Questions FP3 (539 questions)

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OCR FP3 2009 June Q2
5 marks Standard +0.8
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.
  1. Write down the inverse of \(5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
  2. Prove the closure property for the group.
  3. \(Z\) 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
8 marks Standard +0.8
3 A line \(l\) has equation \(\frac { x - 6 } { - 4 } = \frac { y + 7 } { 8 } = \frac { z + 10 } { 7 }\) and a plane \(p\) has equation \(3 x - 4 y - 2 z = 8\).
  1. Find the point of intersection of \(l\) and \(p\).
  2. 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
8 marks Challenging +1.2
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.
  1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left( \frac { 1 + x } { 1 - x } \right) ^ { \frac { 1 } { 2 } }\).
  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
9 marks Challenging +1.2
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = \mathrm { e } ^ { 3 x }$$
  1. Find the complementary function.
  2. Explain briefly why there is no particular integral of either of the forms \(y = k \mathrm { e } ^ { 3 x }\) or \(y = k x \mathrm { e } ^ { 3 x }\).
  3. 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
9 marks Standard +0.8
6 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 5 \\ - 2 \end{array} \right)\).
  1. Express the equation of \(\Pi _ { 1 }\) in the form r.n \(= p\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . \left( \begin{array} { r } 7 \\ 17 \\ - 3 \end{array} \right) = 21\).
  2. 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
14 marks Challenging +1.3
7
  1. Use de Moivre's theorem to prove that $$\tan 3 \theta \equiv \frac { \tan \theta \left( 3 - \tan ^ { 2 } \theta \right) } { 1 - 3 \tan ^ { 2 } \theta } .$$
  2. (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 \(\tan \frac { 1 } { 12 } \pi = 2 - \sqrt { 3 }\).
  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
15 marks Challenging +1.8
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 \(a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }\).
  1. Prove that \(a = p a p\) and that \(p = a p a\).
  2. Find the order of each of the elements \(p ^ { 2 } , a , a p , a p ^ { 2 }\).
  3. Prove that \(\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}\) is a subgroup of \(Q\).
  4. Determine whether \(Q\) is a commutative group.
OCR FP3 2016 June Q1
6 marks Standard +0.3
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
4 marks Standard +0.8
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
10 marks Challenging +1.2
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
5 marks Standard +0.3
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
8 marks Standard +0.8
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
10 marks Standard +0.3
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
17 marks Challenging +1.8
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 MEI FP3 2011 June Q1
24 marks Challenging +1.2
1 The points \(\mathrm { A } ( 2 , - 1,3 ) , \mathrm { B } ( - 2 , - 7,7 )\) and \(\mathrm { C } ( 7,5,1 )\) are three vertices of a tetrahedron ABCD .
The plane ABD has equation \(x + 4 y + 7 z = 19\).
The plane ACD has equation \(2 x - y + 2 z = 11\).
  1. Find the shortest distance from \(B\) to the plane \(A C D\).
  2. Find an equation for the line AD .
  3. Find the shortest distance from C to the line AD .
  4. Find the shortest distance between the lines \(A D\) and \(B C\).
  5. Given that the tetrahedron ABCD has volume 20, find the coordinates of the two possible positions for the vertex \(D\).
OCR MEI FP3 2011 June Q2
24 marks Challenging +1.8
2 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x\).
  1. Sketch the section of \(S\) given by \(y = - 3\), and sketch the section of \(S\) given by \(x = - 6\). Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
  2. From your sketches in part (i), deduce that \(( - 6 , - 3 , - 324 )\) is a stationary point on \(S\), and state the nature of this stationary point.
  3. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\), and hence find the coordinates of the other three stationary points on \(S\).
  4. Show that there are exactly two values of \(k\) for which the plane with equation $$120 x - z = k$$ is a tangent plane to \(S\), and find these values of \(k\).
OCR MEI FP3 2011 June Q3
24 marks Challenging +1.8
3
    1. Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\). The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
    2. Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
    3. Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
    1. Show that the normal to the ellipse at the point with parameter \(\theta\) has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
    2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
    3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
      (A) at the point \(( 2,0 )\),
      (B) at the point \(( 0,1 )\).
OCR MEI FP3 2011 June Q4
24 marks Challenging +1.8
4
  1. Show that the set \(G = \{ 1,3,4,5,9 \}\), under the binary operation of multiplication modulo 11 , is a group. You may assume associativity.
  2. Explain why any two groups of order 5 must be isomorphic to each other. The set \(H = \left\{ 1 , \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 4 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 6 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 8 } { 5 } \pi \mathrm { j } } \right\}\) is a group under the binary operation of multiplication of complex numbers.
  3. Specify an isomorphism between the groups \(G\) and \(H\). The set \(K\) consists of the 25 ordered pairs \(( x , y )\), where \(x\) and \(y\) are elements of \(G\). The set \(K\) is a group under the binary operation defined by $$\left( x _ { 1 } , y _ { 1 } \right) \left( x _ { 2 } , y _ { 2 } \right) = \left( x _ { 1 } x _ { 2 } , y _ { 1 } y _ { 2 } \right)$$ where the multiplications are carried out modulo 11 ; for example, \(( 3,5 ) ( 4,4 ) = ( 1,9 )\).
  4. Write down the identity element of \(K\), and find the inverse of the element \(( 9,3 )\).
  5. Explain why \(( x , y ) ^ { 5 } = ( 1,1 )\) for every element \(( x , y )\) in \(K\).
  6. Deduce that all the elements of \(K\), except for one, have order 5. State which is the exceptional element.
  7. A subgroup of \(K\) has order 5 and contains the element (9, 3). List the elements of this subgroup.
  8. Determine how many subgroups of \(K\) there are with order 5 .
OCR MEI FP3 2011 June Q5
24 marks Challenging +1.2
5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows. On the first day, the bond is owned by Boyles or Cayleys, with probabilities \(0.4,0.6\) respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities \(0.07,0.8,0.13\) respectively. If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities \(0.15,0.75,0.1\) respectively. If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
  3. Find the probability that the bond is owned by Boyles on the 10th day.
  4. Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
  5. Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
  6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
  7. Find the probability that the bond is eventually owned by Alpha. The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
  8. Find the new probabilities for the ownership of the bond on the first day.
OCR MEI FP3 2007 June Q1
24 marks Challenging +1.2
1 Three planes \(P , Q\) and \(R\) have the following equations. $$\begin{array} { l l } \text { Plane } P : & 8 x - y - 14 z = 20 \\ \text { Plane } Q : & 6 x + 2 y - 5 z = 26 \\ \text { Plane } R : & 2 x + y - z = 40 \end{array}$$ The line of intersection of the planes \(P\) and \(Q\) is \(K\).
The line of intersection of the planes \(P\) and \(R\) is \(L\).
  1. Show that \(K\) and \(L\) are parallel lines, and find the shortest distance between them.
  2. Show that the shortest distance between the line \(K\) and the plane \(R\) is \(5 \sqrt { 6 }\). The line \(M\) has equation \(\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )\).
  3. Show that the lines \(K\) and \(M\) intersect, and find the coordinates of the point of intersection.
  4. Find the shortest distance between the lines \(L\) and \(M\).
OCR MEI FP3 2007 June Q2
24 marks Challenging +1.3
2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
  3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
  4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.
OCR MEI FP3 2007 June Q3
24 marks Challenging +1.8
3 The curve \(C\) has equation \(y = \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } \ln x\), and \(a\) is a constant with \(a \geqslant 1\).
  1. Show that the length of the arc of \(C\) for which \(1 \leqslant x \leqslant a\) is \(\frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 4 } \ln a - \frac { 1 } { 2 }\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(\boldsymbol { y }\)-axis.
  3. Show that the radius of curvature of \(C\) at the point where \(x = a\) is \(a \left( a + \frac { 1 } { 4 a } \right) ^ { 2 }\).
  4. Find the centre of curvature corresponding to the point \(\left( 1 , \frac { 1 } { 2 } \right)\) on \(C\). \(C\) is one member of the family of curves defined by \(y = p x ^ { 2 } - p ^ { 2 } \ln x\), where \(p\) is a parameter.
  5. Find the envelope of this family of curves.
OCR MEI FP3 2007 June Q4
24 marks Challenging +1.8
4
  1. Prove that, for a group of order 10, every proper subgroup must be cyclic. The set \(M = \{ 1,2,3,4,5,6,7,8,9,10 \}\) is a group under the binary operation of multiplication modulo 11.
  2. Show that \(M\) is cyclic.
  3. List all the proper subgroups of \(M\). The group \(P\) of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for \(P\) is given below.
    ABCDEFGHIJ
    ACJGHABIFED
    BFEHGBADCJI
    CGDIFCJEBAH
    DJCBEDGFIHA
    EABCDEFGHIJ
    FHIDCFEJABG
    GIHEBGDAJCF
    HDGJAHIBEFC
    IEFAJIHCDGB
    JBAFIJCHGDE
    One of these transformations is the identity transformation, some are rotations and the rest are reflections.
  4. Identify which transformation is the identity, which are rotations and which are reflections.
  5. State, giving a reason, whether \(P\) is isomorphic to \(M\).
  6. Find the order of each element of \(P\).
  7. List all the proper subgroups of \(P\).
OCR MEI FP3 2007 June Q5
24 marks Challenging +1.2
5 A computer is programmed to generate a sequence of letters. The process is represented by a Markov chain with four states, as follows. The first letter is \(A , B , C\) or \(D\), with probabilities \(0.4,0.3,0.2\) and 0.1 respectively.
After \(A\), the next letter is either \(C\) or \(D\), with probabilities 0.8 and 0.2 respectively.
After \(B\), the next letter is either \(C\) or \(D\), with probabilities 0.1 and 0.9 respectively.
After \(C\), the next letter is either \(A\) or \(B\), with probabilities 0.4 and 0.6 respectively.
After \(D\), the next letter is either \(A\) or \(B\), with probabilities 0.3 and 0.7 respectively.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Use your calculator to find \(\mathbf { P } ^ { 4 }\) and \(\mathbf { P } ^ { 7 }\). (Give elements correct to 4 decimal places.)
  3. Find the probability that the 8th letter is \(C\).
  4. Find the probability that the 12th letter is the same as the 8th letter.
  5. By investigating the behaviour of \(\mathbf { P } ^ { n }\) when \(n\) is large, find the probability that the ( \(n + 1\) )th letter is \(A\) when
    (A) \(n\) is a large even number,
    (B) \(n\) is a large odd number. The program is now changed. The initial probabilities and the transition probabilities are the same as before, except for the following. After \(D\), the next letter is \(A , B\) or \(D\), with probabilities \(0.3,0.6\) and 0.1 respectively.
  6. Write down the new transition matrix \(\mathbf { Q }\).
  7. Verify that \(\mathbf { Q } ^ { n }\) approaches a limit as \(n\) becomes large, and hence write down the equilibrium probabilities for \(A , B , C\) and \(D\).
  8. When \(n\) is large, find the probability that the \(( n + 1 )\) th, \(( n + 2 )\) th and \(( n + 3 )\) th letters are DDD.
OCR MEI FP3 2016 June Q1
24 marks Challenging +1.2
1 Positions in space around an aerodrome are modelled by a coordinate system with a point on the runway as the origin, O . The \(x\)-axis is east, the \(y\)-axis is north and the \(z\)-axis is vertically upwards. Units of distance are kilometres. Units of time are hours.
At time \(t = 0\), an aeroplane, P , is at \(( 3,4,8 )\) and is travelling in a direction \(\left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) at a constant speed of \(900 \mathrm { kmh } ^ { - 1 }\).
  1. Find the least distance of the path of P from the point O . At time \(t = 0\), a second aeroplane, Q , is at \(( 80,40,10 )\). It is travelling in a straight line towards the point O . Its speed is constant at \(270 \mathrm { kmh } ^ { - 1 }\).
  2. Show that the shortest distance between the paths of the two aeroplanes is 2.24 km correct to three significant figures.
  3. By finding the points on the paths where the shortest distance occurs and the times at which the aeroplanes are at these points, show that in fact the aeroplanes are never this close.
  4. A third aeroplane, R , is at position \(( 29,19,5.5 )\) at time \(t = 0\) and is travelling at \(285 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a direction \(\left( \begin{array} { c } 18 \\ 6 \\ 1 \end{array} \right)\). Given that Q is in the process of landing and cannot change course, show that R needs to be instructed to alter course or change speed.