Questions — OCR FP3 (140 questions)

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OCR FP3 2007 June Q5
5
  1. Use de Moivre's theorem to prove that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1 .$$
  2. 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.
  1. 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 \(= p\).
  2. 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 \(= p\).
  3. Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
  4. State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).
OCR FP3 2007 June Q8
8
  1. 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.
  2. 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 \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
  1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
  2. 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.
\(*\)\(a\)\(b\)\(c\)\(d\)
\(a\)\(d\)\(a\)\(b\)\(c\)
\(b\)\(a\)\(b\)\(c\)\(d\)
\(c\)\(b\)\(c\)\(d\)\(a\)
\(d\)\(c\)\(d\)\(a\)\(b\)
  1. Find the order of each element of \(G\).
  2. Write down a proper subgroup of \(G\).
  3. Is the group \(G\) cyclic? Give a reason for your answer.
  4. 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
  1. the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), correct to the nearest degree,
  2. 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 \(- \pi < \theta \leqslant \pi\).
  1. Express \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
  2. Find the cube roots of \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
  3. 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 $$\frac { x - 5 } { 1 } = \frac { y - 1 } { - 1 } = \frac { z - 5 } { - 2 } \quad \text { and } \quad \frac { x - 1 } { - 4 } = \frac { y - 11 } { - 14 } = \frac { z - 2 } { 2 } .$$
  1. Find the exact value of the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
  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 $$\mathbf { Q } = \left( \begin{array} { l l } 1 & 1
0 & 1 \end{array} \right)$$
  1. Prove that each matrix \(\mathbf { A }\) must be of the form \(\left( \begin{array} { l l } a & b
    0 & a \end{array} \right)\).
  2. 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 \(S\).
  3. Prove that \(S\) is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)
OCR FP3 Specimen Q7
7
  1. Prove that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), then \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\).
  2. 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
8
  1. 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 $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 \mathrm { e } ^ { - 2 x }$$
  2. Find the solution of this differential equation for which \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
  3. 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
1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).
  1. In each case, write down the smallest possible value of \(n\) :
    (a) if \(G\) is cyclic,
    (b) if \(G\) has a proper subgroup of order 3,
    (c) if \(G\) has at least two elements of order 2 .
  2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).
OCR FP3 2009 January Q2
2
  1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
  2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.
OCR FP3 2009 January Q3
3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$
  1. Find the direction of the common perpendicular to the lines.
  2. Find the shortest distance between the lines.
OCR FP3 2009 January Q4
4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
OCR FP3 2009 January Q5
5 The variables \(x\) and \(y\) are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$
  1. Use the substitution \(y = u - \frac { 1 } { x }\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2009 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{bc975428-c594-427b-a32e-268412b3cd26-3_554_825_264_660} The cuboid \(O A B C D E F G\) shown in the diagram has \(\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }\), and \(M\) is the mid-point of \(G F\).
  1. Find the equation of the plane \(A C G E\), giving your answer in the form r.n \(= p\).
  2. The plane \(O E F C\) has equation \(\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0\). Find the acute angle between the planes \(O E F C\) and \(A C G E\).
  3. The line \(A M\) meets the plane \(O E F C\) at the point \(W\). Find the ratio \(A W : W M\).
OCR FP3 2009 January Q7
7
  1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
    (a) Prove that the set of real numbers, together with the operation \(*\), forms a group.
    (b) State, with a reason, whether the group is commutative.
    (c) Prove that there are no elements of order 2.
  2. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.
OCR FP3 2009 January Q8
8
  1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
  2. Replace \(\theta\) by ( \(\frac { 1 } { 2 } \pi - \theta\) ) in the identity in part (i) to obtain a similar identity for \(\cos ^ { 6 } \theta\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta\).
OCR FP3 2010 January Q1
1 Determine whether the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 1 } = \frac { z + 4 } { 2 } \quad \text { and } \quad \frac { x + 3 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 5 } { 4 }$$ intersect or are skew.
\(2 \quad H\) denotes the set of numbers of the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.
  1. Show that the product of any two members of \(H\) is a member of \(H\). It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
  2. State the identity element of the group.
  3. Find the inverse of \(a + b \sqrt { 5 }\).
  4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse.
OCR FP3 2010 January Q3
3 Use the integrating factor method to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \mathrm { e } ^ { - 3 x }$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2010 January Q4
4
  1. Write down, in cartesian form, the roots of the equation \(z ^ { 4 } = 16\).
  2. Hence solve the equation \(w ^ { 4 } = 16 ( 1 - w ) ^ { 4 }\), giving your answers in cartesian form.
OCR FP3 2010 January Q5
5 A regular tetrahedron has vertices at the points $$A \left( 0,0 , \frac { 2 } { 3 } \sqrt { 6 } \right) , \quad B \left( \frac { 2 } { 3 } \sqrt { 3 } , 0,0 \right) , \quad C \left( - \frac { 1 } { 3 } \sqrt { 3 } , 1,0 \right) , \quad D \left( - \frac { 1 } { 3 } \sqrt { 3 } , - 1,0 \right) .$$
  1. Obtain the equation of the face \(A B C\) in the form $$x + \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$ (Answers which only verify the given equation will not receive full credit.)
  2. Give a geometrical reason why the equation of the face \(A B D\) can be expressed as $$x - \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$
  3. Hence find the cosine of the angle between two faces of the tetrahedron.
OCR FP3 2010 January Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 y = 8 \cos 4 x$$
  1. Find the complementary function of the differential equation.
  2. Given that there is a particular integral of the form \(y = p x \sin 4 x\), where \(p\) is a constant, find the general solution of the equation.
  3. Find the solution of the equation for which \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).