Questions Further Additional Pure

write down an integral which gives $A$ and
find the value of $k$.

OCR Further Additional Pure Specimen Q2

2 Find the volume of tetrahedron OABC , where O is the origin, $\mathrm { A } = ( 2,3,1 ) , \mathrm { B } = ( - 4,2,5 )$ and $\mathrm { C } = ( 1,4,4 )$.

OCR Further Additional Pure Specimen Q3

3 Given $z = x \sin y + y \cos x$, show that $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } + z = 0$.

OCR Further Additional Pure Specimen Q4

Solve the recurrence relation $u _ { n + 2 } = 4 u _ { n + 1 } - 4 u _ { n }$ for $n \geq 0$, given that $u _ { 0 } = 1$ and $u _ { 1 } = 1$.
Show that each term of the sequence $\left\{ u _ { n } \right\}$ is an integer.

OCR Further Additional Pure Specimen Q5

5 In this question you must show detailed reasoning.
It is given that $I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta$ for $n \geq 0$.

Prove that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$ for $n \geq 2$.
(a) Evaluate $I _ { 1 }$.
(b) Use the reduction formula to determine the exact value of $\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta$.

OCR Further Additional Pure Specimen Q6

6 A surface $S$ has equation $z = \mathrm { f } ( x , y )$, where $\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y$. It is given that $S$ has a single stationary point, $P$.

(a) Determine the coordinates of $P$.
(b) Determine the nature of $P$.
Find the equation of the tangent plane to $S$ at the point $Q ( 1,2,38 )$.

OCR Further Additional Pure Specimen Q7

7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, $t$ years after their introduction, is an integer denoted by $N _ { t }$. The initial number of breeding pairs is given by $N _ { 0 }$. An initial discrete population model is proposed for $N _ { t }$. $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$

(a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
(b) Show that $N _ { t + 1 } - N _ { t } < 150 - N _ { t }$ when $N _ { t }$ lies between 0 and 150 .
(c) Hence find the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when $N _ { 0 } \in ( 0,150 )$. An alternative discrete population model is proposed for $N _ { t }$. $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
(a) Given that $N _ { 0 } = 8$, find the value of $N _ { 4 }$ for each of the two models.
(b) Which of the two models gives values for $N _ { t }$ with the more appropriate level of precision?

OCR Further Additional Pure Specimen Q8

8 The set $X$ consists of all $2 \times 2$ matrices of the form $\left( \begin{array} { r r } x & - y
y & x \end{array} \right)$, where $x$ and $y$ are real numbers which are not both zero.

(a) The matrices $\left( \begin{array} { c c } a & - b
b & a \end{array} \right)$ and $\left( \begin{array} { c c } c & - d
d & c \end{array} \right)$ are both elements of $X$. Show that $\left( \begin{array} { c c } a & - b
b & a \end{array} \right) \left( \begin{array} { c c } c & - d
d & c \end{array} \right) = \left( \begin{array} { c c } p & - q
q & p \end{array} \right)$ for some real numbers $p$ and $q$ to be found in terms of $a , b , c$ and $d$.
(b) Prove by contradiction that $p$ and $q$ are not both zero.
Prove that $X$, under matrix multiplication, forms a group $G$. [You may use the result that matrix multiplication is associative.]
Determine a subgroup of $G$ of order 17.

OCR Further Additional Pure Specimen Q9

(a) Prove that $p \equiv \pm 1 ( \bmod 6 )$ for all primes $p > 3$.
(b) Hence or otherwise prove that $p ^ { 2 } - 1 \equiv 0 ( \bmod 24 )$ for all primes $p > 3$.
Given that $p$ is an odd prime, determine the residue of $2 ^ { p ^ { 2 } - 1 }$ modulo $p$.
Let $p$ and $q$ be distinct primes greater than 3 . Prove that $p ^ { q - 1 } + q ^ { p - 1 } \equiv 1 ( \bmod p q )$. \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series.
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OCR Further Additional Pure 2019 June Q8

8 In this question you must show detailed reasoning.

Prove that $2 ( p - 2 ) ^ { p - 2 } \equiv - 1 ( \bmod p )$, where $p$ is an odd prime.
Find two odd prime factors of the number $N = 2 \times 34 ^ { 34 } - 2 ^ { 15 }$. \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}

OCR Further Additional Pure 2021 November Q1

1 In this question you must show detailed reasoning. Express the number $\mathbf { 4 1 7 2 3 } _ { 10 }$ in hexadecimal (base 16).

OCR Further Additional Pure 2018 March Q1

1 Determine the solution of the simultaneous linear congruences $$x \equiv 4 ( \bmod 7 ) , \quad x \equiv 25 ( \bmod 41 ) .$$

OCR Further Additional Pure 2018 March Q2

2 Four points $A , B , C$ and $D$ have coordinates $( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )$ and $( 0,3,7 )$ respectively. Find the volume of tetrahedron $A B C D$.

OCR Further Additional Pure 2018 March Q3

3 The surface $S$ has equation $z = \frac { x } { y } \sin y + \frac { y } { x } \cos x$ where $0 < x \leqslant \pi$ and $0 < y \leqslant \pi$.

Find
- $\frac { \partial z } { \partial x }$,
- $\frac { \partial z } { \partial y }$.
- Determine the equation of the tangent plane to $S$ at the point $A$ where $x = y = \frac { 1 } { 4 } \pi$. Give your answer in the form $a x + b y + c z = d$ where $a , b , c$ and $d$ are exact constants.
- Write down a normal vector to $S$ at $A$.

OCR Further Additional Pure 2018 March Q4

(a) Find all the quadratic residues modulo 11.
(b) Prove that the equation $y ^ { 5 } = x ^ { 2 } + 2017$ has no solution in integers $x$ and $y$.
In this question you must show detailed reasoning. The numbers $M$ and $N$ are given by $$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$ Prove that $M$ is divisible by $N$.

OCR Further Additional Pure 2018 March Q5

(a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that $X _ { 0 } = 12$ and $X _ { 1 } = 1$.
(b) Show that the sequence $\left\{ X _ { n } \right\}$ approaches a geometric sequence as $n$ increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that $X _ { n }$ represents the profit (in $\pounds$ ) at the end of year $n$.
(a) Determine the number of years taken for the projected profit to exceed one million pounds.
(b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
(a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
(b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
(c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next $X _ { n }$ in the process.

OCR Further Additional Pure 2018 March Q6

6 In this question you must show detailed reasoning. It is given that $I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t$ for integers $n \geqslant 0$.

Show that $I _ { 1 } = \frac { 7 } { 3 }$.
Prove that, for $n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }$. The curve $C$ is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve $C$ is rotated through $2 \pi$ radians about the $x$-axis, a surface of revolution is formed with surface area $A$.
Determine
- the values of the integers $k$ and $m$ such that $A = k \pi I _ { m }$,
- the exact value of $A$.

OCR Further Additional Pure 2018 March Q7

7 The set $M$ contains all matrices of the form $\mathbf { X } ^ { n }$, where $\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1
1 & 1 \end{array} \right)$ and $n$ is a positive integer.

Show that $M$ contains exactly 12 elements.
Deduce that $M$, together with the operation of matrix multiplication, form a cyclic group $G$.
Determine all the proper subgroups of $G$. \section*{END OF QUESTION PAPER}

OCR Further Additional Pure 2018 September Q1

Write the number $100011 _ { n }$, where $n \geqslant 2$, as a polynomial in $n$.
Show that $n ^ { 2 } + n + 1$ is a factor of this expression.
Hence show that $100011 _ { n }$ is composite in any number base $n \geqslant 2$.

OCR Further Additional Pure 2018 September Q2

2 In this question, you must show detailed reasoning.
A curve is defined parametrically by $x = t ^ { 3 } - 3 t + 1 , y = 3 t ^ { 2 } - 1$, for $0 \leqslant t \leqslant 5$. Find, in exact form,

the length of the curve,
the area of the surface generated when the curve is rotated completely about the $x$-axis.

OCR Further Additional Pure 2018 September Q3

3 The function $w = \mathrm { f } ( x , y , z )$ is given by $\mathrm { f } ( x , y , z ) = x ^ { 2 } y z + 2 x y ^ { 2 } z + 3 x y z ^ { 2 } - 24 x y z$, for $x , y , z \neq 0$.

(a) Find
- $\mathrm { f } _ { x }$,
- $\mathrm { f } _ { y }$,
- $\mathrm { f } _ { z }$.
  (b) Hence find the values of $a , b , c$ and $d$ for which $w$ has a stationary value when $d = \mathrm { f } ( a , b , c )$.
- You are given that this stationary value is a local minimum of $w$. Find values of $x , y$ and $z$ which show that it is not a global minimum of $w$.

OCR Further Additional Pure 2018 September Q4

4 The points $A , B , C$ and $P$ have coordinates ( $a , 0,0$ ), ( $0 , b , 0$ ), ( $0,0 , c$ ) and ( $a , b , c$ ) respectively, where $a , b$ and $c$ are positive constants.
The plane $\Pi$ contains $A , B$ and $C$.

(a) Use the scalar triple product to determine
- the volume of tetrahedron $O A B C$,
- the volume of tetrahedron PABC.
  (b) Hence show that the distance from $P$ to $\Pi$ is twice the distance from $O$ to $\Pi$.
- (a) Determine a vector which is normal to $\Pi$.
  (b) Hence determine, in terms of $a , b$ and $c$ only, the distance from $P$ to $\Pi$. consisting of the powers of a particular, non-singular $2 \times 2$ real matrix $\mathbf { M } = \left( \begin{array} { l l } a & b
  c & d \end{array} \right)$, under the operation of
  matrix multiplication. matrix multiplication.
- Explain why such a group is only possible if $\operatorname { det } ( \mathbf { M } ) = 1$ or - 1 .
- Write down values of $a , b , c$ and $d$ that would give a suitable matrix $\mathbf { M }$ for which $\mathbf { M } ^ { 6 } = \mathbf { I }$ and
Student Q observes that their class has already found a group of order 6 in a previous task; a group
Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular $2 \times 2$ real matrix $\mathbf { M } = \left( \begin{array} { l l } a & b
c & d \end{array} \right)$, under the operation of
matrix multiplication. $\operatorname { det } ( \mathbf { M } ) = 1$.
Explain why such a group is only possible if $\operatorname { det } ( \mathbf { M } ) = 1$ or - 1 . Student Q believes that it is possible to construct a rational function f in the form $\mathrm { f } ( x ) = \frac { a x + b } { c x + d }$ so that the group of functions is isomorphic to the matrix group which is generated by the matrix $\mathbf { M }$ of part (iii).
(a) Write down and simplify the function f that, according to Student Q , corresponds to $\mathbf { M }$.
(b) By calculating $\mathbf { M } ^ { 3 }$, show that Student Q's suggestion does not work.
(c) Find a different function $f$ that will satisfy the requirements of the task.

OCR Further Additional Pure 2018 September Q7

7 The members of the family of the sequences $\left\{ u _ { n } \right\}$ satisfy the recurrence relation $$u _ { n + 1 } = 10 u _ { n } - u _ { n - 1 } \text { for } n \geqslant 1$$

Determine the general solution of (*).
The sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ are members of this family of sequences, corresponding to the initial terms $a _ { 0 } = 1 , a _ { 1 } = 5$ and $b _ { 0 } = 0 , b _ { 1 } = 2$ respectively.
(a) Find the next two terms of each sequence.
(b) Prove that, for all non-negative integers $n , \left( a _ { n } \right) ^ { 2 } - 6 \left( b _ { n } \right) ^ { 2 } = 1$.
(c) Determine $\lim _ { n \rightarrow \infty } \left( \frac { a _ { n } } { b _ { n } } \right)$. \section*{OCR} Oxford Cambridge and RSA

OCR Further Additional Pure 2018 December Q1

1 A surface has equation $z = x \tan y$ for $- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi$.

Find
- $\frac { \partial z } { \partial x }$,
- $\frac { \partial z } { \partial y }$.
- Find in cartesian form, the equation of the tangent plane to the surface at the point where $x = 1$ and $y = \frac { 1 } { 4 } \pi$.

OCR Further Additional Pure 2018 December Q2

2 A sequence $\left\{ u _ { n } \right\}$ is given by $u _ { n + 1 } = 4 u _ { n } + 1$ for $n \geqslant 1$ and $u _ { 1 } = 3$.

Find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
Solve the recurrence system (*).
1. Prove by induction that each term of the sequence can be written in the form $( 10 m + 3 )$ where $m$ is an integer.
2. Show that no term of the sequence is a square number.

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