10 \\ \left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15 \end{aligned}$$
Determine which one of the following statements about the solution of the equations is true.

OCR Further Pure Core AS 2021 November Q9

13 marks Challenging +1.2

9 The points $P ( 3,5 , - 21 )$ and $Q ( - 1,3 , - 16 )$ are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point $M$ on the ceiling of the tunnel midway between $P$ and $Q$ to horizontal ground level (where the $z$-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$, where $\mathbf { a }$ is the position vector of $M$. You are given that $\mathbf { b } = \left( \begin{array} { l } 1 \\ \mathrm {~s} \\ \mathrm { t } \end{array} \right)$ where $s$ and $t$ are real numbers.

Show that $\mathrm { S } = 2.5 \mathrm { t } - 2$.
Show that at the point where the ventilation shaft reaches the ground $\lambda = \frac { \mathrm { C } } { \mathrm { t } }$, where $c$ is a constant to be determined.
Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
Explain what the fact that $\mathbf { b } \times \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right) \neq \mathbf { O }$ means about the direction of the ventilation shaft.

OCR Further Discrete AS 2021 November Q1

10 marks Challenging +1.2

1 A set consists of five distinct non-integer values, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E . The set is partitioned into non-empty subsets and there are at least two subsets in each partition.

Show that there are 15 different partitions into two subsets.
Show that there are 25 different partitions into three subsets.
Calculate the total number of different partitions. The numbers 12, 24, 36, 48, 60, 72, 84 and 96 are marked on a number line. The number line is then cut into pieces by making cuts at $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E , where $0 < \mathrm { A } < \mathrm { B } < \mathrm { C } < \mathrm { D } < \mathrm { E } < 100$.
Explain why there must be at least one piece with two or more of the numbers 12, 24, 36, 48, 60, 72, 84 and 96.

OCR Further Discrete AS 2021 November Q2

11 marks Moderate -0.3

2 Seven items need to be packed into bins. Each bin has capacity 30 kg . The sizes of the items, in kg, in the order that they are received, are as follows.
12
23
15
18
8
7
5

Find the packing that results using each of these algorithms.
1. The next-fit method
2. The first-fit method
3. The first-fit decreasing method
A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity $M \mathrm {~kg}$, where $M$ is an integer.
The item of size 23 kg can be split into two items, of sizes $x \mathrm {~kg}$ and $( 23 - x ) \mathrm { kg }$, where $x$ may be any integer value you choose from 1 to 11. No other item can be split.
Determine the smallest value of $M$ for which four bins are needed to pack these eight items. Explain your reasoning carefully.

OCR Further Discrete AS 2021 November Q3

10 marks Standard +0.3

3 The diagram shows a simplified map of the main streets in a small town. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
There are no traffic lights at junctions X and Y .
The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.

Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
Complete the copy of the table in the Printed Answer Booklet.
Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .
Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c).

OCR Further Discrete AS 2021 November Q5

11 marks Moderate -0.8

5

Determine the smallest value of $M$ for which four bins are needed to pack these eight items. Explain your reasoning carefully. 3 The diagram shows a simplified map of the main streets in a small town. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
There are no traffic lights at junctions X and Y .
The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.

Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c). 4 Li and Mia play a game in which they simultaneously play one of the strategies $\mathrm { X } , \mathrm { Y }$ and Z . The tables show the points won by each player for each combination of strategies.
A negative entry means that the player loses that number of points.

	Mia
	X	Y	Z
\multirow{3}{*}{Li}	X	5	- 6	0
\cline { 2 - 5 }	Y	- 2	3	4
\cline { 2 - 5 }	Z	- 1	4	8
\cline { 2 - 5 }

	Mia
	X	Y	Z
\multirow{2}{*}{Li}	X	4
\cline { 2 - 5 }	Y	11		5
\cline { 2 - 5 }	Z	10	5	1
\cline { 2 - 5 }

The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.

Complete the table in the Printed Answer Booklet to show the points won by Mia.
Convert the game into a zero-sum game, giving the pay-offs for Li .

Use dominance to reduce the pay-off matrix for the game to a $2 \times 2$ table. You do not need to explain the dominance. Mia knows that Li will choose his play-safe strategy.

Determine which strategy Mia should choose to maximise her points. 5 A linear programming problem is formulated as below. Maximise $\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }$ subject to $2 x + 3 y \geqslant 12$ $x + y \leqslant 10$ $5 x + 2 y \leqslant 30$ $x \geqslant 0 , y \geqslant 0$

Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
Hence determine the maximum value of the objective. The constraint $x + y \leqslant 10$ is changed to $x + y \leqslant k$, the other constraints are unchanged.

Determine, algebraically, the value of $k$ for which the maximum value of $P$ is 3 . Do not draw on the graph from part (a) and do not use the spare grid.

Determine, algebraically, the other value of $k$ for which there is a (non-optimal) vertex of the feasible region at which $P = 3$.
Do not draw on the graph from part (a) and do not use the spare grid.

OCR Further Discrete AS 2021 November Q18

1 marks Moderate -0.8

18
8
7
5

Find the packing that results using each of these algorithms.
1. The next-fit method
2. The first-fit method
3. The first-fit decreasing method
A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity $M \mathrm {~kg}$, where $M$ is an integer.
The item of size 23 kg can be split into two items, of sizes $x \mathrm {~kg}$ and $( 23 - x ) \mathrm { kg }$, where $x$ may be any integer value you choose from 1 to 11. No other item can be split.

Determine the smallest value of $M$ for which four bins are needed to pack these eight items. Explain your reasoning carefully. 3 The diagram shows a simplified map of the main streets in a small town. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
There are no traffic lights at junctions X and Y .
The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.

Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
Complete the copy of the table in the Printed Answer Booklet.
Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .

Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c). 4 Li and Mia play a game in which they simultaneously play one of the strategies $\mathrm { X } , \mathrm { Y }$ and Z . The tables show the points won by each player for each combination of strategies.
A negative entry means that the player loses that number of points.

	Mia
	X	Y	Z
\multirow{3}{*}{Li}	X	5	- 6	0
\cline { 2 - 5 }	Y	- 2	3	4
\cline { 2 - 5 }	Z	- 1	4	8
\cline { 2 - 5 }

	Mia
	X	Y	Z
\multirow{2}{*}{Li}	X	4
\cline { 2 - 5 }	Y	11		5
\cline { 2 - 5 }	Z	10	5	1
\cline { 2 - 5 }

The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.

Complete the table in the Printed Answer Booklet to show the points won by Mia.
Convert the game into a zero-sum game, giving the pay-offs for Li .

Use dominance to reduce the pay-off matrix for the game to a $2 \times 2$ table. You do not need to explain the dominance. Mia knows that Li will choose his play-safe strategy.

Determine which strategy Mia should choose to maximise her points. 5 A linear programming problem is formulated as below. Maximise $\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }$ subject to $2 x + 3 y \geqslant 12$ $x + y \leqslant 10$ $5 x + 2 y \leqslant 30$ $x \geqslant 0 , y \geqslant 0$

Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
Hence determine the maximum value of the objective. The constraint $x + y \leqslant 10$ is changed to $x + y \leqslant k$, the other constraints are unchanged.

Determine, algebraically, the value of $k$ for which the maximum value of $P$ is 3 . Do not draw on the graph from part (a) and do not use the spare grid.

Determine, algebraically, the other value of $k$ for which there is a (non-optimal) vertex of the feasible region at which $P = 3$.
Do not draw on the graph from part (a) and do not use the spare grid. 6 Sarah is having some work done on her garden.
The table below shows the activities involved, their durations and their immediate predecessors. These durations and immediate predecessors are known to be correct.

Activity	Immediate predecessors	Duration (hours)
A Clear site	-	4
B Mark out new design	A	1
C Buy materials, turf, plants and trees	-	3
D Lay paths	B, C	1
E Build patio	B, C	2
F Plant trees	D	1
G Lay turf	D, E	1
H Finish planting	F, G	1

Use a suitable model to determine the following.

Sarah needs the work to be completed as quickly as possible. There will be at least one activity happening at all times, but it may not always be possible to do all the activities that are needed at the same time.

Determine the earliest and latest times at which building the patio (activity E) could start. There needs to be a 2-hour break after laying the paths (activity D). During this time other activities that do not depend on activity D can still take place.

Describe how you would adapt your model to incorporate the 2-hour break.

OCR Further Additional Pure AS 2024 June Q1

2 marks Easy -1.2

1 In this question you must show detailed reasoning. The number $N$ is written as 28 A 3 B in base-12 form. Express $N$ in decimal (base-10) form.

OCR Further Additional Pure AS 2024 June Q2

6 marks Standard +0.3

2 The points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to the origin $O$. It is given that $\mathbf { a } = \left( \begin{array} { c } 2 \\ 4 \\ 3 \lambda \end{array} \right)$ and $\mathbf { b } = \left( \begin{array} { r } \lambda \\ - 4 \\ 6 \end{array} \right)$, where $\lambda$ is a real parameter.

In the case when $\lambda = 3$, determine the area of triangle $O A B$.
Determine the value of $\lambda$ for which $\mathbf { a } \times \mathbf { b } = \mathbf { 0 }$.

OCR Further Additional Pure AS 2024 June Q3

12 marks Standard +0.8

3 The surface $S$ has equation $z = f ( x , y )$, where $f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }$ for all real values of $x$ and $y$. You are given that $S$ has a stationary point at the origin, $O$, and a second stationary point at the point $P ( a , b , c )$, where $\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )$. \begin{enumerate}[label=(\alph*)] \item Determine the values of $a , b$ and $c$. \item Throughout this part, take the values of $a$ and $b$ to be those found in part (a).

Evaluate $\mathrm { f } _ { x }$ at the points $\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )$ and $\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )$.
Evaluate $\mathrm { f } _ { y }$ at the points $\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )$ and $\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )$.
Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of $S$, given by

OCR Further Additional Pure AS 2024 June Q4

5 marks Standard +0.8

4 The first five terms of the Fibonacci sequence, $\left\{ \mathrm { F } _ { \mathrm { n } } \right\}$, where $n \geqslant 1$, are $F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , F _ { 4 } = 3$ and $F _ { 5 } = 5$.

Use the recurrence definition of the Fibonacci sequence, $\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }$, to express $\mathrm { F } _ { \mathrm { n } + 4 }$ in terms of $\mathrm { F } _ { \mathrm { n } }$ and $\mathrm { F } _ { \mathrm { n } - 1 }$.
Hence prove by induction that $\mathrm { F } _ { \mathrm { n } }$ is a multiple of 3 when $n$ is a multiple of 4 .

OCR Further Additional Pure AS 2024 June Q5

14 marks Challenging +1.2

5 The set $S$ consists of all $2 \times 2$ matrices having determinant 1 or - 1 . For instance, the matrices $\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)$ and $\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)$ are elements of $S$. It is given that $\times _ { \mathbf { M } }$ is the operation of matrix multiplication.

State the identity element of $S$ under $\times _ { \mathbf { M } }$. The group $G$ is generated by $\mathbf { P }$, under $\times _ { \mathbf { M } }$.
Determine the order of $G$. The group $H$ is generated by $\mathbf { Q }$ and $\mathbf { R }$, also under $\times _ { \mathbf { M } }$.
1. By finding each element of $H$, determine the order of $H$.
2. List all the proper subgroups of $H$.
State whether each of the following statements is true or false. Give a reason for each of your answers.

OCR Further Additional Pure AS 2024 June Q6

9 marks Challenging +1.8

6 For positive integers $n$, let $f ( n ) = 1 + 2 ^ { n } + 4 ^ { n }$.

1. Given that $n$ is a multiple of 3 , but not of 9 , use the division algorithm to write down the two possible forms that $n$ can take.
2. Show that when $n$ is a multiple of 3 , but not of 9 , $f ( n )$ is a multiple of 73 .
Determine the value of $\mathrm { f } ( n )$, modulo 73 , in the case when $n$ is a multiple of 9 .

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