Questions — OCR FP3 (140 questions)

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OCR FP3 2016 June Q1
1 In this question, give all non-real numbers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 < \theta < 2 \pi\).
  1. Solve \(z ^ { 5 } = 1\).
  2. Hence, or otherwise, solve \(z ^ { 5 } + 32 = 0\). Sketch an Argand diagram showing the roots.
OCR FP3 2016 June Q2
2 Find the shortest distance between the lines \(\mathbf { r } = \left( \begin{array} { l } 2
1
0 \end{array} \right) + \lambda \left( \begin{array} { c } 1
2
- 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { c } - 1
1
2 \end{array} \right) + \mu \left( \begin{array} { l } 3
0
1 \end{array} \right)\).
OCR FP3 2016 June Q3
3 The differential equation $$\frac { 2 } { y } - \frac { x } { y ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } }$$ is to be solved subject to the condition \(y = 1\) when \(x = 1\).
  1. Show that \(y = \frac { 1 } { u }\) transforms the differential equation into $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } + 2 u = \frac { 1 } { x ^ { 2 } } .$$
  2. Find \(y\) in terms of \(x\).
OCR FP3 2016 June Q4
4 Let \(A\) be the set of non-zero integers.
  1. Show that \(A\) does not form a group under multiplication.
  2. State the largest subset of \(A\) which forms a group under multiplication. Show that this is a group.
OCR FP3 2016 June Q5
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 85 \cos x .$$
OCR FP3 2016 June Q6
6 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations $$\mathbf { r } \cdot \left( \begin{array} { l } 1
2
1 \end{array} \right) = 3 \text { and } \mathbf { r } \cdot \left( \begin{array} { l } 2
1
4 \end{array} \right) = 5$$ respectively. They intersect in the line \(l\).
  1. Find cartesian equations of \(l\). The plane \(\Pi _ { 3 }\) has equation \(\mathbf { r } . \left( \begin{array} { c } 1
    5
    - 1 \end{array} \right) = 1\).
  2. Show that \(\Pi _ { 3 }\) is parallel to \(l\) but does not contain it.
  3. Verify that \(( 2,0,1 )\) lies on planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\). Hence write down a vector equation of the line of intersection of these planes.
OCR FP3 2016 June Q8
8 A non-commutative multiplicative group \(G\) of order eight has the elements $$\left\{ e , a , a ^ { 2 } , a ^ { 3 } , b , a b , a ^ { 2 } b , a ^ { 3 } b \right\}$$ where \(e\) is the identity and \(a ^ { 4 } = b ^ { 2 } = e\).
  1. Show that \(b a \neq a ^ { n }\) for any integer \(n\).
  2. Prove, by contradiction, that \(b a \neq a ^ { 2 } b\) and also that \(b a \neq a b\). Deduce that \(b a = a ^ { 3 } b\).
  3. Prove that \(b a ^ { 2 } = a ^ { 2 } b\).
  4. Construct group tables for the three subgroups of \(G\) of order four. \section*{END OF QUESTION PAPER}
OCR FP3 2011 January Q5
  1. Find the general solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = - 2 x + 13 .$$
  2. Find the particular solution for which \(y = - \frac { 7 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
  3. Write down the function to which \(y\) approximates when \(x\) is large and positive.
    \(6 Q\) is a multiplicative group of order 12.
  4. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a ^ { 2 } = r ^ { 3 }\). Find the orders of the elements \(a , a ^ { 2 } , a ^ { 3 }\) and \(r ^ { 2 }\). The table below shows the number of elements of \(Q\) with each possible order.
    Order of element12346
    Number of elements11262
    \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.
  5. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\).
OCR FP3 2007 June Q7
  1. Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
  2. Write down the seven roots of the equation \(z ^ { 7 } = 1\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) and show their positions in an Argand diagram.
  3. Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors.
OCR FP3 2013 June Q2
  1. Write down the operation table and, assuming associativity, show that \(G\) is a group.
  2. State the order of each element.
  3. Find all the proper subgroups of \(G\). The group \(H\) consists of the set \(\{ 1,3,7,9 \}\) with the operation of multiplication modulo 10 .
  4. Explaining your reasoning, determine whether \(H\) is isomorphic to \(G\).
OCR FP3 2016 June Q7
  1. Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
  2. Hence show that, for \(\sin 2 \theta \neq 0\), $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$
OCR FP3 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 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 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 \(l _ { 2 }\).
  5. Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
  6. Write down the seven roots of the equation \(z ^ { 7 } = 1\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) and show their positions in an Argand diagram.
  7. Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors. 8
  8. 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.
  9. Find the particular solution for which \(y = 2\) when \(x = \pi\). 9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
  10. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
  11. 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 \(n \in \mathbb { Z }\). 1 (a) A group \(G\) of order 6 has the combination table shown below.
    \(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(e\)\(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(a\)\(a\)\(b\)\(e\)\(r\)\(p\)\(q\)
    \(b\)\(b\)\(e\)\(a\)\(q\)\(r\)\(p\)
    \(p\)\(p\)\(q\)\(r\)\(e\)\(a\)\(b\)
    \(q\)\(q\)\(r\)\(p\)\(b\)\(e\)\(a\)
    \(r\)\(r\)\(p\)\(q\)\(a\)\(b\)\(e\)
  12. State, with a reason, whether or not \(G\) is commutative.
  13. State the number of subgroups of \(G\) which are of order 2 .
  14. List the elements of the subgroup of \(G\) which is of order 3 .
    (b) 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 \(c ^ { 5 }\). 2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x$$ 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 \(\mathbf { r }\).
  15. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } = \lambda \mathbf { a }\), where \(0 \leqslant \lambda \leqslant 1\).
  16. Given that \(P\) is a point on the line \(A B\), use a property of the vector product to explain why \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }\).
  17. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }\). 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 \(\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)\) 5
  18. 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 \(x = \frac { 1 } { 4 } \pi\).
  19. Find the solution of the differential equation in this case.
  20. Write down a function to which \(y\) approximates when \(x\) is large and positive. 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 \(x + 2 y - 2 z = - 1\).
  21. Find, in either order, the coordinates of \(B\) and the length of \(A B\).
  22. Find the acute angle between the planes \(A C D\) and \(B C D\).
  23. (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 \(0 < \theta < \frac { 1 } { 2 } \pi\).
  24. Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right)$$
  25. 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. \section*{Jan 2008} 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.
  26. Write down the identity element for each of groups \(A , B , C\) and \(D\).
  27. 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.
  28. Prove the closure property for group \(D\).
  29. 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. }1 (a) A cyclic multiplicative group \(G\) has order 12. The identity element of \(G\) is \(e\) and another element is \(r\), with order 12.
  30. Write down, in terms of \(e\) and \(r\), the elements of the subgroup of \(G\) which is of order 4.
  31. Explain briefly why there is no proper subgroup of \(G\) in which two of the elements are \(e\) and \(r\).
    (b) A group \(H\) has order \(m n p\), where \(m , n\) and \(p\) are prime. State the possible orders of proper subgroups of \(H\). 2 Find the acute angle between the line with equation \(\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )\) and the plane with equation \(\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\). 3
  32. Use the substitution \(z = x + y\) to show that the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y + 3 } { x + y - 1 }$$ may be written in the form \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \frac { 2 ( z + 1 ) } { z - 1 }\).
  33. Hence find the general solution of the differential equation (A). 4
  34. By expressing \(\cos \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\cos ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
  35. Hence solve the equation \(\cos 5 \theta + 5 \cos 3 \theta + 9 \cos \theta = 0\) for \(0 \leqslant \theta \leqslant \pi\). 5 Two lines have equations $$\frac { x - k } { 2 } = \frac { y + 1 } { - 5 } = \frac { z - 1 } { - 3 } \quad \text { and } \quad \frac { x - k } { 1 } = \frac { y + 4 } { - 4 } = \frac { z } { - 2 }$$ where \(k\) is a constant.
  36. Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\).
  37. For the case \(k = 1\), find the equation of the plane in which the lines lie, giving your answer in the form \(a x + b y + c z = d\). 6 The operation ○ on real numbers is defined by \(a \circ b = a | b |\).
  38. Show that ∘ is not commutative.
  39. Prove that ∘ is associative.
  40. Determine whether the set of real numbers, under the operation ∘, forms a group. \section*{June 2008}
OCR FP3 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 \(n \in \mathbb { Z }\). 1 (a) A group \(G\) of order 6 has the combination table shown below. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.
  3. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). 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 \(m\).
  4. Show that \(l\) and \(m\) are in the same direction.
  5. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
  6. 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. 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 \(( 1,0 )\).
  7. Prove that \(*\) is associative.
  8. Find all the elements which commute with \(( 1,1 )\).
  9. 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 \(n\).
  10. Find all self-inverse elements.
  11. Give a reason why the elements \(( x , y )\), under the operation \(*\), do not form a group.