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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.
OCR MEI FP3 2015 June Q1
24 marks Standard +0.8
1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).
  1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
  2. Find the equation of the plane ABC in cartesian form.
  3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
  4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
  5. Find the volume of the tetrahedron ABCD .
OCR MEI FP3 2015 June Q2
24 marks Challenging +1.2
2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).
  1. Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
  2. A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
    (A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
    (B) Deduce that the stationary point A is a local minimum.
    (C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B .
  3. The point Q with coordinates \(( 1,1,53 )\) lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
  4. The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\). Find the coordinates of R .
OCR MEI FP3 2015 June Q3
24 marks Challenging +1.8
3 Fig. 3 shows an ellipse with parametric equations \(x = a \cos \theta , y = b \sin \theta\), for \(0 \leqslant \theta \leqslant 2 \pi\), where \(0 < b \leqslant a\).
The curve meets the positive \(x\)-axis at A and the positive \(y\)-axis at B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e032f23-0549-4adc-bfae-59333108fab5-4_668_1255_477_404} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the radius of curvature at A is \(\frac { b ^ { 2 } } { a }\) and find the corresponding centre of curvature.
  2. Write down the radius of curvature and the centre of curvature at B .
  3. Find the relationship between \(a\) and \(b\) if the radius of curvature at B is equal to the radius of curvature at A . What does this mean geometrically?
  4. Show that the arc length from A to B can be expressed as $$b \int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 1 + \lambda ^ { 2 } \sin ^ { 2 } \theta } d \theta$$ where \(\lambda ^ { 2 }\) is to be determined in terms of \(a\) and \(b\).
    Evaluate this integral in the case \(a = b\) and comment on your answer.
  5. Find the cartesian equation of the evolute of the ellipse.
OCR MEI FP3 2015 June Q4
24 marks Challenging +1.8
4 M is the set of all \(2 \times 2\) matrices \(\mathrm { m } ( a , b )\) where \(a\) and \(b\) are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b \\ 0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$
  1. Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
  2. Determine whether the group is commutative. The set \(\mathrm { N } _ { k }\) consists of all \(2 \times 2\) matrices \(\mathrm { m } ( k , b )\) where \(k\) is a fixed positive integer and \(b\) can take any integer value.
  3. Prove that \(\mathrm { N } _ { k }\) is closed under matrix multiplication if and only if \(k = 1\). Now consider the set P consisting of the matrices \(\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )\) and \(\mathrm { m } ( 1,3 )\). The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
  4. Show that \(( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )\).
  5. Construct the group combination table for P . The group R consists of the set \(\{ e , a , b , c \}\) combined under the operation *. The identity element is \(e\), and elements \(a , b\) and \(c\) are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
  6. Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}
OCR MEI FP3 2015 June Q5
24 marks Standard +0.8
5 An inspector has three factories, A, B, C, to check. He spends each day in one of the factories. He chooses the factory to visit on a particular day according to the following rules.
  • If he is in A one day, then the next day he will never choose A but he is equally likely to choose B or C .
  • If he is in B one day, then the next day he is equally likely to choose \(\mathrm { A } , \mathrm { B }\) or C .
  • If he is in C one day, then the next day he will never choose A but he is equally likely to choose B or C .
    1. Write down the transition matrix, \(\mathbf { P }\).
    2. On Day 1 the inspector chooses A.
      (A) Find the probability that he will choose A on Day 4.
      (B) Find the probability that the factory he chooses on Day 7 is the same factory that he chose on Day 2.
    3. Find the equilibrium probabilities and explain what they mean.
The inspector is not satisfied with the number of times he visits A so he changes the rules as follows.
Still not satisfied, the inspector changes the rules as follows.
The new transition matrix is \(\mathbf { R }\).
  • On Day 15 he visits C . Find the first subsequent day for which the probability that he visits B is less than 0.1.
  • Show that in this situation there is an absorbing state, explaining what this means. \section*{END OF QUESTION PAPER}
    •
    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)