Questions - OCR Further Additional Pure

Determine $\mathrm { f } _ { x } , \mathrm { f } _ { y } , \mathrm { f } _ { x x } , \mathrm { f } _ { y y } , \mathrm { f } _ { x y }$ and $\mathrm { f } _ { y x }$.
1. Verify that $z$ has a stationary point at $\left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi , \frac { 1 } { 4 } \pi ^ { 2 } \right)$.
2. Determine the nature of this stationary point.

OCR Further Additional Pure 2019 June Q3

Solve $7 x \equiv 6 ( \bmod 19 )$.
Show that the following simultaneous linear congruences have no solution. $$x \equiv 3 ( \bmod 4 ) , x \equiv 4 ( \bmod 6 )$$

OCR Further Additional Pure 2019 June Q4

Solve the second-order recurrence relation $T _ { n + 2 } + 2 T _ { n } = - 87$ given that $T _ { 0 } = - 27$ and $T _ { 1 } = 27$.
Determine the value of $T _ { 20 }$.

OCR Further Additional Pure 2019 June Q5

5 The group $G$ consists of a set $S$ together with $\times _ { 80 }$, the operation of multiplication modulo 80. It is given that $S$ is the smallest set which contains the element 11 .

By constructing the Cayley table for $G$, determine all the elements of $S$. The Cayley table for a second group, $H$, also with the operation $\times _ { 80 }$, is shown below.
\cline { 2 - 5 } \multicolumn{1}{c|}{$\times _ { 80 }$} 1 9 31 39
1 1 9 31 39
9 9 1 39 31
31 31 39 1 9
39 39 31 9 1
Use the two Cayley tables to explain why $G$ and $H$ are not isomorphic.
1. List
  - all the proper subgroups of $G$,
all the proper subgroups of $H$.
(ii) Use your answers to (c) (i) to give another reason why $G$ and $H$ are not isomorphic.

OCR Further Additional Pure 2019 June Q6

For the vectors $\mathbf { p } = \left( \begin{array} { l } 1
2
3 \end{array} \right) , \mathbf { q } = \left( \begin{array} { r } 3
1
- 1 \end{array} \right)$ and $\mathbf { r } = \left( \begin{array} { r } 2
- 4
5 \end{array} \right)$, calculate
- $\mathbf { p } \cdot \mathbf { q } \times \mathbf { r }$,
- $\mathbf { p } \times ( \mathbf { q } \times \mathbf { r } )$,
- $( \mathbf { p } \times \mathbf { q } ) \times \mathbf { r }$.
- State whether the vector product is associative for three-dimensional column vectors with real components. Justify your answer.
It is given that $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ are three-dimensional column vectors with real components.
Explain geometrically why the vector $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )$ must be expressible in the form $\lambda \mathbf { b } + \mu \mathbf { c }$, where $\lambda$ and $\mu$ are scalar constants. It is given that the following relationship holds for $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$.
$\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) = ( \mathbf { a } \cdot \mathbf { c } ) \mathbf { b } - ( \mathbf { a } \cdot \mathbf { b } ) \mathbf { c }$
Find an expression for ( $\mathbf { a } \times \mathbf { b ) } \times \mathbf { c }$ in the form of (*).

OCR Further Additional Pure 2019 June Q7

7 The points $P \left( \frac { 1 } { 2 } , \frac { 13 } { 24 } \right)$ and $Q \left( \frac { 3 } { 2 } , \frac { 31 } { 24 } \right)$ lie on the curve $y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }$.
The area of the surface generated when arc $P Q$ is rotated completely about the $x$-axis is denoted by $A$.

Find the exact value of $A$. Give your answer as a rational multiple of $\pi$. Student X finds an approximation to $A$ by modelling the arc $P Q$ as the straight line segment $P Q$, then rotating this line segment completely about the $x$-axis to form a surface.
Find the approximation to $A$ obtained by student X . Give your answer as a rational multiple of $\pi$. Student Y finds a second approximation to $A$ by modelling the original curve as the line $y = M$, where $M$ is the mean value of the function $\mathrm { f } ( x ) = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }$, then rotating this line completely about the $x$-axis to form a surface.
Find the approximation to $A$ obtained by student Y . Give your answer correct to four decimal places.

OCR Further Additional Pure 2022 June Q1

1 The surface $E$ has equation $z = \sqrt { 500 - 3 x ^ { 2 } - 2 y ^ { 2 } }$.

Determine the values of $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ at the point $P$ on $E$ with coordinates $( 11 , - 8,3 )$.
Find the equation of the tangent plane to $E$ at $P$, giving your answer in the form $\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }$ where $a , b , c$ and $d$ are integers.

OCR Further Additional Pure 2022 June Q2

2 Consider the integers $a$ and $b$, where, for each integer $n , \mathrm { a } = 7 \mathrm { n } + 4$ and $\mathrm { b } = 8 \mathrm { n } + 5$. Let $\mathrm { h } = \mathrm { hcf } ( \mathrm { a } , \mathrm { b } )$.

Determine all possible values of $h$.
Find all values of $n$ for which $a$ and $b$ are not co-prime.

OCR Further Additional Pure 2022 June Q3

3 The irrational number $\phi = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )$ plays a significant role in the sequence of Fibonacci numbers given by $\mathrm { F } _ { 0 } = 0 , \mathrm {~F} _ { 1 } = 1$ and $\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }$ for $n \geqslant 1$. Prove by induction that, for each positive integer $n , \phi ^ { n } = \mathrm { F } _ { \mathrm { n } } \times \phi + \mathrm { F } _ { \mathrm { n } - 1 }$.

OCR Further Additional Pure 2022 June Q4

4 Let $N$ be the number 15824578 .

1. Use a standard divisibility test to show that $N$ is a multiple of 11 .
2. A student uses the following test for divisibility by 7 . \begin{displayquote} 'Throw away' multiples of 7 that appear either individually or within a pair of consecutive digits of the test number.
  Stop when the number obtained is $0,1,2,3,4,5$ or 6 .
  The test number is only divisible by 7 if that obtained number is 0 . \end{displayquote} For example, for the number $N$, they first 'throw away' the " 7 " in the tens column, leaving the number $N _ { 1 } = 15824508$. At the second stage, they 'throw away' the " 14 " from the left-hand pair of digits of $N _ { 1 }$, leaving $N _ { 2 } = 01824508$; and so on, until a number is obtained which is $0,1,2,3,4,5$ or 6 .
  - Justify the validity of this process.
Continue the student's test to show that $7 \mid N$.
(iii) Given that $N = 11 \times 1438598$, explain why 7| 1438598 .
Let $\mathrm { M } = \mathrm { N } ^ { 2 }$.
1. Express $N$ in the unique form 101a + b for positive integers $a$ and $b$, with $0 \leqslant b < 101$.
2. Hence write $M$ in the form $\mathrm { M } \equiv \mathrm { r } ( \bmod 101 )$, where $0 < r < 101$.
3. Deduce the order of $N$ modulo 101.

OCR Further Additional Pure 2022 June Q5

5 You are given the variable point $A ( 3 , - 8 , t )$, where $t$ is a real parameter, and the fixed point $B ( 1,2 , - 2 )$.

Using only the geometrical properties of the vector product, explain why the statement " $\overrightarrow { \mathrm { OA } } \times \overrightarrow { \mathrm { OB } } = \mathbf { 0 }$ " is false for all values of $t$.
1. Use the vector product to find an expression, in terms of $t$, for the area of triangle $O A B$.
2. Hence determine the value of $t$ for which the area of triangle $O A B$ is a minimum.

OCR Further Additional Pure 2022 June Q6

1 marks

6 In a national park, the number of adults of a given species is carefully monitored and controlled. The number of adults, $n$ months after the start of this project, is $A _ { n }$. Initially, there are 1000 adults. It is predicted that this number will have declined to 960 after one month. The first model for the number of adults is that, from one month to the next, a fixed proportion of adults is lost. In order to maintain a fixed number of adults, the park managers "top up" the numbers by adding a constant number of adults from other parks at the end of each month.

Use this model to express the number of adults as a first-order recurrence system. Instead, it is found that, the proportion of adults lost each month is double the predicted amount, with no change being made to the constant number of adults added each month.
1. Show that the revised recurrence system for $A _ { n }$ is $A _ { 0 } = 1000 , A _ { n + 1 } = 0.92 A _ { n } + 40$. [1]
2. Solve this revised recurrence system.
3. Describe the long-term behaviour of the sequence $\left\{ A _ { n } \right\}$ in this case. A more refined model for the number of adults uses the second-order recurrence system $\mathrm { A } _ { \mathrm { n } + 1 } = 0.9 \mathrm {~A} _ { \mathrm { n } } - 0.1 \mathrm {~A} _ { \mathrm { n } - 1 } + 50$, for $n \geqslant 1$, with $A _ { 0 } = 1000$ and $A _ { 1 } = 920$.
1. Determine the long-term behaviour of the sequence $\left\{ A _ { n } \right\}$ for this more refined model.
2. A criticism of this more refined model is that it does not take account of the fact that the number of adults must be an integer at all times. State a modified form of the second-order recurrence relation for this more refined model that will satisfy this requirement.

OCR Further Additional Pure 2022 June Q7

Differentiate $\left( 16 + t ^ { 2 } \right) ^ { \frac { 3 } { 2 } }$ with respect to $t$. Let $I _ { n } = \int _ { 0 } ^ { 3 } t ^ { n } \sqrt { 16 + t ^ { 2 } } d t$ for integers $n \geqslant 1$.
Show that, for $n \geqslant 3 , \left. ( n + 2 ) \right| _ { n } = 125 \times 3 ^ { n - 1 } - \left. 16 ( n - 1 ) \right| _ { n - 2 }$.
The curve $C$ is defined parametrically by $\mathrm { x } = \mathrm { t } ^ { 4 } \cos \mathrm { t }$, $\mathrm { y } = \mathrm { t } ^ { 4 } \sin \mathrm { t }$, for $0 \leqslant t \leqslant 3$. The length of $C$ is denoted by $L$. Show that $\mathrm { L } = \mathrm { I } _ { 3 }$. (You are not required to evaluate this integral.)

OCR Further Additional Pure 2022 June Q8

Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
Prove that any group, in which every (non-identity) element is self-inverse, is abelian.

A student believes that, if $x$ and $y$ are two distinct, non-identity, self-inverse elements of a group, then the element $x y$ is also self-inverse. The table shown here is the Cayley table for the non-cyclic group of order 6, having elements $i , a , b , c , d$ and $e$, where $i$ is the identity.

	$i$	$a$	$b$	$c$	$d$	$e$
$i$	$i$	$a$	$b$	$c$	$d$	$e$
$a$	$a$	$i$	$d$	$e$	$b$	$c$
$b$	$b$	$e$	$i$	$d$	$c$	$a$
$c$	$c$	$d$	$e$	$i$	$a$	$b$
$d$	$d$	$c$	$a$	$b$	$e$	$i$
$e$	$e$	$b$	$c$	$a$	$i$	$d$

By considering the elements of this group, produce a counter-example which proves that this student is wrong.

A group $G$ has order $4 n + 2$, for some positive integer $n$, and $i$ is the identity element of $G$. Let $x$ and $y$ be two distinct, non-identity, self-inverse elements of $G$. By considering the set $\mathrm { H } = \{ \mathrm { i } , \mathrm { x } , \mathrm { y } , \mathrm { xy } \}$, prove by contradiction that not all elements of $G$ are self-inverse.

OCR Further Additional Pure 2022 June Q9

9 For all real values of $x$ and $y$ the surface $S$ has equation $z = 4 x ^ { 2 } + 4 x y + y ^ { 2 } + 6 x + 3 y + k$, where $k$ is a constant and an integer.

Find $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$.
Determine the smallest value of the integer $k$ for which the whole of $S$ lies above the $x - y$ plane.

OCR Further Additional Pure 2023 June Q1

1 The surface $S$ is defined for all real $x$ and $y$ by the equation $z = x ^ { 2 } + 2 x y$. The intersection of $S$ with the plane $\Pi$ gives a section of the surface. On the axes provided in the Printed Answer Booklet, sketch this section when the equation of $\Pi$ is each of the following.

$x = 1$
$y = 1$

OCR Further Additional Pure 2023 June Q2

2 A curve has equation $\mathrm { y } = \sqrt { 1 + \mathrm { x } ^ { 2 } }$, for $0 \leqslant x \leqslant 1$, where both the $x$ - and $y$-units are in cm. The area of the surface generated when this curve is rotated fully about the $x$-axis is $A \mathrm {~cm} ^ { 2 }$.

Show that $\mathrm { A } = 2 \pi \int _ { 0 } ^ { 1 } \sqrt { 1 + \mathrm { kx } ^ { 2 } } \mathrm { dx }$ for some integer $k$ to be determined. A small component for a car is produced in the shape of this surface. The curved surface area of the component must be $8 \mathrm {~cm} ^ { 2 }$, accurate to within one percent. The engineering process produces such components with a curved surface area accurate to within one half of one percent.
Determine whether all components produced will be suitable for use in the car.

OCR Further Additional Pure 2023 June Q3

3 The points $A$ and $B$ have position vectors $\mathbf { a } = \mathbf { i } + \mathrm { pj } + \mathrm { q } \mathbf { k }$ and $\mathbf { b } = 2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ respectively, relative to the origin $O$.

Determine the value of $p$ and the value of $q$ for which $\mathbf { a } \times \mathbf { b } = 2 \mathbf { i } + 6 \mathbf { j } - 1 \mathbf { 1 } \mathbf { k }$.
The point $C$ has coordinates ( $d , e , f$ ) and the tetrahedron $O A B C$ has volume 7.
1. Using the values of $p$ and $q$ found in part (a), find the possible relationships between $d , e$ and $f$.
2. Explain the geometrical significance of these relationships.

OCR Further Additional Pure 2023 June Q4

4 The sequence $\left\{ A _ { n } \right\}$ is given for all integers $n \geqslant 0$ by $A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }$, where $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x$.

Show that $\left\{ A _ { n } \right\}$ increases monotonically.
Show that $\left\{ \mathrm { A } _ { \mathrm { n } } \right\}$ converges to a limit, $A$, whose exact value should be stated.

OCR Further Additional Pure 2023 June Q5

The group $G$ consists of the set $S = \{ 1,9,17,25 \}$ under $\times _ { 32 }$, the operation of multiplication modulo 32.
1. Complete the Cayley table for $G$ given in the Printed Answer Booklet.
2. Up to isomorphisms, there are only two groups of order 4.
  - $C _ { 4 }$, the cyclic group of order 4
$K _ { 4 }$, the non-cyclic (Klein) group of order 4
1. List the odd quadratic residues modulo 32.
2. Given that $n$ is an odd integer, prove that $n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )$.

OCR Further Additional Pure 2023 June Q6

6 The surface $S$ has equation $z = x \sin y + \frac { y } { x }$ for $x > 0$ and $0 < y < \pi$.

Determine, as a function of $x$ and $y$, the determinant of $\mathbf { H }$, the Hessian matrix of $S$.
Given that $S$ has just one stationary point, $P$, use the answer to part (a) to deduce the nature of $P$.
The coordinates of $P$ are $( \alpha , \beta , \gamma )$. Show that $\beta$ satisfies the equation $\beta + \tan \beta = 0$.

OCR Further Additional Pure 2023 June Q7

7 Binet's formula for the $n$th Fibonacci number is given by $\mathrm { F } _ { \mathrm { n } } = \frac { 1 } { \sqrt { 5 } } \left( \alpha ^ { \mathrm { n } } - \beta ^ { \mathrm { n } } \right)$ for $n \geqslant 0$, where $\alpha$ and $\beta$ (with $\alpha > 0 > \beta$ ) are the roots of $x ^ { 2 } - x - 1 = 0$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Consider the sequence $\left\{ \mathrm { S } _ { \mathrm { n } } \right\}$, where $\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }$ for $n \geqslant 0$.
1. Determine the values of $S _ { 2 }$ and $S _ { 3 }$.
2. Show that $S _ { n + 2 } = S _ { n + 1 } + S _ { n }$ for $n \geqslant 0$.
3. Deduce that $S _ { n }$ is an integer for all $n \geqslant 0$.
A student models the terms of the sequence $\left\{ \mathrm { S } _ { \mathrm { n } } \right\}$ using the formula $\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }$.
1. Explain why this formula is unsuitable for every $n \geqslant 1$.
2. Considering the cases $n$ even and $n$ odd separately, state a modification of the formula $\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }$, other than $\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }$, such that $\mathrm { T } _ { \mathrm { n } } = \mathrm { S } _ { \mathrm { n } }$ for all $n \geqslant 1$.

OCR Further Additional Pure 2023 June Q8

8 Let $f ( n )$ denote the base- $n$ number $2121 _ { n }$ where $n \geqslant 3$.

1. For each $n \geqslant 3$, show that $\mathrm { f } ( n )$ can be written as the product of two positive integers greater than $1 , \mathrm { a } ( n )$ and $\mathrm { b } ( n )$, each of which is a function of $n$.
2. Deduce that $\mathrm { f } ( n )$ is always composite.
Let $h$ be the highest common factor of $\mathrm { a } ( n )$ and $\mathrm { b } ( n )$.
1. Prove that $h$ is either 1 or 5 .
2. Find a value of $n$ for which $h = 5$.

OCR Further Additional Pure 2023 June Q9

9 The set $C$ consists of the set of all complex numbers excluding 1 and - 1 . The operation ⊕ is defined on the elements of $C$ by $\mathrm { a } \oplus \mathrm { b } = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { ab } + 1 }$ where $\mathrm { a } , \mathrm { b } \in \mathrm { C }$.

Determine the identity element of $C$ under ⊕.
For each element $x$ in $C$ show that it has an inverse element in $C$.
Show that $\oplus$ is associative on $C$.
Explain why $( C , \oplus )$ is not a group.
Find a subset, $D$, of $C$ such that $( D , \oplus )$ is a group of order 3 . \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its 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 OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
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	\(i\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(i\)	\(i\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(a\)	\(a\)	\(i\)	\(d\)	\(e\)	\(b\)	\(c\)
\(b\)	\(b\)	\(e\)	\(i\)	\(d\)	\(c\)	\(a\)
\(c\)	\(c\)	\(d\)	\(e\)	\(i\)	\(a\)	\(b\)
\(d\)	\(d\)	\(c\)	\(a\)	\(b\)	\(e\)	\(i\)
\(e\)	\(e\)	\(b\)	\(c\)	\(a\)	\(i\)	\(d\)

\cline { 2 - 5 } \multicolumn{1}{c\|}{\(\times _ { 80 }\)}	1	9	31	39
1	1	9	31	39
9	9	1	39	31
31	31	39	1	9
39	39	31	9	1

Questions — OCR Further Additional Pure (85 questions)

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