Questions — OCR FP3 (140 questions)

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OCR FP3 2010 January Q7
7
  1. Solve the equation \(\cos 6 \theta = 0\), for \(0 < \theta < \pi\).
  2. By using de Moivre's theorem, show that $$\cos 6 \theta \equiv \left( 2 \cos ^ { 2 } \theta - 1 \right) \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 1 \right)$$
  3. Hence find the exact value of $$\cos \left( \frac { 1 } { 12 } \pi \right) \cos \left( \frac { 5 } { 12 } \pi \right) \cos \left( \frac { 7 } { 12 } \pi \right) \cos \left( \frac { 11 } { 12 } \pi \right)$$ justifying your answer.
OCR FP3 2010 January Q8
8 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 1 } { 2 - 2 x }\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\). The function g is defined by \(\mathrm { g } ( x ) = \mathrm { ff } ( x )\).
  1. Show that \(\mathrm { g } ( x ) = \frac { 1 - x } { 1 - 2 x }\) and that \(\operatorname { gg } ( x ) = x\). It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where e : \(x \mapsto x\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\).
  2. State the orders of the elements f and g .
  3. The inverse of the element f is denoted by h . Find \(\mathrm { h } ( x )\).
  4. Construct the operation table for the elements e, f, g, h of the group \(K\).
OCR FP3 2011 January Q1
1
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
  2. Find the particular solution for which \(y = 1\) when \(x = 0\).
OCR FP3 2011 January Q2
2 Two intersecting lines, lying in a plane \(p\), have equations $$\frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 3 } \quad \text { and } \quad \frac { x - 1 } { - 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 4 } .$$
  1. Obtain the equation of \(p\) in the form \(2 x - y + z = 3\).
  2. Plane \(q\) has equation \(2 x - y + z = 21\). Find the distance between \(p\) and \(q\).
OCR FP3 2011 January Q3
3
  1. Express \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) and show that $$\sin ^ { 4 } \theta \equiv \frac { 1 } { 8 } ( \cos 4 \theta - 4 \cos 2 \theta + 3 )$$
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta\).
OCR FP3 2011 January Q4
4 The cube roots of 1 are denoted by \(1 , \omega\) and \(\omega ^ { 2 }\), where the imaginary part of \(\omega\) is positive.
  1. Show that \(1 + \omega + \omega ^ { 2 } = 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{d12573dd-c0c2-4f0d-8e49-8fdf8d5864a5-2_616_748_1676_699} In the diagram, \(A B C\) is an equilateral triangle, labelled anticlockwise. The points \(A , B\) and \(C\) represent the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) respectively.
  2. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z _ { 1 } - z _ { 3 } = \omega \left( z _ { 3 } - z _ { 2 } \right)\).
  3. Hence show that \(z _ { 1 } + \omega z _ { 2 } + \omega ^ { 2 } z _ { 3 } = 0\).
OCR FP3 2011 January Q7
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 \(m\).
  1. Show that \(l\) and \(m\) are in the same direction.
  2. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
  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.
OCR FP3 2011 January Q8
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 \(( 1,0 )\).
  1. Prove that \(*\) is associative.
  2. Find all the elements which commute with \(( 1,1 )\).
  3. 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 \(n\).
  4. Find all self-inverse elements.
  5. Give a reason why the elements \(( x , y )\), under the operation \(*\), do not form a group.
OCR FP3 2012 January Q1
1 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y } .$$
  1. Use the substitution \(y = u x\), where \(u\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } = \frac { 2 } { u } .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR FP3 2012 January Q2
2
  1. Show that \(\left( z ^ { n } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { n } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 n } - ( 2 \cos \theta ) z ^ { n } + 1\).
  2. Express \(z ^ { 4 } - z ^ { 2 } + 1\) as the product of four factors of the form \(\left( z - e ^ { \mathrm { i } \alpha } \right)\), where \(0 \leqslant \alpha < 2 \pi\).
OCR FP3 2012 January Q3
3 A multiplicative group contains the distinct elements \(e , x\) and \(y\), where \(e\) is the identity.
  1. Prove that \(x ^ { - 1 } y ^ { - 1 } = ( y x ) ^ { - 1 }\).
  2. Given that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) for some integer \(n \geqslant 2\), prove that \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\).
  3. If \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\), does it follow that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) ? Give a reason for your answer.
OCR FP3 2012 January Q4
4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).
  1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
  2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).
OCR FP3 2012 January Q5
5 The variables \(x\) and \(y\) satisfy the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 5 \mathrm { e } ^ { - 2 x }$$
  1. Find the complementary function of the differential equation.
  2. Given that there is a particular integral of the form \(y = p x \mathrm { e } ^ { - 2 x }\), find the constant \(p\).
  3. Find the solution of the equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\).
OCR FP3 2012 January Q6
6 The plane \(\Pi\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1
6
7 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 1
- 1 \end{array} \right) + \mu \left( \begin{array} { r } 2
- 3
- 5 \end{array} \right)\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 7
4
1 \end{array} \right) + t \left( \begin{array} { r } 3
0
- 1 \end{array} \right)\).
  1. Express the equation of \(\Pi\) in the form r.n \(= p\).
  2. Find the point of intersection of \(l\) and \(\Pi\).
  3. The equation of \(\Pi\) may be expressed in the form \(\mathbf { r } = \left( \begin{array} { l } 1
    6
    7 \end{array} \right) + \lambda \left( \begin{array} { r } 2
    - 1
    - 1 \end{array} \right) + \mu \mathbf { c }\), where \(\mathbf { c }\) is perpendicular to \(\left( \begin{array} { r } 2
    - 1
    - 1 \end{array} \right)\). Find \(\mathbf { c }\).
OCR FP3 2012 January Q7
7 The set \(M\) consists of the six matrices \(\left( \begin{array} { l l } 1 & 0
n & 1 \end{array} \right)\), where \(n \in \{ 0,1,2,3,4,5 \}\). It is given that \(M\) forms a group ( \(M , \times\) ) under matrix multiplication, with numerical addition and multiplication both being carried out modulo 6 .
  1. Determine whether ( \(M , \times\) ) is a commutative group, justifying your answer.
  2. Write down the identity element of the group and find the inverse of \(\left( \begin{array} { l l } 1 & 0
    2 & 1 \end{array} \right)\).
  3. State the order of \(\left( \begin{array} { l l } 1 & 0
    3 & 1 \end{array} \right)\) and give a reason why \(( M , \times )\) has no subgroup of order 4.
  4. The multiplicative group \(G\) has order 6. All the elements of \(G\), apart from the identity, have order 2 or 3 . Determine whether \(G\) is isomorphic to ( \(M , \times\) ), justifying your answer.
OCR FP3 2012 January Q8
8
  1. Use de Moivre's theorem to prove that $$\tan 5 \theta \equiv \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Solve the equation \(\tan 5 \theta = 1\), for \(0 \leqslant \theta < \pi\).
  3. Show that the roots of the equation $$t ^ { 4 } - 4 t ^ { 3 } - 14 t ^ { 2 } - 4 t + 1 = 0$$ may be expressed in the form \(\tan \alpha\), stating the exact values of \(\alpha\), where \(0 \leqslant \alpha < \pi\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR FP3 2013 January Q1
1 Two planes have equations $$x + 2 y + 5 z = 12 \text { and } 2 x - y + 3 z = 5$$
  1. Find the acute angle between the planes.
  2. Find a vector equation of the line of intersection of the planes.
OCR FP3 2013 January Q2
2 The elements of a group \(G\) are the complex numbers \(a + b \mathrm { i }\) where \(a , b \in \{ 0,1,2,3,4 \}\). These elements are combined under the operation of addition modulo 5 .
  1. State the identity element and the order of \(G\).
  2. Write down the inverse of \(2 + 4 \mathrm { i }\).
  3. Show that every non-zero element of \(G\) has order 5 .
OCR FP3 2013 January Q3
3 Solve the differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = x ^ { 4 } \mathrm { e } ^ { 2 x }\) for \(y\) in terms of \(x\), given that \(y = 0\) when \(x = 1\).
OCR FP3 2013 January Q4
4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 1
2
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
3
- 1 \end{array} \right) \text { and } \mathbf { r } = \left( \begin{array} { l } 3
0
1 \end{array} \right) + \mu \left( \begin{array} { r } 4
- 1
- 1 \end{array} \right)$$ respectively.
  1. Find the shortest distance between the lines.
  2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and which is parallel to \(l _ { 2 }\).
OCR FP3 2013 January Q5
5
  1. Solve the equation \(z ^ { 5 } = 1\), giving your answers in polar form.
  2. Hence, by considering the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\), show that the roots of $$5 z ^ { 4 } + 10 z ^ { 3 } + 10 z ^ { 2 } + 5 z + 1 = 0$$ can be expressed in the form \(\frac { 1 } { \mathrm { e } ^ { \mathrm { i } \theta } - 1 }\), stating the values of \(\theta\).
OCR FP3 2013 January Q6
6 The differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin k x\) is to be solved, where \(k\) is a constant.
  1. In the case \(k = 2\), by using a particular integral of the form \(a x \cos 2 x + b x \sin 2 x\), find the general solution.
  2. Describe briefly the behaviour of \(y\) when \(x \rightarrow \infty\).
  3. In the case \(k \neq 2\), explain whether \(y\) would exhibit the same behaviour as in part (ii) when \(x \rightarrow \infty\).
OCR FP3 2013 January Q7
7 Let \(S = \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { 2 \mathrm { i } \theta } + \mathrm { e } ^ { 3 \mathrm { i } \theta } + \ldots + \mathrm { e } ^ { 10 \mathrm { i } \theta }\).
  1. (a) Show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$S = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } \left( \mathrm { e } ^ { 10 \mathrm { i } \theta } - 1 \right) } { 2 \mathrm { i } \sin \left( \frac { 1 } { 2 } \theta \right) }$$ (b) State the value of \(S\) for \(\theta = 2 n \pi\), where \(n\) is an integer.
  2. Hence show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = \frac { \sin \left( \frac { 21 } { 2 } \theta \right) } { 2 \sin \left( \frac { 1 } { 2 } \theta \right) } - \frac { 1 } { 2 }$$
  3. Hence show that \(\theta = \frac { 1 } { 11 } \pi\) is a root of \(\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = 0\) and find another root in the interval \(0 < \theta < \frac { 1 } { 4 } \pi\).
OCR FP3 2013 January Q8
8 A multiplicative group \(H\) has the elements \(\left\{ e , a , a ^ { 2 } , a ^ { 3 } , w , a w , a ^ { 2 } w , a ^ { 3 } w \right\}\) where \(e\) is the identity, elements \(a\) and \(w\) have orders 4 and 2 respectively and \(w a = a ^ { 3 } w\).
  1. Show that \(w a ^ { 2 } = a ^ { 2 } w\) and also that \(w a ^ { 3 } = a w\).
  2. Hence show that each of \(a w , a ^ { 2 } w\) and \(a ^ { 3 } w\) has order 2 .
  3. Find two non-cyclic subgroups of \(H\) of order 4, and show that they are not cyclic.
OCR FP3 2012 June Q1
1 The plane \(p\) has equation \(\mathbf { r } . ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } ) = 4\) and the line \(l _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { j } - \mathbf { k } + t ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The line \(l _ { 2 }\) is parallel to \(p\) and perpendicular to \(l _ { 1 }\), and passes through the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\). Find the equation of \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).