Questions FP3 (473 questions)

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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 MEI FP3 2011 June Q1
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
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
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
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
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
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
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
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
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
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
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.
OCR MEI FP3 2016 June Q2
2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).
  1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
  2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
  3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
  4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
  5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).
OCR MEI FP3 2016 June Q3
3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{07eaad51-dc00-44d2-8bff-8652d62902ec-4_634_1294_388_386} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the values of \(t\) where the curve cuts the \(y\)-axis are \(t = 0 , \pm \frac { 1 } { \sqrt { 3 } }\). Write down the corresponding values of \(y\).
  2. Find the radius and centre of curvature when \(t = \frac { 1 } { \sqrt { 3 } }\). The arc of the curve given by \(0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }\) is denoted by \(C\).
  3. Find the length of \(C\).
  4. Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).
OCR MEI FP3 2016 June Q4
4
  1. The elements of the set \(P = \{ 1,3,9,11 \}\) are combined under the binary operation, *, defined as multiplication modulo 16.
    1. Demonstrate associativity for the elements \(3,9,11\) in that order. Assuming associativity holds in general, show that \(P\) forms a group under the binary operation *.
    2. Write down the order of each element.
    3. Write down all subgroups of \(P\).
    4. Show that the group in part (i) is cyclic.
  2. Now consider a group of order 4 containing the identity element \(e\) and the two distinct elements, \(a\) and \(b\), where \(a ^ { 2 } = b ^ { 2 } = e\). Construct the composition table. Show that the group is non-cyclic.
  3. Now consider the four matrices \(\mathbf { I } , \mathbf { X } , \mathbf { Y }\) and \(\mathbf { Z }\) where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
    0 & 1 \end{array} \right) , \mathbf { X } = \left( \begin{array} { r r } 1 & 0
    0 & - 1 \end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r } - 1 & 0
    0 & 1 \end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r } - 1 & 0
    0 & - 1 \end{array} \right) .$$ The group G consists of the set \(\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}\) with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
  4. The distinct elements \(\{ p , q , r , s \}\) are combined under the binary operation \({ } ^ { \circ }\). You are given that \(p ^ { \circ } q = r\) and \(q ^ { \circ } p = s\). By reference to the group axioms, prove that \(\{ p , q , r , s \}\) is not a group under \({ } ^ { \circ }\). Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}
OCR MEI FP3 2016 June Q5
5 Each day that Adam is at work he carries out one of three tasks A, B or C. Each task takes a whole day. Adam chooses the task to carry out on each day according to the following set of three rules.
  1. If, on any given day, he has worked on task A then the next day he will choose task A with probability 0.75 , and tasks B and C with equal probability.
  2. If, on any given day, he has worked on task B then the next day he will choose task B or task C with equal probability but will never choose task A .
  3. If, on any given day, he has worked on task C then the next day he will choose task A with probability \(p\) and tasks B and C with equal probability.
    1. Write down the transition matrix.
    2. Over a long period Adam carries out the tasks \(\mathrm { A } , \mathrm { B }\) and C with equal frequency. Find the value of \(p\).
    3. On day 1 Adam chooses task A . Find the probability that he also chooses task A on day 5 .
    Adam decides to change rule 3 as follows.
    If, on any given day, he has worked on task C then the next day he will choose tasks \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) with probabilities \(0.4,0.3,0.3\) respectively.
  4. On day 1 Adam chooses task A. Find the probability that he chooses the same task on day 7 as he did on day 4 .
  5. On a particular day, Adam chooses task A. Find the expected number of consecutive further days on which he will choose A. Adam changes all three rules again as follows.
    • If he works on A one day then on the next day he chooses C .
    • If he works on B one day then on the next day he chooses A or C each with probability 0.5.
    • If he works on C one day then on the next day he chooses A or B each with probability 0.5 .
    • Find the long term probabilities for each task.
AQA FP3 2008 January Q1
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x ^ { 2 } - y ^ { 2 }$$ and $$y ( 2 ) = 1$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.1\), to obtain an approximation to \(y ( 2.1 )\).
  2. Use the formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with your answer to part (a), to obtain an approximation to \(y ( 2.2 )\).
AQA FP3 2008 January Q2
2 The diagram shows a sketch of part of the curve \(C\) whose polar equation is \(r = 1 + \tan \theta\). The point \(O\) is the pole.
\includegraphics[max width=\textwidth, alt={}, center]{0c177d90-02ae-4e91-bc9d-d0c7051799b8-3_561_629_406_772} The points \(P\) and \(Q\) on the curve are given by \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) respectively.
  1. Show that the area of the region bounded by the curve \(C\) and the lines \(O P\) and \(O Q\) is $$\frac { 1 } { 2 } \sqrt { 3 } + \ln 2$$ (6 marks)
  2. Hence find the area of the shaded region bounded by the line \(P Q\) and the arc \(P Q\) of \(C\).
AQA FP3 2008 January Q3
3
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 5$$
  2. Hence express \(y\) in terms of \(x\), given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) when \(x = 0\).
AQA FP3 2008 January Q4
4
  1. Explain why \(\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\) is an improper integral.
  2. Find \(\int x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2008 January Q5
5 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 x } { x ^ { 2 } + 1 } y = x$$ given that \(y = 1\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2008 January Q6
6 A curve \(C\) has polar equation $$r ^ { 2 } \sin 2 \theta = 8$$
  1. Find the cartesian equation of \(C\) in the form \(y = \mathrm { f } ( x )\).
  2. Sketch the curve \(C\).
  3. The line with polar equation \(r = 2 \sec \theta\) intersects \(C\) at the point \(A\). Find the polar coordinates of \(A\).
AQA FP3 2008 January Q7
7
    1. Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
    2. State the range of values of \(x\) for which this expansion is valid.
    1. Given that \(y = \ln \cos x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
    2. Find the value of \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) when \(x = 0\).
    3. Hence, by using Maclaurin's theorem, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \cos x\) are $$- \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 12 }$$
  3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln ( 1 + 2 x ) } { x ^ { 2 } - \ln \cos x } \right]$$
AQA FP3 2008 January Q8
8
  1. Given that \(x = \mathrm { e } ^ { t }\) and that \(y\) is a function of \(x\), show that:
    1. \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t }\);
    2. \(\quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t }\).
  2. Hence find the general solution of the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = 0$$
AQA FP3 2009 January Q1
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x ^ { 2 } + y ^ { 2 } } { x + y }$$ and $$y ( 1 ) = 3$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.2\), to obtain an approximation to \(y ( 1.2 )\).
  2. Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.2\), to obtain an approximation to \(y ( 1.2 )\), giving your answer to four decimal places.