Questions - OCR FP3 - A-Level Maths

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OCR FP3 2007 January Q1

5 marks Standard +0.3

Show that the set of numbers $\{ 3,5,7 \}$, under multiplication modulo 8, does not form a group.
The set of numbers $\{ 3,5,7 , a \}$, under multiplication modulo 8 , forms a group. Write down the value of $a$.
State, justifying your answer, whether or not the group in part (ii) is isomorphic to the multiplicative group $\left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}$, where $e$ is the identity and $r ^ { 4 } = e$.

OCR FP3 2007 January Q2

5 marks Standard +0.8

2 Find the equation of the line of intersection of the planes with equations $$\mathbf { r } . ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 4 \quad \text { and } \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } ) = 6 ,$$ giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.

OCR FP3 2007 January Q3

7 marks Standard +0.3

Solve the equation $z ^ { 2 } - 6 z + 36 = 0$, and give your answers in the form $r ( \cos \theta \pm \mathrm { i } \sin \theta )$, where $r > 0$ and $0 \leqslant \theta \leqslant \pi$.
Given that $Z$ is either of the roots found in part (i), deduce the exact value of $Z ^ { - 3 }$.

OCR FP3 2007 January Q4

9 marks Standard +0.8

4 The variables $x$ and $y$ are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$

Use the substitution $y = x z$, where $z$ is a function of $x$, to obtain the differential equation $$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
Hence show by integration that the general solution of the differential equation (A) may be expressed in the form $x ^ { 2 } \left( x ^ { 2 } - 2 y ^ { 2 } \right) = k$, where $k$ is a constant.

OCR FP3 2007 January Q5

10 marks Challenging +1.8

5 A multiplicative group $G$ of order 9 has distinct elements $p$ and $q$, both of which have order 3 . The group is commutative, the identity element is $e$, and it is given that $q \neq p ^ { 2 }$.

Write down the elements of a proper subgroup of $G$
1. which does not contain $q$,
2. which does not contain $p$.
3. Find the order of each of the elements $p q$ and $p q ^ { 2 }$, justifying your answers.
4. State the possible order(s) of proper subgroups of $G$.
5. Find two proper subgroups of $G$ which are distinct from those in part (i), simplifying the elements.

OCR FP3 2007 January Q6

10 marks Standard +0.3

6 The variables $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y = 2 x + 1$$ Find

the complementary function,
the general solution. In a particular case, it is given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.
Find the solution of the differential equation in this case.
Write down the function to which $y$ approximates when $x$ is large and positive.

OCR FP3 2007 January Q7

13 marks Standard +0.3

7 The position vectors of the points $A , B , C , D , G$ are given by $$\mathbf { a } = 6 \mathbf { i } + 4 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { c } = \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { d } = 3 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { g } = 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$$ respectively.

The line through $A$ and $G$ meets the plane $B C D$ at $M$. Write down the vector equation of the line through $A$ and $G$ and hence show that the position vector of $M$ is $2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }$.
Find the value of the ratio $A G : A M$.
Find the position vector of the point $P$ on the line through $C$ and $G$, such that $\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }$.
Verify that $P$ lies in the plane $A B D$.

OCR FP3 2007 January Q8

13 marks Challenging +1.2

Use de Moivre's theorem to find an expression for $\tan 4 \theta$ in terms of $\tan \theta$.
Deduce that $\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }$.
Hence show that one of the roots of the equation $x ^ { 2 } - 6 x + 1 = 0$ is $\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)$.
Hence find the value of $\operatorname { cosec } ^ { 2 } \left( \frac { 1 } { 8 } \pi \right) + \operatorname { cosec } ^ { 2 } \left( \frac { 3 } { 8 } \pi \right)$, justifying your answer.

OCR FP3 2007 June Q1

3 marks Moderate -0.8

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 $z$.

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

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

8 marks Standard +0.8

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

10 marks Standard +0.8

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

10 marks Standard +0.3

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

12 marks Standard +0.3

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.
1. The numbers $3 ^ { 2 n }$, where $n \in \mathbb { Z }$.
2. The numbers $3 ^ { n }$, where $n \in \mathbb { Z }$ and $n \geqslant 0$.
3. The numbers $3 ^ { \left( \pm n ^ { 2 } \right) }$, where $n \in \mathbb { Z }$. 4

OCR FP3 Specimen Q1

5 marks Moderate -0.3

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

6 marks Standard +0.8

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

8 marks Standard +0.8

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

9 marks Standard +0.3

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

9 marks Standard +0.8

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

10 marks Challenging +1.2

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

10 marks Challenging +1.2

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

15 marks Challenging +1.2

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 FP3 2009 January Q1

5 marks Standard +0.8

1 In this question $G$ is a group of order $n$, where $3 \leqslant n < 8$.

In each case, write down the smallest possible value of $n$ :
1. if $G$ is cyclic,
2. if $G$ has a proper subgroup of order 3,
3. if $G$ has at least two elements of order 2 .
4. Another group has the same order as $G$, but is not isomorphic to $G$. Write down the possible value(s) of $n$.

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

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

Questions — OCR FP3 (182 questions)

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OCR FP3 2007 January Q1

OCR FP3 2007 January Q2

OCR FP3 2007 January Q3

OCR FP3 2007 January Q4

OCR FP3 2007 January Q5

OCR FP3 2007 January Q6

OCR FP3 2007 January Q7

OCR FP3 2007 January Q8

OCR FP3 2007 June Q1

OCR FP3 2007 June Q2

OCR FP3 2007 June Q3

OCR FP3 2007 June Q4

OCR FP3 2007 June Q5

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OCR FP3 2007 June Q8

OCR FP3 2007 June Q9

OCR FP3 Specimen Q1

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OCR FP3 Specimen Q8

OCR FP3 2009 January Q1