Questions FP3 (539 questions)

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OCR FP3 2007 June Q5
8 marks Standard +0.8
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
10 marks Standard +0.8
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
10 marks Standard +0.3
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
12 marks Standard +0.3
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.
    1. The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
    2. The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
    3. The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\). 4
OCR FP3 Specimen Q1
5 marks Moderate -0.3
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
6 marks Standard +0.8
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
8 marks Standard +0.8
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
9 marks Standard +0.3
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
9 marks Standard +0.8
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
10 marks Challenging +1.2
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
10 marks Challenging +1.2
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
15 marks Challenging +1.2
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 MEI FP3 2006 June Q1
24 marks Challenging +1.2
1 Four points have coordinates \(\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )\) and \(\mathrm { D } ( 0,9 , k )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\).
  2. For the case when AB is parallel to CD ,
    (A) state the value of \(k\),
    (B) find the shortest distance between the parallel lines AB and CD ,
    (C) find, in the form \(a x + b y + c z + d = 0\), the equation of the plane containing AB and CD .
  3. When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of \(k\).
  4. Find the value of \(k\) for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.
OCR MEI FP3 2006 June Q2
24 marks Challenging +1.2
2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).
  1. Find a normal vector at the point \(( x , y , z )\) on the surface.
  2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
  3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
  4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
  5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.
OCR MEI FP3 2006 June Q3
24 marks Challenging +1.8
3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).
  1. Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the equation of the normal to \(C\) at the point with parameter \(t\) is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
  4. Find the cartesian equation of the envelope of the normals to \(C\).
  5. The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).
OCR MEI FP3 2006 June Q4
24 marks Challenging +1.2
\(\mathbf { 4 }\) The group \(G\) consists of the 8 complex matrices \(\{ \mathbf { I } , \mathbf { J } , \mathbf { K } , \mathbf { L } , - \mathbf { I } , - \mathbf { J } , - \mathbf { K } , - \mathbf { L } \}\) under matrix multiplication, where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } \mathrm { j } & 0 \\ 0 & - \mathrm { j } \end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 0 & \mathrm { j } \\ \mathrm { j } & 0 \end{array} \right)$$
  1. Copy and complete the following composition table for \(G\).
    IJKL-I-J-K\(- \mathbf { L }\)
    IIJKL-I-J-K-L
    JJ-IL-K-JI-LK
    KK-L-I
    LLK
    -I-I-J
    -J-JI
    -K-KL
    -L-L-K
    (Note that \(\mathbf { J K } = \mathbf { L }\) and \(\mathbf { K J } = - \mathbf { L }\).)
  2. State the inverse of each element of \(G\).
  3. Find the order of each element of \(G\).
  4. Explain why, if \(G\) has a subgroup of order 4, that subgroup must be cyclic.
  5. Find all the proper subgroups of \(G\).
  6. Show that \(G\) is not isomorphic to the group of symmetries of a square.
OCR MEI FP3 2006 June Q5
24 marks Challenging +1.2
5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division. This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 . The following transition probabilities apply to this team from 2007 onwards.
  • When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
  • When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
  • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .
This process is modelled as a Markov chain with three states corresponding to the three divisions.
  1. Write down the transition matrix.
  2. Determine in which division the team is most likely to be playing in 2014.
  3. Find the equilibrium probabilities for each division for this team. In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.
    -The transition probabilities from Divisions 1 and 2 remain the same as before.
    • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
    The team plays in Division 2 in 2015.
    The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.
  4. Write down the transition matrix which applies from 2015.
  5. Find the probability that the team is still playing in the league in 2020.
  6. Find the first year for which the probability that the team is out of the league is greater than 0.5 .
OCR MEI FP3 2008 June Q1
24 marks Challenging +1.8
1 A tetrahedron ABCD has vertices \(\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )\) and \(\mathrm { D } ( 5,4,8 )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\), and hence find the equation of the plane ABC .
  2. Find the shortest distance from \(D\) to the plane \(A B C\).
  3. Find the shortest distance between the lines AB and CD .
  4. Find the volume of the tetrahedron ABCD . The plane \(P\) with equation \(3 x - 2 z + 5 = 0\) contains the point B , and meets the lines AC and AD at E and F respectively.
  5. Find \(\lambda\) and \(\mu\) such that \(\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }\) and \(\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }\). Deduce that E is between A and C , and that F is between A and D.
  6. Hence, or otherwise, show that \(P\) divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.
OCR MEI FP3 2008 June Q2
24 marks Challenging +1.2
2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
  2. Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
  3. The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
  4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
  5. Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).
OCR MEI FP3 2008 June Q3
24 marks Challenging +1.2
3 The curve \(C\) has parametric equations \(x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }\), for \(t \geqslant 0\).
  1. Show that \(\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }\). Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 2\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the curvature at a general point on \(C\) is \(\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }\).
  4. Find the coordinates of the centre of curvature corresponding to the point on \(C\) where \(t = 1\).
OCR MEI FP3 2008 June Q4
24 marks Standard +0.8
4 A binary operation * is defined on real numbers \(x\) and \(y\) by $$x * y = 2 x y + x + y$$ You may assume that the operation \(*\) is commutative and associative.
  1. Explain briefly the meanings of the terms 'commutative' and 'associative'.
  2. Show that \(x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }\). The set \(S\) consists of all real numbers greater than \(- \frac { 1 } { 2 }\).
  3. (A) Use the result in part (ii) to show that \(S\) is closed under the operation *.
    (B) Show that \(S\), with the operation \(*\), is a group.
  4. Show that \(S\) contains no element of order 2 . The group \(G = \{ 0,1,2,4,5,6 \}\) has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
  5. Show that \(4 \circ 6 = 2\). The composition table for \(G\) is as follows.
    \(\circ\)012456
    0012456
    1140625
    2205164
    4461502
    5526041
    6654210
  6. Find the order of each element of \(G\).
  7. List all the subgroups of \(G\).
OCR MEI FP3 2008 June Q5
24 marks Challenging +1.2
5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes \(A , B\) and \(C\). The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route \(A\) was used on the previous day, route \(A\), \(B\) or \(C\) will be used, with probabilities \(0.1,0.4,0.5\) respectively.
If route \(B\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.7,0.2,0.1\) respectively.
If route \(C\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.1,0.6,0.3\) respectively. The situation is modelled as a Markov chain with three states.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Find the probability that route \(B\) is used on the 7th day.
  3. Find the probability that the same route is used on the 7th and 8th days.
  4. Find the probability that the route used on the 10th day is the same as that used on the 7th day.
  5. Given that \(\mathbf { P } ^ { n } \rightarrow \mathbf { Q }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { Q }\) (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix \(\mathbf { Q }\). The computer program is now to be changed, so that the long-run probabilities for routes \(A , B\) and \(C\) will become \(0.4,0.2\) and 0.4 respectively. The transition probabilities after routes \(A\) and \(B\) remain the same as before.
  6. Find the new transition probabilities after route \(C\).
  7. A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day.
OCR MEI FP3 2010 June Q1
24 marks Challenging +1.2
1 Four points have coordinates $$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$ where \(p\) is a constant.
  1. Find the perpendicular distance from C to the line AB .
  2. Find \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\) in terms of \(p\), and show that the shortest distance between the lines AB and CD is $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
  3. Find, in terms of \(p\), the volume of the tetrahedron ABCD .
  4. State the value of \(p\) for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case. Option 2: Multi-variable calculus
OCR MEI FP3 2010 June Q2
24 marks Challenging +1.2
2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 2 \\ 1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
  2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
  3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
  4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
  5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
  6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).
OCR MEI FP3 2010 June Q3
24 marks Challenging +1.8
3 A curve \(C\) has equation \(y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } }\), for \(x \geqslant 0\).
  1. Show that the arc of \(C\) for which \(0 \leqslant x \leqslant a\) has length \(a ^ { \frac { 1 } { 2 } } + \frac { 1 } { 3 } a ^ { \frac { 3 } { 2 } }\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Find the coordinates of the centre of curvature corresponding to the point \(\left( 4 , - \frac { 2 } { 3 } \right)\) on \(C\). The curve \(C\) is one member of the family of curves defined by $$y = p ^ { 2 } x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } p ^ { 3 } x ^ { \frac { 3 } { 2 } } \quad ( \text { for } x \geqslant 0 )$$ where \(p\) is a parameter (and \(p > 0\) ).
  4. Find the equation of the envelope of this family of curves.