Questions FP3 - A-Level Maths

AQA FP3 2007 January Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln \left( 1 + x ^ { 2 } + y \right)$$ and $$y ( 1 ) = 0.6$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.05$, to obtain an approximation to $y ( 1.05 )$, giving your answer to four decimal places.
Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.05$, to obtain an approximation to $y ( 1.05 )$, giving your answer to four decimal places.

AQA FP3 2007 January Q2

2 A curve has polar equation $r ( 1 - \sin \theta ) = 4$. Find its cartesian equation in the form $y = \mathrm { f } ( x )$.

AQA FP3 2007 January Q3

3

Show that $x ^ { 2 }$ is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
Solve this differential equation, given that $y = 1$ when $x = 2$.

AQA FP3 2007 January Q4

4

Explain why $\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x$ is an improper integral.
(1 mark)
Use integration by parts to find $\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x$.
(3 marks)
Show that $\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x$ exists and find its value.
(4 marks)

AQA FP3 2007 January Q5

5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 6 + 5 \sin x$$ (12 marks)

AQA FP3 2007 January Q6

6 The function f is defined by $\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }$.

1. Find f'''(x).
2. Using Maclaurin's theorem, show that, for small values of $x$, $$\mathrm { f } ( x ) \approx 1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
Use the expansion of $\mathrm { e } ^ { x }$ together with the result in part (a)(ii) to show that, for small values of $x$, $$\mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \approx 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where $k$ is a rational number to be found.
Write down the first four terms in the expansion, in ascending powers of $x$, of $\mathrm { e } ^ { 2 x }$.
Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } - \mathrm { e } ^ { 2 x } } { 1 - \cos x }$$ (4 marks)

AQA FP3 2007 January Q7

7 A curve $C$ has polar equation $$r = 6 + 4 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ The diagram shows a sketch of the curve $C$, the pole $O$ and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{0d894ac0-8d96-4182-8454-c306e1fdad8f-4_599_866_612_587}

Calculate the area of the region bounded by the curve $C$.
The point $P$ is the point on the curve $C$ for which $\theta = \frac { 2 \pi } { 3 }$. The point $Q$ is the point on $C$ for which $\theta = \pi$.
Show that $Q P$ is parallel to the line $\theta = \frac { \pi } { 2 }$.
The line $P Q$ intersects the curve $C$ again at a point $R$. The line $R O$ intersects $C$ again at a point $S$.
1. Find, in surd form, the length of $P S$.
2. Show that the angle $O P S$ is a right angle.

AQA FP3 2007 June Q1

1

Find the value of the constant $k$ for which $k x ^ { 2 } \mathrm { e } ^ { 5 x }$ is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 10 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 6 \mathrm { e } ^ { 5 x }$$
Hence find the general solution of this differential equation.

AQA FP3 2007 June Q2

2 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } + 3 }$$ and $$y ( 1 ) = 2$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.1$, to obtain an approximation to $y ( 1.1 )$, giving your answer to four decimal places.
Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.1$, to obtain an approximation to $y ( 1.1 )$, giving your answer to four decimal places.

AQA FP3 2007 June Q3

3 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$ given that $y = 3$ when $x = 0$.

AQA FP3 2007 June Q4

4

Show that $( \cos \theta + \sin \theta ) ^ { 2 } = 1 + \sin 2 \theta$.
A curve has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( x + y ) ^ { 4 }$$ Given that $r \geqslant 0$, show that the polar equation of the curve is $$r = 1 + \sin 2 \theta$$
The curve with polar equation $$r = 1 + \sin 2 \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f90167c3-2ffd-464a-b2d2-9f86a8d64887-3_389_611_1062_708}
1. Find the two values of $\theta$ for which $r = 0$.
2. Find the area of one of the loops.

AQA FP3 2007 June Q5

5

A differential equation is given by $$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$ Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + x$$ transforms this differential equation into $$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$ (4 marks)
Find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$ giving your answer in the form $u = \mathrm { f } ( x )$.
Hence find the general solution of the differential equation $$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$ giving your answer in the form $y = \mathrm { g } ( x )$.

AQA FP3 2007 June Q6

6

The function f is defined by $$\mathrm { f } ( x ) = \ln \left( 1 + \mathrm { e } ^ { x } \right)$$ Use Maclaurin's theorem to show that when $\mathrm { f } ( x )$ is expanded in ascending powers of $x$ :
1. the first three terms are $$\ln 2 + \frac { 1 } { 2 } x + \frac { 1 } { 8 } x ^ { 2 }$$
2. the coefficient of $x ^ { 3 }$ is zero.
Hence write down the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right)$.
Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \ldots$$ to write down the first three terms in the expansion, in ascending powers of $x$, of $\ln \left( 1 - \frac { x } { 2 } \right)$.
Use your answers to parts (b) and (c) to find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right) + \ln \left( 1 - \frac { x } { 2 } \right) } { x - \sin x } \right]$$

AQA FP3 2007 June Q7

7

Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
Use the substitution $u = x \mathrm { e } ^ { - x } + 1$ to find $\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x$.
Hence evaluate $\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x$, showing the limiting process used.

OCR FP3 Q3

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 8 y = \mathrm { e } ^ { 3 x }$$

OCR FP3 Q5

5

Use de Moivre's theorem to prove that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
Hence find the largest positive root of the equation $$64 x ^ { 6 } - 96 x ^ { 4 } + 36 x ^ { 2 } - 3 = 0$$ giving your answer in trigonometrical form.

OCR FP3 Q6

6 Lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\frac { x - 3 } { 2 } = \frac { y - 4 } { - 1 } = \frac { z + 1 } { 1 } \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 2 }$$ respectively.

Find the equation of the plane $\Pi _ { 1 }$ which contains $l _ { 1 }$ and is parallel to $l _ { 2 }$, giving your answer in the form r.n $= p$.
Find the equation of the plane $\Pi _ { 2 }$ which contains $l _ { 2 }$ and is parallel to $l _ { 1 }$, giving your answer in the form r.n $= p$.
Find the distance between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
State the relationship between the answer to part (iii) and the lines $l _ { 1 }$ and $l _ { 2 }$.
Show that $\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1$.
Write down the seven roots of the equation $z ^ { 7 } = 1$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$ and show their positions in an Argand diagram.
Hence express $z ^ { 7 } - 1$ as the product of one real linear factor and three real quadratic factors. 8
Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$ expressing $y$ in terms of $x$ in your answer.
Find the particular solution for which $y = 2$ when $x = \pi$. 9 The set $S$ consists of the numbers $3 ^ { n }$, where $n \in \mathbb { Z }$. ( $\mathbb { Z }$ denotes the set of integers $\{ 0 , \pm 1 , \pm 2 , \ldots \}$.)
Prove that the elements of $S$, under multiplication, form a commutative group $G$. (You may assume that addition of integers is associative and commutative.)

Determine whether or not each of the following subsets of $S$, under multiplication, forms a subgroup of $G$, justifying your answers.
(a) The numbers $3 ^ { 2 n }$, where $n \in \mathbb { Z }$.
(b) The numbers $3 ^ { n }$, where $n \in \mathbb { Z }$ and $n \geqslant 0$.
(c) The numbers $3 ^ { \left( \pm n ^ { 2 } \right) }$, where $n \in \mathbb { Z }$. 1 (a) A group $G$ of order 6 has the combination table shown below.

	$e$	$a$	$b$	$p$	$q$	$r$
$e$	$e$	$a$	$b$	$p$	$q$	$r$
$a$	$a$	$b$	$e$	$r$	$p$	$q$
$b$	$b$	$e$	$a$	$q$	$r$	$p$
$p$	$p$	$q$	$r$	$e$	$a$	$b$
$q$	$q$	$r$	$p$	$b$	$e$	$a$
$r$	$r$	$p$	$q$	$a$	$b$	$e$

State, with a reason, whether or not $G$ is commutative.
State the number of subgroups of $G$ which are of order 2 .
List the elements of the subgroup of $G$ which is of order 3 .
(b) A multiplicative group $H$ of order 6 has elements $e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }$, where $e$ is the identity. Write down the order of each of the elements $c ^ { 3 } , c ^ { 4 }$ and $c ^ { 5 }$. 2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x$$ 3 Two fixed points, $A$ and $B$, have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to the origin $O$, and a variable point $P$ has position vector $\mathbf { r }$.
Give a geometrical description of the locus of $P$ when $\mathbf { r }$ satisfies the equation $\mathbf { r } = \lambda \mathbf { a }$, where $0 \leqslant \lambda \leqslant 1$.
Given that $P$ is a point on the line $A B$, use a property of the vector product to explain why $( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }$.
Give a geometrical description of the locus of $P$ when $\mathbf { r }$ satisfies the equation $\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }$. 4 The integrals $C$ and $S$ are defined by $$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$ By considering $C + \mathrm { i } S$ as a single integral, show that $$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right)$$ and obtain a similar expression for $S$.
(You may assume that the standard result for $\int \mathrm { e } ^ { k x } \mathrm {~d} x$ remains true when $k$ is a complex constant, so that $\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)$ 5
Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$ expressing $y$ in terms of $x$ in your answer. In a particular case, it is given that $y = \frac { 2 } { \pi }$ when $x = \frac { 1 } { 4 } \pi$.
Find the solution of the differential equation in this case.
Write down a function to which $y$ approximates when $x$ is large and positive. 6 A tetrahedron $A B C D$ is such that $A B$ is perpendicular to the base $B C D$. The coordinates of the points $A , C$ and $D$ are $( - 1 , - 7,2 ) , ( 5,0,3 )$ and $( - 1,3,3 )$ respectively, and the equation of the plane $B C D$ is $x + 2 y - 2 z = - 1$.
Find, in either order, the coordinates of $B$ and the length of $A B$.
Find the acute angle between the planes $A C D$ and $B C D$.
(a) Verify, without using a calculator, that $\theta = \frac { 1 } { 8 } \pi$ is a solution of the equation $\sin 6 \theta = \sin 2 \theta$.
(b) By sketching the graphs of $y = \sin 6 \theta$ and $y = \sin 2 \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, or otherwise, find the other solution of the equation $\sin 6 \theta = \sin 2 \theta$ in the interval $0 < \theta < \frac { 1 } { 2 } \pi$.
Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right)$$
Hence show that one of the solutions obtained in part (i) satisfies $\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )$, and justify which solution it is. \section*{Jan 2008} 8 Groups $A , B , C$ and $D$ are defined as follows:
A: the set of numbers $\{ 2,4,6,8 \}$ under multiplication modulo 10 ,
$B$ : the set of numbers $\{ 1,5,7,11 \}$ under multiplication modulo 12 ,
$C$ : the set of numbers $\left\{ 2 ^ { 0 } , 2 ^ { 1 } , 2 ^ { 2 } , 2 ^ { 3 } \right\}$ under multiplication modulo 15,
$D$ : the set of numbers $\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.$, where $m$ and $n$ are integers $\}$ under multiplication.
Write down the identity element for each of groups $A , B , C$ and $D$.
Determine in each case whether the groups $$\begin{aligned} & A \text { and } B ,
& B \text { and } C ,
& A \text { and } C \end{aligned}$$ are isomorphic or non-isomorphic. Give sufficient reasons for your answers.
Prove the closure property for group $D$.
Elements of the set $\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.$, where $m$ and $n$ are integers $\}$ are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }1 (a) A cyclic multiplicative group $G$ has order 12. The identity element of $G$ is $e$ and another element is $r$, with order 12.
Write down, in terms of $e$ and $r$, the elements of the subgroup of $G$ which is of order 4.
Explain briefly why there is no proper subgroup of $G$ in which two of the elements are $e$ and $r$.
(b) A group $H$ has order $m n p$, where $m , n$ and $p$ are prime. State the possible orders of proper subgroups of $H$. 2 Find the acute angle between the line with equation $\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$ and the plane with equation $\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )$. 3
Use the substitution $z = x + y$ to show that the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y + 3 } { x + y - 1 }$$ may be written in the form $\frac { \mathrm { d } z } { \mathrm {~d} x } = \frac { 2 ( z + 1 ) } { z - 1 }$.
Hence find the general solution of the differential equation (A). 4
By expressing $\cos \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$, show that $$\cos ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
Hence solve the equation $\cos 5 \theta + 5 \cos 3 \theta + 9 \cos \theta = 0$ for $0 \leqslant \theta \leqslant \pi$. 5 Two lines have equations $$\frac { x - k } { 2 } = \frac { y + 1 } { - 5 } = \frac { z - 1 } { - 3 } \quad \text { and } \quad \frac { x - k } { 1 } = \frac { y + 4 } { - 4 } = \frac { z } { - 2 }$$ where $k$ is a constant.
Show that, for all values of $k$, the lines intersect, and find their point of intersection in terms of $k$.
For the case $k = 1$, find the equation of the plane in which the lines lie, giving your answer in the form $a x + b y + c z = d$. 6 The operation ○ on real numbers is defined by $a \circ b = a | b |$.
Show that ∘ is not commutative.
Prove that ∘ is associative.
Determine whether the set of real numbers, under the operation ∘, forms a group. \section*{June 2008}

OCR FP3 Q9

9 The set $S$ consists of the numbers $3 ^ { n }$, where $n \in \mathbb { Z }$. ( $\mathbb { Z }$ denotes the set of integers $\{ 0 , \pm 1 , \pm 2 , \ldots \}$.)

Prove that the elements of $S$, under multiplication, form a commutative group $G$. (You may assume that addition of integers is associative and commutative.)
Determine whether or not each of the following subsets of $S$, under multiplication, forms a subgroup of $G$, justifying your answers.
(a) The numbers $3 ^ { 2 n }$, where $n \in \mathbb { Z }$.
(b) The numbers $3 ^ { n }$, where $n \in \mathbb { Z }$ and $n \geqslant 0$.
(c) The numbers $3 ^ { \left( \pm n ^ { 2 } \right) }$, where $n \in \mathbb { Z }$. 1 (a) A group $G$ of order 6 has the combination table shown below. $G$ and $H$ are the non-cyclic groups of order 4 and 6 respectively.
Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups $G$ and $H$. Hence explain why there are no non-cyclic proper subgroups of $Q$. 7 Three planes $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$ have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ respectively. Planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ intersect in a line $l$; planes $\Pi _ { 2 }$ and $\Pi _ { 3 }$ intersect in a line $m$.
Show that $l$ and $m$ are in the same direction.
Write down what you can deduce about the line of intersection of planes $\Pi _ { 1 }$ and $\Pi _ { 3 }$.
By considering the cartesian equations of $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$, or otherwise, determine whether or not the three planes have a common line of intersection. 8 The operation $*$ is defined on the elements $( x , y )$, where $x , y \in \mathbb { R }$, by $$( a , b ) * ( c , d ) = ( a c , a d + b ) .$$ It is given that the identity element is $( 1,0 )$.
Prove that $*$ is associative.
Find all the elements which commute with $( 1,1 )$.
It is given that the particular element $( m , n )$ has an inverse denoted by $( p , q )$, where $$( m , n ) * ( p , q ) = ( p , q ) * ( m , n ) = ( 1,0 ) .$$ Find $( p , q )$ in terms of $m$ and $n$.
Find all self-inverse elements.
Give a reason why the elements $( x , y )$, under the operation $*$, do not form a group.

OCR MEI FP3 2015 June Q1

1 The point A has coordinates $( 2,5,4 )$ and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8
25
43 \end{array} \right) + \lambda \left( \begin{array} { c } 4
15
25 \end{array} \right)$$ You are given that $\mathrm { AB } = \mathrm { AC } = 15$.

Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
Find the equation of the plane ABC in cartesian form.
Show that the plane containing the line BC and perpendicular to the plane ABC has equation $5 y - 3 z + 4 = 0$. The point D has coordinates $( 1,1,3 )$.
Show that $| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }$ and hence find the shortest distance between the lines $B C$ and $A D$.
Find the volume of the tetrahedron ABCD .

OCR MEI FP3 2015 June Q2

2 A surface has equation $z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60$.

Show that the point $\mathrm { A } ( 8,4 , - 4 )$ is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
A point P with coordinates $( 8 + h , 4 + k , p )$ lies on the surface.
(A) Show that $p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )$.
(B) Deduce that the stationary point A is a local minimum.
(C) By considering sections of the surface near to B in each of the planes $x = 0$ and $y = 0$, investigate the nature of the stationary point B .
The point Q with coordinates $( 1,1,53 )$ lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
The tangent plane at the point R has equation $6 x + 6 y + z = \lambda$ where $\lambda \neq 65$. Find the coordinates of R .

OCR MEI FP3 2015 June Q3

3 Fig. 3 shows an ellipse with parametric equations $x = a \cos \theta , y = b \sin \theta$, for $0 \leqslant \theta \leqslant 2 \pi$, where $0 < b \leqslant a$.
The curve meets the positive $x$-axis at A and the positive $y$-axis at B . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0e032f23-0549-4adc-bfae-59333108fab5-4_668_1255_477_404} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Show that the radius of curvature at A is $\frac { b ^ { 2 } } { a }$ and find the corresponding centre of curvature.
Write down the radius of curvature and the centre of curvature at B .
Find the relationship between $a$ and $b$ if the radius of curvature at B is equal to the radius of curvature at A . What does this mean geometrically?
Show that the arc length from A to B can be expressed as $$b \int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 1 + \lambda ^ { 2 } \sin ^ { 2 } \theta } d \theta$$ where $\lambda ^ { 2 }$ is to be determined in terms of $a$ and $b$.
Evaluate this integral in the case $a = b$ and comment on your answer.
Find the cartesian equation of the evolute of the ellipse.

OCR MEI FP3 2015 June Q4

4 M is the set of all $2 \times 2$ matrices $\mathrm { m } ( a , b )$ where $a$ and $b$ are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b
0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$

Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
Determine whether the group is commutative. The set $\mathrm { N } _ { k }$ consists of all $2 \times 2$ matrices $\mathrm { m } ( k , b )$ where $k$ is a fixed positive integer and $b$ can take any integer value.
Prove that $\mathrm { N } _ { k }$ is closed under matrix multiplication if and only if $k = 1$. Now consider the set P consisting of the matrices $\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )$ and $\mathrm { m } ( 1,3 )$. The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
Show that $( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )$.
Construct the group combination table for P . The group R consists of the set $\{ e , a , b , c \}$ combined under the operation *. The identity element is $e$, and elements $a , b$ and $c$ are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}

OCR MEI FP3 2015 June Q5

5 An inspector has three factories, A, B, C, to check. He spends each day in one of the factories. He chooses the factory to visit on a particular day according to the following rules.

If he is in A one day, then the next day he will never choose A but he is equally likely to choose B or C .
If he is in B one day, then the next day he is equally likely to choose $\mathrm { A } , \mathrm { B }$ or C .
If he is in C one day, then the next day he will never choose A but he is equally likely to choose B or C .
1. Write down the transition matrix, $\mathbf { P }$.
2. On Day 1 the inspector chooses A.
  (A) Find the probability that he will choose A on Day 4.
  (B) Find the probability that the factory he chooses on Day 7 is the same factory that he chose on Day 2.
3. Find the equilibrium probabilities and explain what they mean.

The inspector is not satisfied with the number of times he visits A so he changes the rules as follows.

If he is in A one day, then the next day he will choose $\mathrm { A } , \mathrm { B } , \mathrm { C }$, with probabilities $0.8,0.1,0.1$, respectively.
If he is in B or C one day, then the probabilities for choosing the factory the next day remain as before.
Write down the new transition matrix, $\mathbf { Q }$, and find the new equilibrium probabilities.
On a particular day, the inspector visits factory A. Find the expected number of consecutive further days on which he will visit factory A.

Still not satisfied, the inspector changes the rules as follows.

If he is in A one day, then the next day he will choose $\mathrm { A } , \mathrm { B } , \mathrm { C }$, with probabilities $1,0,0$, respectively.
If he is in B or C one day, then the probabilities for choosing the factory the next day remain as before.

The new transition matrix is $\mathbf { R }$.

On Day 15 he visits C . Find the first subsequent day for which the probability that he visits B is less than 0.1.

Show that in this situation there is an absorbing state, explaining what this means. \section*{END OF QUESTION PAPER}

	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
\(e\)	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
\(a\)	\(a\)	\(b\)	\(e\)	\(r\)	\(p\)	\(q\)
\(b\)	\(b\)	\(e\)	\(a\)	\(q\)	\(r\)	\(p\)
\(p\)	\(p\)	\(q\)	\(r\)	\(e\)	\(a\)	\(b\)
\(q\)	\(q\)	\(r\)	\(p\)	\(b\)	\(e\)	\(a\)
\(r\)	\(r\)	\(p\)	\(q\)	\(a\)	\(b\)	\(e\)

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