Questions — OCR FP3 (140 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR FP3 2007 January Q1
1
  1. Show that the set of numbers \(\{ 3,5,7 \}\), under multiplication modulo 8, does not form a group.
  2. The set of numbers \(\{ 3,5,7 , a \}\), under multiplication modulo 8 , forms a group. Write down the value of \(a\).
  3. 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
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
3
  1. 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 \(0 \leqslant \theta \leqslant \pi\).
  2. Given that \(Z\) is either of the roots found in part (i), deduce the exact value of \(Z ^ { - 3 }\).
OCR FP3 2007 January Q4
4 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$
  1. Use the substitution \(y = x z\), where \(z\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
  2. 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
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 \(q \neq p ^ { 2 }\).
  1. Write down the elements of a proper subgroup of \(G\)
    (a) which does not contain \(q\),
    (b) which does not contain \(p\).
  2. Find the order of each of the elements \(p q\) and \(p q ^ { 2 }\), justifying your answers.
  3. State the possible order(s) of proper subgroups of \(G\).
  4. Find two proper subgroups of \(G\) which are distinct from those in part (i), simplifying the elements.
OCR FP3 2007 January Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y = 2 x + 1$$ Find
  1. the complementary function,
  2. the general solution. In a particular case, it is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
  3. Find the solution of the differential equation in this case.
  4. Write down the function to which \(y\) approximates when \(x\) is large and positive.
OCR FP3 2007 January Q7
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.
  1. 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 \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\).
  2. Find the value of the ratio \(A G : A M\).
  3. Find the position vector of the point \(P\) on the line through \(C\) and \(G\), such that \(\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }\).
  4. Verify that \(P\) lies in the plane \(A B D\).
OCR FP3 2007 January Q8
8
  1. Use de Moivre's theorem to find an expression for \(\tan 4 \theta\) in terms of \(\tan \theta\).
  2. Deduce that \(\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }\).
  3. Hence show that one of the roots of the equation \(x ^ { 2 } - 6 x + 1 = 0\) is \(\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).
  4. 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 2008 January Q1
1
  1. A group \(G\) of order 6 has the combination table shown below.
    \(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(e\)\(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(a\)\(a\)\(b\)\(e\)\(r\)\(p\)\(q\)
    \(b\)\(b\)\(e\)\(a\)\(q\)\(r\)\(p\)
    \(p\)\(p\)\(q\)\(r\)\(e\)\(a\)\(b\)
    \(q\)\(q\)\(r\)\(p\)\(b\)\(e\)\(a\)
    \(r\)\(r\)\(p\)\(q\)\(a\)\(b\)\(e\)
    1. State, with a reason, whether or not \(G\) is commutative.
    2. State the number of subgroups of \(G\) which are of order 2 .
    3. List the elements of the subgroup of \(G\) which is of order 3 .
  2. A multiplicative group \(H\) of order 6 has elements \(e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }\), where \(e\) is the identity. Write down the order of each of the elements \(c ^ { 3 } , c ^ { 4 }\) and \(c ^ { 5 }\).
OCR FP3 2008 January Q2
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x .$$
OCR FP3 2008 January Q3
3 Two fixed points, \(A\) and \(B\), have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\), and a variable point \(P\) has position vector \(\mathbf { r }\).
  1. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } = \lambda \mathbf { a }\), where \(0 \leqslant \lambda \leqslant 1\).
  2. Given that \(P\) is a point on the line \(A B\), use a property of the vector product to explain why \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }\).
  3. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }\).
OCR FP3 2008 January Q4
4 The integrals \(C\) and \(S\) are defined by $$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$ By considering \(C + \mathrm { i } S\) as a single integral, show that $$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right) ,$$ and obtain a similar expression for \(S\).
(You may assume that the standard result for \(\int \mathrm { e } ^ { k x } \mathrm {~d} x\) remains true when \(k\) is a complex constant, so that \(\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)\)
OCR FP3 2008 January Q5
5
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$ expressing \(y\) in terms of \(x\) in your answer. In a particular case, it is given that \(y = \frac { 2 } { \pi }\) when \(x = \frac { 1 } { 4 } \pi\).
  2. Find the solution of the differential equation in this case.
  3. Write down a function to which \(y\) approximates when \(x\) is large and positive.
OCR FP3 2008 January Q6
6 A tetrahedron \(A B C D\) is such that \(A B\) is perpendicular to the base \(B C D\). The coordinates of the points \(A , C\) and \(D\) are \(( - 1 , - 7,2 ) , ( 5,0,3 )\) and \(( - 1,3,3 )\) respectively, and the equation of the plane \(B C D\) is \(x + 2 y - 2 z = - 1\).
  1. Find, in either order, the coordinates of \(B\) and the length of \(A B\).
  2. Find the acute angle between the planes \(A C D\) and \(B C D\).
  3. (a) Verify, without using a calculator, that \(\theta = \frac { 1 } { 8 } \pi\) is a solution of the equation \(\sin 6 \theta = \sin 2 \theta\).
    (b) By sketching the graphs of \(y = \sin 6 \theta\) and \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), or otherwise, find the other solution of the equation \(\sin 6 \theta = \sin 2 \theta\) in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\).
  4. Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right) .$$
  5. Hence show that one of the solutions obtained in part (i) satisfies \(\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )\), and justify which solution it is.
OCR FP3 2008 January Q8
8 Groups \(A , B , C\) and \(D\) are defined as follows:
\(A\) : the set of numbers \(\{ 2,4,6,8 \}\) under multiplication modulo 10 ,
\(B\) : the set of numbers \(\{ 1,5,7,11 \}\) under multiplication modulo 12 ,
\(C\) : the set of numbers \(\left\{ 2 ^ { 0 } , 2 ^ { 1 } , 2 ^ { 2 } , 2 ^ { 3 } \right\}\) under multiplication modulo 15,
\(D\) : the set of numbers \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) under multiplication.
  1. Write down the identity element for each of groups \(A , B , C\) and \(D\).
  2. 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.
  3. Prove the closure property for group \(D\).
  4. 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
1
  1. 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 \(a + \mathrm { i } b\).
  2. 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
2
  1. 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 \(0 \leqslant \theta < 2 \pi\).
  2. 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 \(\frac { x } { 2 } = \frac { y + 4 } { 3 } = \frac { z + 9 } { 5 }\).
  1. Find the cartesian equation of the plane which is parallel to \(l _ { 1 }\) and which contains the points \(( 2,1,5 )\) and \(( 0 , - 1,5 )\).
  2. Write down the position vector of a point on \(l _ { 1 }\) with parameter \(t\).
  3. 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 \(\frac { x - a } { p } = \frac { y - b } { q } = \frac { z - c } { r }\).
  4. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin x$$
  5. 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 $$\begin{aligned} & C = 1 + \cos \theta + \cos 2 \theta + \cos 3 \theta + \cos 4 \theta + \cos 5 \theta
& S = \quad \sin \theta + \sin 2 \theta + \sin 3 \theta + \sin 4 \theta + \sin 5 \theta \end{aligned}$$
  1. Show that \(C + \mathrm { i } S = \frac { \mathrm { e } ^ { 3 \mathrm { i } \theta } - \mathrm { e } ^ { - 3 \mathrm { i } \theta } } { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { i } \theta } } \mathrm { e } ^ { \frac { 5 } { 2 } \mathrm { i } \theta }\).
  2. Deduce that \(C = \sin 3 \theta \cos \frac { 5 } { 2 } \theta \operatorname { cosec } \frac { 1 } { 2 } \theta\) and write down the corresponding expression for \(S\).
  3. 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\(e\)\(r\)\(r ^ { 2 }\)\(r ^ { 3 }\)
arar\(a r ^ { 2 }\)\(a r ^ { 3 }\)\(a r ^ { 4 }\)\(а\)
\(a r ^ { 2 }\)\(a r ^ { 2 }\)\(a r ^ { 3 }\)\(a r ^ { 4 }\)\(a\)arT
\(a r ^ { 3 }\)\(a r ^ { 3 }\)\(a r ^ { 4 }\)\(а\)ar\(a r ^ { 2 }\)
\(a r ^ { 4 }\)\(a r ^ { 4 }\)\(а\)ar\(a r ^ { 2 }\)\(a r ^ { 3 }\)
  1. Give a reason why \(D\) is not commutative.
  2. Write down the orders of any possible proper subgroups of \(D\).
  3. List the elements of a proper subgroup which contains
    (a) the element \(a\),
    (b) the element \(r\).
  4. Determine the order of each of the elements \(r ^ { 3 }\), \(a r\) and \(a r ^ { 2 }\).
  5. 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
1
  1. By writing \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(z z ^ { * } = | z | ^ { 2 }\).
  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.
\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(t\)\(s\)\(p\)\(r\)\(q\)
\(q\)\(s\)\(p\)\(q\)\(t\)\(r\)
\(r\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(s\)\(r\)\(t\)\(s\)\(q\)\(p\)
\(t\)\(q\)\(r\)\(t\)\(p\)\(s\)
  1. Verify that \(q ( s t ) = ( q s ) t\).
  2. 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 \(G\).
  3. 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 \(d\).