Questions - OCR FP3 - A-Level Maths

OCR FP3 2015 June Q7

Use de Moivre's theorem to show that $\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$.
Hence find the exact roots of $t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0$.

OCR FP3 2015 June Q8

8 Let $G$ be any multiplicative group. $H$ is a subset of $G$. $H$ consists of all elements $h$ such that $h g = g h$ for every element $g$ in $G$.

Prove that $H$ is a subgroup of $G$. Now consider the case where $G$ is given by the following table:

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

Show that $H$ consists of just the identity element.

OCR FP3 2009 June Q1

1 Find the cube roots of $\frac { 1 } { 2 } \sqrt { 3 } + \frac { 1 } { 2 } \mathrm { i }$, giving your answers in the form $\cos \theta + \mathrm { i } \sin \theta$, where $0 \leqslant \theta < 2 \pi$.

OCR FP3 2009 June Q2

2 It is given that the set of complex numbers of the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ for $- \pi < \theta \leqslant \pi$ and $r > 0$, under multiplication, forms a group.

Write down the inverse of $5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }$.
Prove the closure property for the group.
$Z$ denotes the element $\mathrm { e } ^ { \mathrm { i } \gamma }$, where $\frac { 1 } { 2 } \pi < \gamma < \pi$. Express $Z ^ { 2 }$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$, where $- \pi < \theta < 0$.

OCR FP3 2009 June Q3

3 A line $l$ has equation $\frac { x - 6 } { - 4 } = \frac { y + 7 } { 8 } = \frac { z + 10 } { 7 }$ and a plane $p$ has equation $3 x - 4 y - 2 z = 8$.

Find the point of intersection of $l$ and $p$.
Find the equation of the plane which contains $l$ and is perpendicular to $p$, giving your answer in the form $a x + b y + c z = d$.

OCR FP3 2009 June Q4

4 The differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { 1 - x ^ { 2 } } y = ( 1 - x ) ^ { \frac { 1 } { 2 } } , \quad \text { where } | x | < 1$$ can be solved by the integrating factor method.

Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as $\left( \frac { 1 + x } { 1 - x } \right) ^ { \frac { 1 } { 2 } }$.
Hence find the solution of the differential equation for which $y = 2$ when $x = 0$, giving your answer in the form $y = \mathrm { f } ( x )$.

OCR FP3 2009 June Q5

5 The variables $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = \mathrm { e } ^ { 3 x }$$

Find the complementary function.
Explain briefly why there is no particular integral of either of the forms $y = k \mathrm { e } ^ { 3 x }$ or $y = k x \mathrm { e } ^ { 3 x }$.
Given that there is a particular integral of the form $y = k x ^ { 2 } \mathrm { e } ^ { 3 x }$, find the value of $k$.

OCR FP3 2009 June Q6

6 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { l } 2
2
1 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
0 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 5
- 2 \end{array} \right)$.

Express the equation of $\Pi _ { 1 }$ in the form r.n $= p$. The plane $\Pi _ { 2 }$ has equation $\mathbf { r } . \left( \begin{array} { r } 7
17
- 3 \end{array} \right) = 21$.
Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.

OCR FP3 2009 June Q7

Use de Moivre's theorem to prove that $$\tan 3 \theta \equiv \frac { \tan \theta \left( 3 - \tan ^ { 2 } \theta \right) } { 1 - 3 \tan ^ { 2 } \theta } .$$
(a) By putting $\theta = \frac { 1 } { 12 } \pi$ in the identity in part (i), show that $\tan \frac { 1 } { 12 } \pi$ is a solution of the equation $$t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0 .$$ (b) Hence show that $\tan \frac { 1 } { 12 } \pi = 2 - \sqrt { 3 }$.
Use the substitution $t = \tan \theta$ to show that $$\int _ { 0 } ^ { 2 - \sqrt { 3 } } \frac { t \left( 3 - t ^ { 2 } \right) } { \left( 1 - 3 t ^ { 2 } \right) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t = a \ln b$$ where $a$ and $b$ are positive constants to be determined.

OCR FP3 2009 June Q8

8 A multiplicative group $Q$ of order 8 has elements $\left\{ e , p , p ^ { 2 } , p ^ { 3 } , a , a p , a p ^ { 2 } , a p ^ { 3 } \right\}$, where $e$ is the identity. The elements have the properties $p ^ { 4 } = e$ and $a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }$.

Prove that $a = p a p$ and that $p = a p a$.
Find the order of each of the elements $p ^ { 2 } , a , a p , a p ^ { 2 }$.
Prove that $\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}$ is a subgroup of $Q$.
Determine whether $Q$ is a commutative group.

OCR FP3 2010 June Q1

1 The line $l _ { 1 }$ passes through the points $( 0,0,10 )$ and $( 7,0,0 )$ and the line $l _ { 2 }$ passes through the points $( 4,6,0 )$ and $( 3,3,1 )$. Find the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.

OCR FP3 2010 June Q2

2 A multiplicative group with identity $e$ contains distinct elements $a$ and $r$, with the properties $r ^ { 6 } = e$ and $a r = r ^ { 5 } a$.

Prove that r ar $= a$.
Prove, by induction or otherwise, that $r ^ { n } a r ^ { n } = a$ for all positive integers $n$.

OCR FP3 2010 June Q3

3 In this question, $w$ denotes the complex number $\cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi$.

Express $w ^ { 2 } , w ^ { 3 }$ and $w ^ { * }$ in polar form, with arguments in the interval $0 \leqslant \theta < 2 \pi$.
The points in an Argand diagram which represent the numbers $$1 , \quad 1 + w , \quad 1 + w + w ^ { 2 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 }$$ are denoted by $A , B , C , D , E$ respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.)
Write down a polynomial equation of degree 5 which is satisfied by $w$.

OCR FP3 2010 June Q4

Use the substitution $y = x z$ to find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = x \cos \left( \frac { y } { x } \right)$$ giving your answer in a form without logarithms. (You may quote an appropriate result given in the List of Formulae (MF1).)
Find the solution of the differential equation for which $y = \pi$ when $x = 4$.

OCR FP3 2010 June Q5

5 Convergent infinite series $C$ and $S$ are defined by $$\begin{gathered} C = 1 + \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots
S = \quad \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots \end{gathered}$$

Show that $C + \mathrm { i } S = \frac { 2 } { 2 - \mathrm { e } ^ { \mathrm { i } \theta } }$.
Hence show that $C = \frac { 4 - 2 \cos \theta } { 5 - 4 \cos \theta }$, and find a similar expression for $S$.
Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 17 y = 17 x + 36$$
Show that, when $x$ is large and positive, the solution approximates to a linear function, and state its equation.

OCR FP3 2010 June Q7

7 A line $l$ has equation $\mathbf { r } = \left( \begin{array} { r } - 7
- 3
0 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 2
3 \end{array} \right)$. A plane $\Pi$ passes through the points $( 1,3,5 )$ and ( $5,2,5$ ), and is parallel to $l$.

Find an equation of $\Pi$, giving your answer in the form r.n $= p$.
Find the distance between $l$ and $\Pi$.
Find an equation of the line which is the reflection of $l$ in $\Pi$, giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.

OCR FP3 2010 June Q8

8 A set of matrices $M$ is defined by $$A = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) , \quad B = \left( \begin{array} { c c } \omega & 0
0 & \omega ^ { 2 } \end{array} \right) , \quad C = \left( \begin{array} { c c } \omega ^ { 2 } & 0
0 & \omega \end{array} \right) , \quad D = \left( \begin{array} { c c } 0 & 1
1 & 0 \end{array} \right) , \quad E = \left( \begin{array} { c c } 0 & \omega ^ { 2 }
\omega & 0 \end{array} \right) , \quad F = \left( \begin{array} { c c } 0 & \omega
\omega ^ { 2 } & 0 \end{array} \right) ,$$ where $\omega$ and $\omega ^ { 2 }$ are the complex cube roots of 1 . It is given that $M$ is a group under matrix multiplication.

Write down the elements of a subgroup of order 2.
Explain why there is no element $X$ of the group, other than $A$, which satisfies the equation $X ^ { 5 } = A$.
By finding $B E$ and $E B$, verify the closure property for the pair of elements $B$ and $E$.
Find the inverses of $B$ and $E$.
Determine whether the group $M$ is isomorphic to the group $N$ which is defined as the set of numbers $\{ 1,2,4,8,7,5 \}$ under multiplication modulo 9 . Justify your answer clearly.

OCR FP3 2011 June Q1

1 A line $l$ has equation $\frac { x - 1 } { 5 } = \frac { y - 6 } { 6 } = \frac { z + 3 } { - 7 }$ and a plane $p$ has equation $x + 2 y - z = 40$.

Find the acute angle between $l$ and $p$.
Find the perpendicular distance from the point $( 1,6 , - 3 )$ to $p$.

OCR FP3 2011 June Q2

2 It is given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$, where $0 < \theta < 2 \pi$, and $w = \frac { 1 + z } { 1 - z }$.

Prove that $w = \mathrm { i } \cot \frac { 1 } { 2 } \theta$.
Sketch separate Argand diagrams to show the locus of $z$ and the locus of $w$. You should show the direction in which each locus is described when $\theta$ increases in the interval $0 < \theta < 2 \pi$.

OCR FP3 2011 June Q3

3 The variables $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 4 y = 5 \cos 3 x$$

Find the complementary function.
Hence, or otherwise, find the general solution.
Find the approximate range of values of $y$ when $x$ is large and positive.

OCR FP3 2011 June Q4

4 A group $G$, of order 8, is generated by the elements $a , b , c . G$ has the properties $$a ^ { 2 } = b ^ { 2 } = c ^ { 2 } = e , \quad a b = b a , \quad b c = c b , \quad c a = a c ,$$ where $e$ is the identity.

Using these properties and basic group properties as necessary, prove that $a b c = c b a$. The operation table for $G$ is shown below.

	$e$	$a$	$b$	$c$	$b c$	ca	$a b$	$a b c$
$e$	$e$	$a$	$b$	$c$	$b c$	ca	$a b$	$a b c$
$a$	$a$	$e$	$a b$	ca	$a b c$	$c$	$b$	$b c$
$b$	$b$	$a b$	$e$	$b c$	$c$	$a b c$	$a$	ca
c	$c$	ca	$b c$	$e$	$b$	$a$	$a b c$	$a b$
$b c$	$b c$	$a b c$	$c$	$b$	$e$	$a b$	ca	$a$
ca	ca	c	$a b c$	$a$	$a b$	$e$	$b c$	$b$
$a b$	$a b$	$b$	$a$	$a b c$	ca	bc	$e$	$c$
$a b c$	$a b c$	$b c$	ca	$a b$	$a$	$b$	$c$	$e$

List all the subgroups of order 2 .
List five subgroups of order 4.
Determine whether all the subgroups of $G$ which are of order 4 are isomorphic.

OCR FP3 2011 June Q5

5 The substitution $y = u ^ { k }$, where $k$ is an integer, is to be used to solve the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } y ^ { 2 }$$ by changing it into an equation (B) in the variables $u$ and $x$.

Show that equation (B) may be written in the form $$\frac { \mathrm { d } u } { \mathrm {~d} x } + \frac { 3 } { k x } u = \frac { 1 } { k } x u ^ { k + 1 }$$
Write down the value of $k$ for which the integrating factor method may be used to solve equation (B).
Using this value of $k$, solve equation (B) and hence find the general solution of equation (A), giving your answer in the form $y = \mathrm { f } ( x )$.
(a) The set of polynomials $\{ a x + b \}$, where $a , b \in \mathbb { R }$, is denoted by $P$. Assuming that the associativity property holds, prove that $P$, under addition, is a group.
(b) The set of polynomials $\{ a x + b \}$, where $a , b \in \{ 0,1,2 \}$, is denoted by $Q$. It is given that $Q$, under addition modulo 3 , is a group, denoted by $( Q , + ( \bmod 3 ) )$.
State the order of the group.
Write down the inverse of the element $2 x + 1$.
$\mathrm { q } ( x ) = a x + b$ is any element of $Q$ other than the identity. Find the order of $\mathrm { q } ( x )$ and hence determine whether $( Q , + ( \bmod 3 ) )$ is a cyclic group.

OCR FP3 2011 June Q7

7 (In this question, the notation $\triangle A B C$ denotes the area of the triangle $A B C$.)
The points $P , Q$ and $R$ have position vectors $p \mathbf { i } , q \mathbf { j }$ and $r \mathbf { k }$ respectively, relative to the origin $O$, where $p , q$ and $r$ are positive. The points $O , P , Q$ and $R$ are joined to form a tetrahedron.

Draw a sketch of the tetrahedron and write down the values of $\triangle O P Q , \triangle O Q R$ and $\triangle O R P$.
Use the definition of the vector product to show that $\frac { 1 } { 2 } | \overrightarrow { R P } \times \overrightarrow { R Q } | = \Delta P Q R$.
Show that $( \triangle O P Q ) ^ { 2 } + ( \triangle O Q R ) ^ { 2 } + ( \triangle O R P ) ^ { 2 } = ( \triangle P Q R ) ^ { 2 }$.
Use de Moivre's theorem to express $\cos 4 \theta$ as a polynomial in $\cos \theta$.
Hence prove that $\cos 4 \theta \cos 2 \theta \equiv 16 \cos ^ { 6 } \theta - 24 \cos ^ { 4 } \theta + 10 \cos ^ { 2 } \theta - 1$.
Use part (ii) to show that the only roots of the equation $\cos 4 \theta \cos 2 \theta = 1$ are $\theta = n \pi$, where $n$ is an integer.
Show that $\cos 4 \theta \cos 2 \theta = - 1$ only when $\cos \theta = 0$.

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

	\(e\)	\(a\)	\(b\)	\(c\)	\(b c\)	ca	\(a b\)	\(a b c\)
\(e\)	\(e\)	\(a\)	\(b\)	\(c\)	\(b c\)	ca	\(a b\)	\(a b c\)
\(a\)	\(a\)	\(e\)	\(a b\)	ca	\(a b c\)	\(c\)	\(b\)	\(b c\)
\(b\)	\(b\)	\(a b\)	\(e\)	\(b c\)	\(c\)	\(a b c\)	\(a\)	ca
c	\(c\)	ca	\(b c\)	\(e\)	\(b\)	\(a\)	\(a b c\)	\(a b\)
\(b c\)	\(b c\)	\(a b c\)	\(c\)	\(b\)	\(e\)	\(a b\)	ca	\(a\)
ca	ca	c	\(a b c\)	\(a\)	\(a b\)	\(e\)	\(b c\)	\(b\)
\(a b\)	\(a b\)	\(b\)	\(a\)	\(a b c\)	ca	bc	\(e\)	\(c\)
\(a b c\)	\(a b c\)	\(b c\)	ca	\(a b\)	\(a\)	\(b\)	\(c\)	\(e\)

Questions — OCR FP3 (140 questions)

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