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AQA FP3 2006 January Q5
17 marks Standard +0.3
5
  1. The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x \ln x + \frac { y } { x }$$ and $$y ( 1 ) = 1$$
    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 \(y ( 1.1 )\).
    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)(i) to obtain an approximation to \(y ( 1.2 )\), giving your answer to three decimal places.
    1. Show that \(\frac { 1 } { x }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 1 } { x } y = x \ln x$$
    2. Solve this differential equation, given that \(y = 1\) when \(x = 1\).
    3. Calculate the value of \(y\) when \(x = 1.2\), giving your answer to three decimal places.
AQA FP3 2006 January Q6
16 marks Challenging +1.2
6
  1. A circle \(C _ { 1 }\) has cartesian equation \(x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36\). Show that the polar equation of \(C _ { 1 }\) is \(r = 12 \sin \theta\).
  2. A curve \(C _ { 2 }\) with polar equation \(r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651} Calculate the area bounded by \(C _ { 2 }\).
  3. The circle \(C _ { 1 }\) intersects the curve \(C _ { 2 }\) at the points \(P\) and \(Q\). Find, in surd form, the area of the quadrilateral \(O P M Q\), where \(M\) is the centre of the circle and \(O\) is the pole.
    (6 marks)
AQA FP3 2007 January Q1
9 marks Standard +0.3
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln \left( 1 + x ^ { 2 } + y \right)$$ and $$y ( 1 ) = 0.6$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
  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 \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
AQA FP3 2007 January Q2
6 marks Standard +0.8
2 A curve has polar equation \(r ( 1 - \sin \theta ) = 4\). Find its cartesian equation in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2007 January Q3
9 marks Standard +0.3
3
  1. Show that \(x ^ { 2 }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
  2. Solve this differential equation, given that \(y = 1\) when \(x = 2\).
AQA FP3 2007 January Q4
8 marks Challenging +1.2
4
  1. Explain why \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) is an improper integral.
    (1 mark)
  2. Use integration by parts to find \(\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\).
    (3 marks)
  3. Show that \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) exists and find its value.
    (4 marks)
AQA FP3 2007 January Q5
12 marks Challenging +1.2
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 6 + 5 \sin x$$ (12 marks)
AQA FP3 2007 January Q6
16 marks Standard +0.8
6 The function f is defined by \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\).
    1. Find f'''(x).
    2. Using Maclaurin's theorem, show that, for small values of \(x\), $$\mathrm { f } ( x ) \approx 1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
  2. Use the expansion of \(\mathrm { e } ^ { x }\) together with the result in part (a)(ii) to show that, for small values of \(x\), $$\mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \approx 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where \(k\) is a rational number to be found.
  3. Write down the first four terms in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { 2 x }\).
  4. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } - \mathrm { e } ^ { 2 x } } { 1 - \cos x }$$ (4 marks)
AQA FP3 2007 January Q7
15 marks Challenging +1.8
7 A curve \(C\) has polar equation $$r = 6 + 4 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ The diagram shows a sketch of the curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0d894ac0-8d96-4182-8454-c306e1fdad8f-4_599_866_612_587}
  1. Calculate the area of the region bounded by the curve \(C\).
  2. The point \(P\) is the point on the curve \(C\) for which \(\theta = \frac { 2 \pi } { 3 }\). The point \(Q\) is the point on \(C\) for which \(\theta = \pi\).
    Show that \(Q P\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).
  3. The line \(P Q\) intersects the curve \(C\) again at a point \(R\). The line \(R O\) intersects \(C\) again at a point \(S\).
    1. Find, in surd form, the length of \(P S\).
    2. Show that the angle \(O P S\) is a right angle.
AQA FP3 2007 June Q1
10 marks Standard +0.3
1
  1. Find the value of the constant \(k\) for which \(k x ^ { 2 } \mathrm { e } ^ { 5 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 10 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 6 \mathrm { e } ^ { 5 x }$$
  2. Hence find the general solution of this differential equation.
AQA FP3 2007 June Q2
9 marks Standard +0.3
2 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } + 3 }$$ and $$y ( 1 ) = 2$$
  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 \(y ( 1.1 )\), giving your answer to four decimal places.
  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 \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\), giving your answer to four decimal places.
AQA FP3 2007 June Q3
8 marks Standard +0.3
3 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$ given that \(y = 3\) when \(x = 0\).
AQA FP3 2007 June Q4
14 marks Challenging +1.2
4
  1. Show that \(( \cos \theta + \sin \theta ) ^ { 2 } = 1 + \sin 2 \theta\).
  2. A curve has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( x + y ) ^ { 4 }$$ Given that \(r \geqslant 0\), show that the polar equation of the curve is $$r = 1 + \sin 2 \theta$$
  3. The curve with polar equation $$r = 1 + \sin 2 \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f90167c3-2ffd-464a-b2d2-9f86a8d64887-3_389_611_1062_708}
    1. Find the two values of \(\theta\) for which \(r = 0\).
    2. Find the area of one of the loops.
AQA FP3 2007 June Q5
12 marks Challenging +1.2
5
  1. A differential equation is given by $$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$ Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + x$$ transforms this differential equation into $$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$ (4 marks)
  2. Find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$ giving your answer in the form \(u = \mathrm { f } ( x )\).
  3. Hence find the general solution of the differential equation $$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$ giving your answer in the form \(y = \mathrm { g } ( x )\).
AQA FP3 2007 June Q6
15 marks Standard +0.8
6
  1. The function f is defined by $$\mathrm { f } ( x ) = \ln \left( 1 + \mathrm { e } ^ { x } \right)$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
    1. the first three terms are $$\ln 2 + \frac { 1 } { 2 } x + \frac { 1 } { 8 } x ^ { 2 }$$
    2. the coefficient of \(x ^ { 3 }\) is zero.
  2. Hence write down the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right)\).
  3. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \ldots$$ to write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln \left( 1 - \frac { x } { 2 } \right)\).
  4. Use your answers to parts (b) and (c) to find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right) + \ln \left( 1 - \frac { x } { 2 } \right) } { x - \sin x } \right]$$
AQA FP3 2007 June Q7
7 marks Challenging +1.8
7
  1. Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
  2. Use the substitution \(u = x \mathrm { e } ^ { - x } + 1\) to find \(\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing the limiting process used.
AQA D1 Q3
Easy -1.8
3
    1. State the number of edges in a minimum spanning tree of a network with 10 vertices.
    2. State the number of edges in a minimum spanning tree of a network with \(n\) vertices.
  2. The following network has 10 vertices: \(A , B , \ldots , J\). The numbers on each edge represent the distances, in miles, between pairs of vertices. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-004_1294_1118_785_445}
    1. Use Kruskal's algorithm to find the minimum spanning tree for the network.
    2. State the length of your spanning tree.
    3. Draw your spanning tree.
AQA D1 Q4
Moderate -0.3
4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-005_1349_1395_408_294}
  1. On the feasible region, find:
    1. the maximum value of \(2 x + 3 y\);
    2. the maximum value of \(3 x + 2 y\);
    3. the minimum value of \(- 2 x + y\).
  2. Find the 5 inequalities that define the feasible region.
AQA D1 Q5
Moderate -0.8
5 [Figure 1, printed on the insert, is provided for use in this question.]
The network shows the times, in minutes, to travel between 10 towns. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-006_412_1561_568_233}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
    (6 marks)
  2. State the corresponding route.
    (1 mark)
AQA D1 Q7
Moderate -0.8
7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-007_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).
  1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
  2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
  3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.
AQA D1 2006 January Q1
7 marks Moderate -0.8
1
  1. Draw a bipartite graph representing the following adjacency matrix.
    (2 marks)
    \(\boldsymbol { U }\)\(V\)\(\boldsymbol { W }\)\(\boldsymbol { X }\)\(\boldsymbol { Y }\)\(\boldsymbol { Z }\)
    \(\boldsymbol { A }\)101010
    \(\boldsymbol { B }\)010100
    \(\boldsymbol { C }\)010001
    \(\boldsymbol { D }\)000100
    \(\boldsymbol { E }\)001011
    \(\boldsymbol { F }\)000110
  2. Given that initially \(A\) is matched to \(W , B\) is matched to \(X , C\) is matched to \(V\), and \(E\) is matched to \(Y\), use the alternating path algorithm, from this initial matching, to find a complete matching. List your complete matching.
AQA D1 2006 January Q2
5 marks Easy -1.2
2 Use the quicksort algorithm to rearrange the following numbers into ascending order. Indicate clearly the pivots that you use. $$\begin{array} { l l l l l l l l } 18 & 23 & 12 & 7 & 26 & 19 & 16 & 24 \end{array}$$
AQA D1 2006 January Q3
15 marks Easy -2.0
3
    1. State the number of edges in a minimum spanning tree of a network with 10 vertices.
    2. State the number of edges in a minimum spanning tree of a network with \(n\) vertices.
  2. The following network has 10 vertices: \(A , B , \ldots , J\). The numbers on each edge represent the distances, in miles, between pairs of vertices. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-03_1294_1118_785_445}
    1. Use Kruskal's algorithm to find the minimum spanning tree for the network.
    2. State the length of your spanning tree.
    3. Draw your spanning tree.
AQA D1 2006 January Q4
8 marks Moderate -0.8
4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-04_1349_1395_408_294}
  1. On the feasible region, find:
    1. the maximum value of \(2 x + 3 y\);
    2. the maximum value of \(3 x + 2 y\);
    3. the minimum value of \(- 2 x + y\).
  2. Find the 5 inequalities that define the feasible region.
AQA D1 2006 January Q5
7 marks Moderate -0.8
5 [Figure 1, printed on the insert, is provided for use in this question.]
The network shows the times, in minutes, to travel between 10 towns. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-05_412_1561_568_233}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
    (6 marks)
  2. State the corresponding route.
    (1 mark)