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AQA FP3 2015 June Q1
5 marks Standard +0.3
1 It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x + y ^ { 2 } } { x }$$ and $$y ( 2 ) = 5$$
  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 \(y ( 2.05 )\).
  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 \(y ( 2.1 )\), giving your answer to three significant figures.
    [0pt] [3 marks]
AQA FP3 2015 June Q2
9 marks Standard +0.8
2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \tan ^ { 3 } x \sec x$$ given that \(y = 2\) when \(x = \frac { \pi } { 3 }\).
[0pt] [9 marks]
AQA FP3 2015 June Q3
8 marks Challenging +1.2
3
    1. Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\).
    2. Hence, or otherwise, find the first two non-zero terms in the expansion of $$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$ in ascending powers of \(x\) and state the range of values of \(x\) for which the expansion is valid.
  2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]\).
AQA FP3 2015 June Q4
7 marks Standard +0.8
4
  1. Explain why \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) is an improper integral.
  2. Evaluate \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2015 June Q5
11 marks Challenging +1.2
5
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$
  2. It is given that \(y = \mathrm { f } ( x )\) is the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$ such that \(\mathrm { f } ( 0 ) = 0\) and \(\mathrm { f } ^ { \prime } ( 0 ) = 0\).
    1. Show that \(f ^ { \prime \prime } ( 0 ) = 0\).
    2. Find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\).
      [0pt] [3 marks]
AQA FP3 2015 June Q6
17 marks Challenging +1.8
6 A differential equation is given by $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
  1. Show that the substitution \(x = \mathrm { e } ^ { 2 t }\) transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 4 t ^ { 2 } + 5 \mathrm { e } ^ { - t }$$
  2. Hence find the general solution of the differential equation $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
    \includegraphics[max width=\textwidth, alt={}]{7b4a1237-bb28-4cba-84b1-35fa91d87408-14_1634_1709_1071_153}
AQA FP3 2015 June Q7
18 marks Challenging +1.3
7 The diagram shows the sketch of a curve \(C _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve \(C _ { 1 }\) is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$
  1. Find the area of the region bounded by the curve \(C _ { 1 }\).
  2. The curve \(C _ { 2 }\) whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve \(C _ { 1 }\) at the pole \(O\) and at the point \(A\). The straight line drawn through \(A\) parallel to the initial line intersects \(C _ { 1 }\) again at the point \(B\).
    1. Find the polar coordinates of \(A\).
    2. Show that the length of \(O B\) is \(\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )\).
    3. Find the length of \(A B\), giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141} \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141} \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}
AQA FP3 2016 June Q1
6 marks Moderate -0.3
1
  1. Find the values of the constants \(a\) and \(b\) for which \(a x + b\) is a particular integral of the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$$
  2. Hence find the general solution of \(2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x\).
    [0pt] [3 marks]
AQA FP3 2016 June Q2
5 marks Standard +0.8
2
  1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
  2. It is given that the first non-zero term in the expansion of $$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$ in ascending powers of \(x\) is \(q x ^ { 5 }\).
    Find the values of the rational numbers \(p\) and \(q\).
AQA FP3 2016 June Q3
12 marks Standard +0.8
3
  1. It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = ( 2 x + 1 ) \ln ( x + y )$$ and $$y ( 0 ) = 2$$ 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 \(y ( 0.1 )\), giving your answer to three decimal places.
  2. It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) \ln ( x + y )$$ and \(y = 2\) when \(x = 0\).
    1. Use implicit differentiation to find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving your answer in terms of \(x\) and \(y\).
    2. Hence find the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(y ( x )\). Give your answer in an exact form.
    3. Use your answer to part (b)(ii) to obtain an approximation to \(y ( 0.1 )\), giving your answer to three decimal places.
      [0pt] [1 mark]
AQA FP3 2016 June Q4
6 marks Challenging +1.2
4
  1. The curve with Cartesian equation \(\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1\) is mapped onto the curve with polar equation \(r = \frac { 10 } { 3 - 2 \cos \theta }\) by a single geometrical transformation. By writing the polar equation as a Cartesian equation in a suitable form, find the values of the constants \(c\) and \(d\).
  2. Hence describe the geometrical transformation referred to in part (a).
    [0pt] [1 mark]
AQA FP3 2016 June Q5
12 marks Challenging +1.2
5
  1. Express \(\frac { 1 } { ( 1 + x ) ( 2 + x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 2 + x }\), where \(A\) and \(B\) are integers.
  2. Use the substitution \(u = \frac { \mathrm { d } y } { \mathrm {~d} x }\) to solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { 1 } { ( 1 + x ) ( 2 + x ) } \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + x } { 1 + x }$$ given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
    [0pt] [11 marks]
AQA FP3 2016 June Q6
7 marks Standard +0.8
6
  1. Use the substitution \(a = \frac { 1 } { p }\) to find \(\lim _ { p \rightarrow \infty } \left[ \frac { \ln p } { p ^ { k } } \right]\), where \(k > 0\).
  2. Evaluate the improper integral \(\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 7 } } \mathrm {~d} x\), showing the limiting process used.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-16_2039_1719_671_148}
AQA FP3 2016 June Q7
10 marks Challenging +1.2
7 Find the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 10 \mathrm { e } ^ { 4 x } + 8 \sin 2 x + 4 \cos 2 x$$ given that \(y = 2.5\) when \(x = 0\) and \(y = \frac { \pi } { 4 }\) when \(x = \frac { \pi } { 4 }\).
[0pt] [10 marks]
AQA FP3 2016 June Q8
17 marks Challenging +1.2
8 The diagram shows the sketch of part of a curve, the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is \(r = 1 + \tan \theta\).
The point \(A\) is the point on the curve at which \(\theta = \frac { \pi } { 3 }\).
The perpendicular, \(A N\), from \(A\) to the initial line intersects the curve at the point \(B\).
  1. Find the exact length of \(O A\).
    1. Given that, at the point \(B , \theta = \alpha\), show that \(( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }\).
    2. Hence, or otherwise, find \(\alpha\) in terms of \(\pi\).
  3. Show that the area of triangle \(O A B\) is \(\frac { 3 + 2 \sqrt { 3 } } { 6 }\).
  4. Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment \(A B\).
    [0pt] [7 marks]
    \includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
    \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA D1 2005 January Q1
4 marks Easy -2.0
1 A student is using the algorithm below.
LINE 10INPUT \(A , B\)
LINE 20LET \(C = A - B\)
LINE 30LET \(D = A + B\)
LINE 40LET \(E = ( D * D ) - ( C * C )\)
LINE 50LET \(F = E / 4\)
LINE 60PRINT \(F\)
LINE 70END
Trace the algorithm in the case where \(A = 5\) and \(B = 3\).
AQA D1 2005 January Q2
7 marks Easy -1.8
2
  1. Use a bubble sort algorithm to rearrange the following numbers into ascending order, showing the new arrangement after each pass. $$\begin{array} { l l l l l l l l } 19 & 3 & 7 & 20 & 2 & 6 & 5 & 15 \end{array}$$
  2. Write down the number of comparisons and the number of swaps during the first pass.
    (2 marks)
AQA D1 2005 January Q3
1 marks Easy -1.2
3 A local council is responsible for gritting roads. The diagram shows the length, in miles, of the roads that have to be gritted. \includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-03_671_686_488_669} Total length \(= 87\) miles The gritter is based at \(A\), and must travel along all the roads, at least once, before returning to \(A\).
  1. Explain why it is not possible to start from \(A\) and, by travelling along each road only once, return to \(A\).
  2. Find an optimal 'Chinese postman' route around the network, starting and finishing at \(A\). State the length of your route.
AQA D1 2005 January Q4
9 marks Easy -1.2
4 The Head Teacher of a school is allocating five teachers, Amy ( \(A\) ), Ben ( \(B\) ), Celia ( \(C\) ), Duncan ( \(D\) ), and Erica ( \(E\) ), to the five posts of Head of Year 7, 8, 9, 10 and 11. The five teachers are asked which year(s) they would be willing to take. This information is shown below. Amy is willing to take Year 7 or Year 8 . Ben is willing to take Year 7, Year 8 or Year 10. Celia is willing to take Year 8, Year 9 or Year 11. Duncan will take only Year 9.
Erica will take only Year 11.
  1. Show this information on a bipartite graph.
  2. Initially the Head Teacher assigns Amy to Year 8, Ben to Year 10, Celia to Year 9 and Erica to Year 11. Demonstrate, by using an alternating path from this initial matching, how each teacher can be matched to a year that they are willing to take.
AQA D1 2005 January Q5
10 marks Moderate -0.8
5 The network shows the lengths, in miles, of roads connecting eleven villages. \includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-04_1100_1575_406_251}
  1. Use Prim's algorithm, starting from \(A\), to find the minimum spanning tree for the network.
  2. State the length of your minimum spanning tree.
  3. Draw your minimum spanning tree.
  4. A student used Kruskal's algorithm to find the same minimum spanning tree. Find the seventh and eighth edges that the student added to his spanning tree.
AQA D1 2005 January Q6
8 marks Standard +0.8
6 [Figure 1, printed on a separate sheet, is provided for use in this question.]
A theme park is built on two levels. The levels are connected by a staircase. There are five rides on each of the levels. The diagram shows the ten rides: \(A , B , \ldots \ldots J\). The numbers on the edges represent the times, in minutes, taken to travel between pairs of rides. \includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-05_984_1593_584_226}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
  2. A new staircase is built connecting rides \(B\) and \(G\). The time taken to travel from \(B\) to \(G\) using this staircase is \(x\) minutes, where \(x\) is an integer. The time taken to travel from \(A\) to \(G\) is reduced, but the time taken to travel from \(A\) to \(J\) is not reduced. Find two inequalities for \(x\) and hence state the value of \(x\).
    (4 marks)
AQA D1 2005 January Q7
16 marks Moderate -0.8
7 Rob delivers bread to six shops \(A , B , C , D , E\) and \(F\). Each day, Rob starts at shop \(A\), travels to each of the other shops before returning to shop \(A\). The table shows the distances, in miles, between the shops.
\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)\(\boldsymbol { E }\)\(\boldsymbol { F }\)
\(\boldsymbol { A }\)-869127
\(\boldsymbol { B }\)8-1014138
\(\boldsymbol { C }\)610-71610
\(\boldsymbol { D }\)9147-155
\(\boldsymbol { E }\)12131615-11
\(\boldsymbol { F }\)7810511-
    1. Find the length of the tour \(A B C D E F A\).
    2. Find the length of the tour obtained by using the nearest neighbour algorithm starting from \(A\).
  2. By deleting \(A\), find a lower bound for the length of a minimum tour.
  3. By deleting \(F\), another lower bound of 45 miles is obtained for the length of a minimum tour. The length of a minimum tour is \(T\) miles. Write down the smallest interval for \(T\) which can be obtained from your answers to parts (a) and (b), and the information given in this part.
    (3 marks)
AQA D1 2005 January Q8
18 marks Moderate -0.8
8 [Figure 2, printed on a separate sheet, is provided for use in this question.]
A bakery makes two types of pizza, large and medium.
Every day the bakery must make at least 40 of each type.
Every day the bakery must make at least 120 in total but not more than 400 pizzas in total.
Each large pizza takes 4 minutes to make, and each medium pizza takes 2 minutes to make. There are four workers available, each for five hours a day, to make the pizzas. The bakery makes a profit of \(\pounds 3\) on each large pizza sold and \(\pounds 1\) on each medium pizza sold.
Each day, the bakery makes and sells \(x\) large pizzas and \(y\) medium pizzas.
The bakery wishes to maximise its profit, \(\pounds P\).
  1. Show that one of the constraints leads to the inequality $$2 x + y \leqslant 600$$
  2. Formulate this situation as a linear programming problem.
  3. On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and an objective line.
  4. Use your diagram to find the maximum daily profit.
  5. The bakery introduces a new pricing structure in which the profit is \(\pounds 2\) on each large pizza sold and \(\pounds 2\) on each medium pizza sold.
    1. Find the new maximum daily profit for the bakery.
    2. Write down the number of different combinations that would give the new maximum daily profit.
AQA D1 2012 January Q1
5 marks Easy -1.2
1 Use a Shell sort to rearrange the following numbers into ascending order, showing the new arrangement after each pass. \(\begin{array} { l l l l l l l l } 37 & 25 & 16 & 12 & 36 & 24 & 13 & 11 \end{array}\) (5 marks) PART REFERENCE REFERENCE
AQA D1 2012 January Q2
5 marks Easy -1.8
2
  1. Draw a bipartite graph representing the following adjacency matrix.
    \cline { 2 - 7 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)
    \(\boldsymbol { A }\)110011
    \(\boldsymbol { B }\)010011
    \(\boldsymbol { C }\)100011
    \(\boldsymbol { D }\)011101
    \(\boldsymbol { E }\)000011
    \(\boldsymbol { F }\)000001
  2. Given that \(A , B , C , D , E\) and \(F\) represent six people and that 1, 2, 3, 4, 5 and 6 represent six tasks to which they may be assigned, explain why a complete matching is impossible. \begin{verbatim} QUESTION PART REFERENCE \end{verbatim}