Questions — Edexcel (9685 questions)

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Edexcel D2 Q1
7 marks Moderate -0.8
  1. A team of gardeners is called in to attend to the grounds of a stately home. The three gardeners will each be assigned to one of three areas, the lawns, the hedgerows and the flower beds. The table below shows the estimated time, in hours, it will take each gardener to do each job.
\cline { 2 - 4 } \multicolumn{1}{c|}{}LawnsHedgerowsFlower Beds
Alan44.56
Beth345
Colin3.556
The team wishes to complete the tasks in the least total time.
Formulate this information as a linear programming problem.
  1. State your decision variables.
  2. Write down the objective function in terms of your decision variables.
  3. Write down the constraints and explain what each one represents.
Edexcel D2 Q2
10 marks Moderate -0.3
2. This question should be answered on the sheet provided. A pool player is to play in four tournaments. Some of the tournaments take place simultaneously and the player has to choose one of each of the following: $$\begin{array} { l l } 1 ^ { \text {st } } \text { tournament: } & A , B \text { or } C , \\ 2 ^ { \text {nd } } \text { tournament: } & D , E \text { or } F , \\ 3 ^ { \text {rd } } \text { tournament: } & G , H \text { or } I , \\ 4 ^ { \text {th } } \text { tournament: } & J , K \text { or } L \end{array}$$ Each tournament has six rounds and the player estimates how well he will do in each tournament based on which tournament he plays before it. The table below shows his expectations with each number indicating the round he expects to reach and a "7" indicating he expects to win the tournament.
\multirow{2}{*}{}Expected performance in tournament
ABC\(D\)E\(F\)\(G\)\(H\)IJ\(K\)\(L\)
\multirow{10}{*}{Previous tournament}None533
A637
B554
C755
D533
E356
\(F\)365
G241
H322
\(I\)253
He wishes to choose the tournaments such that his worst performance is as good as possible. Use dynamic programming to find which tournaments he should play.
(10 marks) Turn over
Edexcel D2 Q3
10 marks Moderate -0.5
3. Four people are contributing to the entertainment section of an email magazine. For one issue reviews are required for a film, a musical, a ballet and a concert such that each person reviews one show. The people in charge of the magazine will pay each person's expenses and the cost, in pounds, for each reviewer to attend each show are given below.
FilmMusicalBalletConcert
Andrew5201218
Betty6181516
Carlos421915
Davina5161113
Use the Hungarian algorithm to find an optimal assignment which minimises the total cost. State the total cost of this allocation.
(10 marks)
Edexcel D2 Q4
15 marks Standard +0.8
4. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 4 }III
\multirow{2}{*}{\(A\)}I4\({ } ^ { - } 8\)
\cline { 2 - 4 }II2\({ } ^ { - } 4\)
\cline { 2 - 4 }III\({ } ^ { - } 8\)2
  1. Explain why the game does not have a saddle point.
  2. Using a graphical method, find the optimal strategy for player \(B\).
  3. Find the optimal strategy for player \(A\).
  4. Find the value of the game.
Edexcel D2 Q5
16 marks Moderate -0.5
5. A carpet manufacturer has two warehouses, \(W _ { 1 }\) and \(W _ { 2 }\), which supply carpets for three sales outlets, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). At one point \(S _ { 1 }\) requires 40 rolls of carpet, \(S _ { 2 }\) requires 23 rolls of carpet and \(S _ { 3 }\) requires 37 rolls of carpet. At this point \(W _ { 1 }\) has 45 rolls in stock and \(W _ { 2 }\) has 40 rolls in stock. The following table shows the cost, in pounds, of transporting one roll from each warehouse to each sales outlet:
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(S _ { 1 }\)\(S _ { 2 }\)\(S _ { 3 }\)
\(W _ { 1 }\)8711
\(W _ { 2 }\)91011
The company's manager wishes to supply the 85 rolls that are in stock such that transportation costs are kept to a minimum.
  1. Use the north-west corner rule to obtain an initial solution to the problem.
  2. Calculate improvement indices for the unused routes.
  3. Use the stepping-stone method to obtain an optimal solution. Turn over
Edexcel D2 Q6
17 marks Standard +0.8
6. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{073926c5-03cc-41d4-82bf-315740ead663-6_672_984_322_431} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A band is going on tour to play gigs in six towns, including their home town, \(A\). The network in Figure 1 shows the distances, in miles, between the various towns. The band must begin and end their tour at \(A\) and visit each of the other towns once, and they wish to keep the total distance travelled as small as possible.
  1. By inspection, draw a complete network showing the shortest distances between the towns.
  2. Use your complete network and the nearest neighbour algorithm, starting at \(A\), to find an upper bound for the total distance travelled.
    1. Use your complete network to obtain and draw a minimum spanning tree and hence obtain another upper bound for the total distance travelled.
    2. Improve this upper bound using two shortcuts to find an upper bound below 225 miles.
  4. By deleting \(A\), find a lower bound for the total distance travelled.
  5. State an interval of as small a width as possible within which \(d\), the minimum distance travelled, in miles, must lie. \section*{Please hand this sheet in for marking}
    StagePrevious tournamentCurrent tournament
    \multirow[t]{3}{*}{1}G
    J
    K
    L
    \(H\)
    J
    K
    L
    I
    J
    K
    L
    \multirow[t]{3}{*}{2}D
    G
    H
    I
    \(E\)
    G
    H
    I
    \(F\)
    G
    H
    I
    \multirow[t]{3}{*}{3}A
    D
    E
    F
    \(B\)
    D
    E
    F
    C
    D
    E
    F
    4None
    A
    B
    C
    \section*{Please hand this sheet in for marking}
  6. \includegraphics[max width=\textwidth, alt={}, center]{073926c5-03cc-41d4-82bf-315740ead663-8_684_992_461_427}
  7. \section*{Sheet for answering question 6 (cont.)}
    2. \(\_\_\_\_\)
  10. \(\_\_\_\_\)
Edexcel AS Paper 1 2018 June Q1
4 marks Easy -1.3
  1. Find
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - 6 \sqrt { x } + 1 \right) \mathrm { d } x$$ giving your answer in its simplest form.
Edexcel AS Paper 1 2018 June Q2
5 marks Moderate -0.8
  1. (i) Show that \(x ^ { 2 } - 8 x + 17 > 0\) for all real values of \(x\) (ii) "If I add 3 to a number and square the sum, the result is greater than the square of the original number."
State, giving a reason, if the above statement is always true, sometimes true or never true.
Edexcel AS Paper 1 2018 June Q3
4 marks Easy -1.3
  1. Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
    (b) find \(| \overrightarrow { A B } |\).
Give your answer as a simplified surd.
Edexcel AS Paper 1 2018 June Q4
4 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)
The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.
Edexcel AS Paper 1 2018 June Q5
5 marks Moderate -0.5
  1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.
$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3 \\ & 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3 \\ & 2 \log _ { 2 } ( \sqrt { x } ) = 3 \\ & \log _ { 2 } x = 3 \\ & x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log
  1. Identify two errors made by this student, giving a brief explanation of each.
  2. Write out the correct solution.
Edexcel AS Paper 1 2018 June Q6
7 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-12_599_1084_292_486} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,
  1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
  2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
    State, according to the model,
    1. the maximum possible annual profit,
    2. the selling price of the toy that maximises the annual profit.
Edexcel AS Paper 1 2018 June Q7
6 marks Standard +0.3
  1. In a triangle \(A B C\), side \(A B\) has length 10 cm , side \(A C\) has length 5 cm , and angle \(B A C = \theta\) where \(\theta\) is measured in degrees. The area of triangle \(A B C\) is \(15 \mathrm {~cm} ^ { 2 }\)
    1. Find the two possible values of \(\cos \theta\)
    Given that \(B C\) is the longest side of the triangle,
  2. find the exact length of \(B C\).
Edexcel AS Paper 1 2018 June Q8
9 marks Moderate -0.3
  1. A lorry is driven between London and Newcastle.
In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$
  1. Find, according to this model,
    1. the value of \(v\) that minimises the cost of the journey,
    2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
  2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
  3. State one limitation of this model.
Edexcel AS Paper 1 2018 June Q9
9 marks Standard +0.3
9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
  2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
  3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
    1. \(\mathrm { g } ( x ) \leqslant 0\)
    2. \(\mathrm { g } ( 2 x ) = 0\)
Edexcel AS Paper 1 2018 June Q10
4 marks Moderate -0.5
  1. Prove, from first principles, that the derivative of \(x ^ { 3 }\) is \(3 x ^ { 2 }\)
Edexcel AS Paper 1 2018 June Q11
8 marks Standard +0.3
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$ giving each term in its simplest form. $$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm { f } ( x )\) are 128 and \(36 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).
Edexcel AS Paper 1 2018 June Q12
8 marks Standard +0.3
  1. (a) Show that the equation
$$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2018 June Q13
8 marks Moderate -0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-36_563_1019_244_523} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The value of a rare painting, \(\pounds V\), is modelled by the equation \(V = p q ^ { t }\), where \(p\) and \(q\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1980. The line \(l\) shown in Figure 3 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) since 1st January 1980. The equation of line \(l\) is \(\log _ { 10 } V = 0.05 t + 4.8\)
  1. Find, to 4 significant figures, the value of \(p\) and the value of \(q\).
  2. With reference to the model interpret
    1. the value of the constant \(p\),
    2. the value of the constant \(q\).
  3. Find the value of the painting, as predicted by the model, on 1st January 2010, giving your answer to the nearest hundred thousand pounds.
Edexcel AS Paper 1 2018 June Q14
9 marks Standard +0.3
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the radius of \(C\) The line with equation \(y = k x\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\).
Edexcel AS Paper 1 2018 June Q15
10 marks Standard +0.8
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-44_595_977_242_536} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2019 June Q1
4 marks Easy -1.2
  1. The line \(l _ { 1 }\) has equation \(2 x + 4 y - 3 = 0\)
The line \(l _ { 2 }\) has equation \(y = m x + 7\), where \(m\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,
  1. find the value of \(m\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\).
  2. Find the \(x\) coordinate of \(P\). \includegraphics[max width=\textwidth, alt={}, center]{deba6a2b-1821-4110-bde8-bde18a5f9be9-02_2258_48_313_1980}
Edexcel AS Paper 1 2019 June Q2
8 marks Moderate -0.8
  1. Find, using algebra, all real solutions to the equation
    1. \(16 a ^ { 2 } = 2 \sqrt { a }\)
    2. \(b ^ { 4 } + 7 b ^ { 2 } - 18 = 0\)
Edexcel AS Paper 1 2019 June Q3
6 marks Moderate -0.8
  1. (a) Given that \(k\) is a constant, find
$$\int \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x$$ simplifying your answer.
(b) Hence find the value of \(k\) such that $$\int _ { 0.5 } ^ { 2 } \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x = 8$$
Edexcel AS Paper 1 2019 June Q4
5 marks Moderate -0.8
  1. A tree was planted in the ground.
Its height, \(H\) metres, was measured \(t\) years after planting.
Exactly 3 years after planting, the height of the tree was 2.35 metres.
Exactly 6 years after planting, the height of the tree was 3.28 metres.
Using a linear model,
  1. find an equation linking \(H\) with \(t\). The height of the tree was approximately 140 cm when it was planted.
  2. Explain whether or not this fact supports the use of the linear model in part (a).