Questions FP3 - A-Level Maths

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. }

OCR FP3 2006 June Q1

For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of $1 + 2 \mathrm { i }$, giving your answers in the form $a + \mathrm { i } b$.
For the group of matrices of the form $\left( \begin{array} { l l } a & 0
0 & 0 \end{array} \right)$ under matrix addition, where $a \in \mathbb { R }$, state the identity element and the inverse of $\left( \begin{array} { l l } 3 & 0
0 & 0 \end{array} \right)$.

OCR FP3 2006 June Q2

Given that $z _ { 1 } = 2 \mathrm { e } ^ { \frac { 1 } { 6 } \pi \mathrm { i } }$ and $z _ { 2 } = 3 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }$, express $z _ { 1 } z _ { 2 }$ and $\frac { z _ { 1 } } { z _ { 2 } }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$.
Given that $w = 2 \left( \cos \frac { 1 } { 8 } \pi + \mathrm { i } \sin \frac { 1 } { 8 } \pi \right)$, express $w ^ { - 5 }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$.

OCR FP3 2006 June Q3

3 Find the perpendicular distance from the point with position vector $12 \mathbf { i } + 5 \mathbf { j } + 3 \mathbf { k }$ to the line with equation $\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + t ( 8 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k } )$.

OCR FP3 2006 June Q4

4 Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { x ^ { 2 } y } { 1 + x ^ { 3 } } = x ^ { 2 }$$ for which $y = 1$ when $x = 0$, expressing your answer in the form $y = \mathrm { f } ( x )$.
$5 \quad$ A line $l _ { 1 }$ has equation $\frac { x } { 2 } = \frac { y + 4 } { 3 } = \frac { z + 9 } { 5 }$.

Find the cartesian equation of the plane which is parallel to $l _ { 1 }$ and which contains the points $( 2,1,5 )$ and $( 0 , - 1,5 )$.
Write down the position vector of a point on $l _ { 1 }$ with parameter $t$.
Hence, or otherwise, find an equation of the line $l _ { 2 }$ which intersects $l _ { 1 }$ at right angles and which passes through the point ( $- 5,3,4$ ). Give your answer in the form $\frac { x - a } { p } = \frac { y - b } { q } = \frac { z - c } { r }$.
Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin x$$
Find the solution of the differential equation for which $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { 3 }$ when $x = 0$.

OCR FP3 2006 June Q7

7 The series $C$ and $S$ are defined for $0 < \theta < \pi$ by $$\begin{aligned} & C = 1 + \cos \theta + \cos 2 \theta + \cos 3 \theta + \cos 4 \theta + \cos 5 \theta
& S = \quad \sin \theta + \sin 2 \theta + \sin 3 \theta + \sin 4 \theta + \sin 5 \theta \end{aligned}$$

Show that $C + \mathrm { i } S = \frac { \mathrm { e } ^ { 3 \mathrm { i } \theta } - \mathrm { e } ^ { - 3 \mathrm { i } \theta } } { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { i } \theta } } \mathrm { e } ^ { \frac { 5 } { 2 } \mathrm { i } \theta }$.
Deduce that $C = \sin 3 \theta \cos \frac { 5 } { 2 } \theta \operatorname { cosec } \frac { 1 } { 2 } \theta$ and write down the corresponding expression for $S$.
Hence find the values of $\theta$, in the range $0 < \theta < \pi$, for which $C = S$.

OCR FP3 2006 June Q8

8 A group $D$ of order 10 is generated by the elements $a$ and $r$, with the properties $a ^ { 2 } = e , r ^ { 5 } = e$ and $r ^ { 4 } a = a r$, where $e$ is the identity. Part of the operation table is shown below.

	$e$	$а$	$r$	$r ^ { 2 }$	$r ^ { 3 }$	$r ^ { 4 }$	ar	$a r ^ { 2 }$	$a r ^ { 3 }$	$a r ^ { 4 }$
$e$	$e$	$а$	$r$	$r ^ { 2 }$	$r ^ { 3 }$	$r ^ { 4 }$	ar	$a r ^ { 2 }$	$a r ^ { 3 }$	$a r ^ { 4 }$
$а$	$а$	$e$	ar	$a r ^ { 2 }$	$a r ^ { 3 }$	$a r ^ { 4 }$
$r$	r		$r ^ { 2 }$	$r ^ { 3 }$	$r ^ { 4 }$	$e$
$r ^ { 2 }$	$r ^ { 2 }$		$r ^ { 3 }$	$r ^ { 4 }$	$e$	$r$
$r ^ { 3 }$	$r ^ { 3 }$		$r ^ { 4 }$	$e$	$r$	$r ^ { 2 }$
$r ^ { 4 }$	$r ^ { 4 }$	ar	$e$	$r$	$r ^ { 2 }$	$r ^ { 3 }$
ar	ar		$a r ^ { 2 }$	$a r ^ { 3 }$	$a r ^ { 4 }$	$а$
$a r ^ { 2 }$	$a r ^ { 2 }$		$a r ^ { 3 }$	$a r ^ { 4 }$	$a$	ar			T
$a r ^ { 3 }$	$a r ^ { 3 }$		$a r ^ { 4 }$	$а$	ar	$a r ^ { 2 }$
$a r ^ { 4 }$	$a r ^ { 4 }$		$а$	ar	$a r ^ { 2 }$	$a r ^ { 3 }$

Give a reason why $D$ is not commutative.
Write down the orders of any possible proper subgroups of $D$.
List the elements of a proper subgroup which contains
(a) the element $a$,
(b) the element $r$.
Determine the order of each of the elements $r ^ { 3 }$, $a r$ and $a r ^ { 2 }$.
Copy and complete the section of the table marked $\mathbf { E }$, showing the products of the elements $a r , a r ^ { 2 } , a r ^ { 3 }$ and $a r ^ { 4 }$.

OCR FP3 2007 June Q1

By writing $z$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, show that $z z ^ { * } = | z | ^ { 2 }$.
Given that $z z ^ { * } = 9$, describe the locus of $z$.

OCR FP3 2007 June Q2

2 A line $l$ has equation $\mathbf { r } = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } )$ and a plane $\Pi$ has equation $8 x - 7 y + 10 z = 7$. Determine whether $l$ lies in $\Pi$, is parallel to $\Pi$ without intersecting it, or intersects $\Pi$ at one point.

OCR FP3 2007 June 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 2007 June Q4

4 Elements of the set $\{ p , q , r , s , t \}$ are combined according to the operation table shown below.

	$p$	$q$	$r$	$s$	$t$
$p$	$t$	$s$	$p$	$r$	$q$
$q$	$s$	$p$	$q$	$t$	$r$
$r$	$p$	$q$	$r$	$s$	$t$
$s$	$r$	$t$	$s$	$q$	$p$
$t$	$q$	$r$	$t$	$p$	$s$

Verify that $q ( s t ) = ( q s ) t$.
Assuming that the associative property holds for all elements, prove that the set $\{ p , q , r , s , t \}$, with the operation table shown, forms a group $G$.
A multiplicative group $H$ is isomorphic to the group $G$. The identity element of $H$ is $e$ and another element is $d$. Write down the elements of $H$ in terms of $e$ and $d$.

OCR FP3 2007 June Q5

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 2007 June 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 $l _ { 2 }$.

OCR FP3 2007 June Q8

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 $x = \pi$.

OCR FP3 2007 June 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 $n \in \mathbb { Z }$. 4

OCR FP3 Specimen Q1

1 Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = x ,$$ giving $y$ in terms of $x$ in your answer.

OCR FP3 Specimen Q2

2 The set $S = \{ a , b , c , d \}$ under the binary operation * forms a group $G$ of order 4 with the following operation table.

$*$	$a$	$b$	$c$	$d$
$a$	$d$	$a$	$b$	$c$
$b$	$a$	$b$	$c$	$d$
$c$	$b$	$c$	$d$	$a$
$d$	$c$	$d$	$a$	$b$

Find the order of each element of $G$.
Write down a proper subgroup of $G$.
Is the group $G$ cyclic? Give a reason for your answer.
State suitable values for each of $a , b , c$ and $d$ in the case where the operation $*$ is multiplication of complex numbers.

OCR FP3 Specimen Q3

3 The planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ have equations $\mathbf { r } \cdot ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) = 1$ and $\mathbf { r } \cdot ( 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) = 3$ respectively. Find

the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$, correct to the nearest degree,
the equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.

OCR FP3 Specimen Q4

4 In this question, give your answers exactly in polar form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.

Express $4 ( ( \sqrt { } 3 ) - \mathrm { i } )$ in polar form.
Find the cube roots of $4 ( ( \sqrt { } 3 ) - \mathrm { i } )$ in polar form.
Sketch an Argand diagram showing the positions of the cube roots found in part (ii). Hence, or otherwise, prove that the sum of these cube roots is zero.

OCR FP3 Specimen Q5

5 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\frac { x - 5 } { 1 } = \frac { y - 1 } { - 1 } = \frac { z - 5 } { - 2 } \quad \text { and } \quad \frac { x - 1 } { - 4 } = \frac { y - 11 } { - 14 } = \frac { z - 2 } { 2 } .$$

Find the exact value of the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.
Find an equation for the plane containing $l _ { 1 }$ and parallel to $l _ { 2 }$ in the form $a x + b y + c z = d$.

OCR FP3 Specimen Q6

6 The set $S$ consists of all non-singular $2 \times 2$ real matrices $\mathbf { A }$ such that $\mathbf { A Q } = \mathbf { Q A }$, where $$\mathbf { Q } = \left( \begin{array} { l l } 1 & 1
0 & 1 \end{array} \right)$$

Prove that each matrix $\mathbf { A }$ must be of the form $\left( \begin{array} { l l } a & b
0 & a \end{array} \right)$.
State clearly the restriction on the value of $a$ such that $\left( \begin{array} { l l } a & b
0 & a \end{array} \right)$ is in $S$.
Prove that $S$ is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)

OCR FP3 Specimen Q7

Prove that if $z = \mathrm { e } ^ { \mathrm { i } \theta }$, then $z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$.
Express $\cos ^ { 6 } \theta$ in terms of cosines of multiples of $\theta$, and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 6 } \theta \mathrm {~d} \theta$$

OCR FP3 Specimen Q8

Find the value of the constant $k$ such that $y = k x ^ { 2 } \mathrm { e } ^ { - 2 x }$ is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 \mathrm { e } ^ { - 2 x }$$
Find the solution of this differential equation for which $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.
Use the differential equation to determine the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 0$. Hence prove that $0 < y \leqslant 1$ for $x \geqslant 0$.

OCR MEI FP3 2006 June Q1

1 Four points have coordinates $\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )$ and $\mathrm { D } ( 0,9 , k )$.

Find the vector product $\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }$.
For the case when AB is parallel to CD ,
(A) state the value of $k$,
(B) find the shortest distance between the parallel lines AB and CD ,
(C) find, in the form $a x + b y + c z + d = 0$, the equation of the plane containing AB and CD .
When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of $k$.
Find the value of $k$ for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.

OCR MEI FP3 2006 June Q2

2 A surface has equation $x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0$.

Find a normal vector at the point $( x , y , z )$ on the surface.
Find the equation of the tangent plane to the surface at the point $\mathrm { Q } ( 17,4,1 )$.
The point $( 17 + h , 4 + p , 1 - h )$, where $h$ and $p$ are small, is on the surface and is close to Q . Find an approximate expression for $p$ in terms of $h$.
Show that there is no point on the surface where the normal line is parallel to the $z$-axis.
Find the two values of $k$ for which $5 x - 6 y + 2 z = k$ is a tangent plane to the surface.

	\(e\)	\(а\)	\(r\)	\(r ^ { 2 }\)	\(r ^ { 3 }\)	\(r ^ { 4 }\)	ar	\(a r ^ { 2 }\)	\(a r ^ { 3 }\)	\(a r ^ { 4 }\)
\(e\)	\(e\)	\(а\)	\(r\)	\(r ^ { 2 }\)	\(r ^ { 3 }\)	\(r ^ { 4 }\)	ar	\(a r ^ { 2 }\)	\(a r ^ { 3 }\)	\(a r ^ { 4 }\)
\(а\)	\(а\)	\(e\)	ar	\(a r ^ { 2 }\)	\(a r ^ { 3 }\)	\(a r ^ { 4 }\)
\(r\)	r		\(r ^ { 2 }\)	\(r ^ { 3 }\)	\(r ^ { 4 }\)	\(e\)
\(r ^ { 2 }\)	\(r ^ { 2 }\)		\(r ^ { 3 }\)	\(r ^ { 4 }\)	\(e\)	\(r\)
\(r ^ { 3 }\)	\(r ^ { 3 }\)		\(r ^ { 4 }\)	\(e\)	\(r\)	\(r ^ { 2 }\)
\(r ^ { 4 }\)	\(r ^ { 4 }\)	ar	\(e\)	\(r\)	\(r ^ { 2 }\)	\(r ^ { 3 }\)
ar	ar		\(a r ^ { 2 }\)	\(a r ^ { 3 }\)	\(a r ^ { 4 }\)	\(а\)
\(a r ^ { 2 }\)	\(a r ^ { 2 }\)		\(a r ^ { 3 }\)	\(a r ^ { 4 }\)	\(a\)	ar			T
\(a r ^ { 3 }\)	\(a r ^ { 3 }\)		\(a r ^ { 4 }\)	\(а\)	ar	\(a r ^ { 2 }\)
\(a r ^ { 4 }\)	\(a r ^ { 4 }\)		\(а\)	ar	\(a r ^ { 2 }\)	\(a r ^ { 3 }\)

	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(p\)	\(t\)	\(s\)	\(p\)	\(r\)	\(q\)
\(q\)	\(s\)	\(p\)	\(q\)	\(t\)	\(r\)
\(r\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(s\)	\(r\)	\(t\)	\(s\)	\(q\)	\(p\)
\(t\)	\(q\)	\(r\)	\(t\)	\(p\)	\(s\)

\(*\)	\(a\)	\(b\)	\(c\)	\(d\)
\(a\)	\(d\)	\(a\)	\(b\)	\(c\)
\(b\)	\(a\)	\(b\)	\(c\)	\(d\)
\(c\)	\(b\)	\(c\)	\(d\)	\(a\)
\(d\)	\(c\)	\(d\)	\(a\)	\(b\)

Questions FP3 (473 questions)

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