Questions — Edexcel (10514 questions)

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Edexcel FD1 AS 2019 June Q2
7 marks Moderate -0.3
The following algorithm produces a numerical approximation for the integral $$I = \int_A^B x^4 \, dx$$
Step 1Start
Step 2Input the values of A, B and N
Step 3Let H = (B - A) / N
Step 4Let C = H / 2
Step 5Let D = 0
Step 6Let D = D + A\(^4\) + B\(^4\)
Step 7Let E = A
Step 8Let E = E + H
Step 9If E = B go to Step 12
Step 10Let D = D + 2 × E\(^4\)
Step 11Go to Step 8
Step 12Let F = C × D
Step 13Output F
Step 14Stop
For the case when A = 1, B = 3 and N = 4,
    1. complete the table in the answer book to show the results obtained at each step of the algorithm.
    2. State the final output. [4]
  2. Calculate, to 3 significant figures, the percentage error between the exact value of \(I\) and the value obtained from using the approximation to \(I\) in this case. [3]
Edexcel FD1 AS 2019 June Q3
7 marks Standard +0.3
ActivityImmediately preceding activities
A-
B-
CA
DA
EA
FB, C
GB, C
HD
ID, E, F, G
JD, E, F, G
KG
  1. Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies. [5]
Every activity shown in the precedence table has the same duration.
  1. Explain why activity B cannot be critical. [1]
  2. State which other activities are not critical. [1]
Edexcel FD1 AS 2019 June Q4
10 marks Challenging +1.2
\includegraphics{figure_1} **Figure 1** [The total weight of the network is \(135 + 4x + 2y\)] The weights on the arcs in Figure 1 represent distances. The weights on the arcs CE and GH are given in terms of \(x\) and \(y\), where \(x\) and \(y\) are positive constants and \(7 < x + y < 20\) There are three paths from A to H that have the same minimum length.
  1. Use Dijkstra's algorithm to find \(x\) and \(y\). [7]
An inspection route starting at A and finishing at H is found. The route traverses each arc at least once and is of minimum length.
  1. State the arcs that are traversed twice. [1]
  2. State the number of times that vertex C appears in the inspection route. [1]
  3. Determine the length of the inspection route. [1]
Edexcel FD1 AS 2019 June Q5
10 marks Standard +0.3
Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order. • At least three-fifths of all the flowers must be roses. • For every 2 hydrangeas there must be at most 3 peonies. • The total number of flowers must be exactly 1000 The cost of each rose is £1, the cost of each hydrangea is £5 and the cost of each peony is £4 Ben wants to minimise the cost of the flowers. Let \(x\) represent the number of roses, let \(y\) represent the number of hydrangeas and let \(z\) represent the number of peonies that he will order.
  1. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. [7]
Ben decides to order the minimum number of roses that satisfy his constraints.
    1. Calculate the number of each type of flower that he will order to minimise the cost of the flowers.
    2. Calculate the corresponding total cost of this order. [3]
Edexcel CP1 2021 June Q1
6 marks Moderate -0.3
The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$
  1. Determine
    1. the value of \(k\),
    2. the smallest value of \(\theta\)
    [4] A square \(S\) has vertices at the points with coordinates \((0, 0)\), \((a, -a)\), \((2a, 0)\) and \((a, a)\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S'\) by the transformation represented by \(\mathbf{M}\).
  2. Determine, in terms of \(a\), the area of \(S'\) [2]
Edexcel CP1 2021 June Q2
7 marks Standard +0.3
  1. Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos^2\left(\frac{x}{3}\right)\) in ascending powers of \(x\), up to and including the term in \(x^4\) Give each term in simplest form. [2]
  2. Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$ [3]
  3. Use the integration function on your calculator to evaluate $$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$ Give your answer to 5 decimal places. [1]
  4. Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b). [1]
Edexcel CP1 2021 June Q3
6 marks Standard +0.8
The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]
Edexcel CP1 2021 June Q4
9 marks Standard +0.3
  1. \(\mathbf{A}\) is a 2 by 2 matrix and \(\mathbf{B}\) is a 2 by 3 matrix. Giving a reason for your answer, explain whether it is possible to evaluate
    1. \(\mathbf{AB}\)
    2. \(\mathbf{A} + \mathbf{B}\)
    [2]
  2. Given that $$\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}\begin{pmatrix} 0 & 5 & 0 \\ 2 & 12 & -1 \\ -1 & -11 & 3 \end{pmatrix} = \lambda\mathbf{I}$$ where \(a\), \(b\) and \(\lambda\) are constants,
    1. determine
    2. Hence deduce the inverse of the matrix \(\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}\)
    [3]
  3. Given that $$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leq \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf{M}\) is singular. [4]
Edexcel CP1 2021 June Q5
7 marks Moderate -0.3
  1. Evaluate the improper integral $$\int_1^{\infty} 2e^{-\frac{1}{2}x} dx$$ [3]
  2. The air temperature, \(\theta ^{\circ}C\), on a particular day in London is modelled by the equation $$\theta = 8 - 5\sin\left(\frac{\pi}{12}t\right) - \cos\left(\frac{\pi}{6}t\right) \quad 0 \leq t \leq 24$$ where \(t\) is the number of hours after midnight.
    1. Use calculus to show that the mean air temperature on this day is \(8^{\circ}C\), according to the model. [3] Given that the actual mean air temperature recorded on this day was higher than \(8^{\circ}C\),
    2. explain how the model could be refined. [1]
Edexcel CP1 2021 June Q6
12 marks Standard +0.3
A tourist decides to do a bungee jump from a bridge over a river. One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist. The tourist jumps off the bridge. At time \(t\) seconds after the tourist reaches their lowest point, their vertical displacement is \(x\) metres above a fixed point 30 metres vertically above the river. When \(t = 0\)
  • \(x = -20\)
  • the velocity of the tourist is \(0\text{ms}^{-1}\)
  • the acceleration of the tourist is \(13.6\text{ms}^{-2}\)
In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation $$5k\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + 17x = 0 \quad t \geq 0$$ where \(k\) is a positive constant.
  1. Determine the value of \(k\) [2]
  2. Determine the particular solution to the differential equation. [7]
  3. Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point. [2]
  4. Give a limitation of the model. [1]
Edexcel CP1 2021 June Q7
8 marks Standard +0.8
The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(\mathbf{2i + 3j - 4k}\) is perpendicular to \(\Pi\). [2]
  2. Hence find a Cartesian equation of \(\Pi\). [2]
The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)
  1. determine the possible coordinates of \(A\). [4]
Edexcel CP1 2021 June Q8
9 marks Standard +0.3
Two different colours of paint are being mixed together in a container. The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint. The colour of the paint mixture is now altered by
  • adding red paint to the container at a rate of 2 litres per second
  • adding blue paint to the container at a rate of 1 litre per second
  • pumping fully mixed paint from the container at a rate of 3 litres per second.
Let \(r\) litres be the amount of red paint in the container at time \(t\) seconds after the colour of the paint mixture starts to be altered.
  1. Show that the amount of red paint in the container can be modelled by the differential equation $$\frac{dr}{dt} = 2 - \frac{r}{a}$$ where \(a\) is a positive constant to be determined. [2]
  2. By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. [6] It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.
  3. Use this information to evaluate the model, giving a reason for your answer. [1]
Edexcel CP1 2021 June Q9
11 marks Challenging +1.2
  1. Use a hyperbolic substitution and calculus to show that $$\int \frac{x^2}{\sqrt{x^2 - 1}} dx = \frac{1}{2}\left[x\sqrt{x^2 - 1} + \text{arcosh } x\right] + k$$ where \(k\) is an arbitrary constant. [6]
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac{4}{15}x \text{ arcosh } x \quad x \geq 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
  1. Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac{1}{15}\left[17\ln\left(3 + 2\sqrt{2}\right) - 6\sqrt{2}\right]$$ [5]
Edexcel AEA 2014 June Q1
5 marks Standard +0.8
The function f is given by $$f(x) = \ln(2x - 5), \quad x > 2.5$$
  1. Find \(f^{-1}(x)\). [2] The function g has domain \(x > 2\) and $$g(x) = \ln\left(\frac{x + 10}{x - 2}\right), \quad x > 2$$
  2. Find \(g(x)\) and simplify your answer. [3]
Edexcel AEA 2014 June Q2
6 marks Challenging +1.2
Given that $$3\sin^2 x + 2\sin x = 6\cos x + 9\sin x \cos x$$ and that \(-90° < x < 90°\), find the possible values of \(\tan x\). [6]
Edexcel AEA 2014 June Q3
11 marks Standard +0.8
  1. On separate diagrams sketch the curves with the following equations. On each sketch you should mark the coordinates of the points where the curve crosses the coordinate axes.
    1. \(y = x^2 - 2x - 3\)
    2. \(y = x^2 - 2|x| - 3\)
    3. \(y = x^2 - x - |x| - 3\)
    [7]
  2. Solve the equation $$x^2 - x - |x| - 3 = x + |x|$$ [4]
Edexcel AEA 2014 June Q4
13 marks Hard +2.3
Given that $$(1 + x)^n = 1 + \sum_{r=1}^{\infty} \frac{n(n-1)...(n-r+1)}{1 \times 2 \times ... \times r} x^r \quad (|x| < 1, x \in \mathbb{R}, n \in \mathbb{R})$$
  1. show that $$(1 - x)^{-\frac{1}{2}} = \sum_{r=0}^{\infty} \binom{2r}{r}\left(\frac{x}{4}\right)^r$$ [5]
  2. show that \((9 - 4x^2)^{-\frac{1}{2}}\) can be written in the form \(\sum_{r=0}^{\infty} \binom{2r}{r} \frac{x^{2r}}{3^{r}}\) and give \(q\) in terms of \(r\). [3]
  3. Find \(\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r}{9} \times \left(\frac{x}{3}\right)^{2r-1}\) [3]
  4. Hence find the exact value of $$\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r\sqrt{5}}{9} \times \frac{1}{5^r}$$ giving your answer as a rational number. [2]
Edexcel AEA 2014 June Q5
15 marks Challenging +1.8
The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$
  1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
  2. Find
    1. the length of a side of the square base of \(P\),
    2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
    3. the height of \(P\),
    4. the position vector of \(E\).
    [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
  3. Find the position vector of the other vertex of this octahedron. [3]
Edexcel AEA 2014 June Q6
20 marks Hard +2.3
  1. A curve with equation \(y = f(x)\) has \(f(x) \geq 0\) for \(x \geq a\) and $$A = \int_a^b f(x) \, dx \quad \text{and} \quad V = \pi \int_a^b [f(x)]^2 \, dx$$ where \(a\) and \(b\) are constants with \(b > a\). Use integration by substitution to show that for the positive constants \(r\) and \(h\) $$\pi \int_{a+h}^{b+h} [r + f(x - h)]^2 \, dx = \pi r^2 (b - a) + 2\pi rA + V$$ [3]
  2. % \includegraphics{figure_1} - Shows a curve with vertical asymptotes at x=m and x=n, crossing y-axis at point p Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac{2}{\sqrt{3}\cos x + \sin x}\) This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \((0, p)\). (a) Find the value of \(p\), the value of \(m\) and the value of \(n\). [4] (b) Show that the equation of \(C\) can be written in the form \(y = r + f(x - h)\) and specify the function \(f\) and the constants \(r\) and \(h\). [4] The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{3}\) is rotated through \(2\pi\) radians about the \(x\)-axis. (c) Find the volume of the solid formed. [9]
Edexcel AEA 2014 June Q7
23 marks Hard +2.3
% \includegraphics{figure_2} - Shows a circular tower with center T at (0,1), a goat at point G attached to the base at O, with string along arc OA then tangent AG A circular tower stands in a large horizontal field of grass. A goat is attached to one end of a string and the other end of the string is attached to the fixed point \(O\) at the base of the tower. Taking the point \(O\) as the origin \((0, 0)\), the centre of the base of the tower is at the point \(T(0, 1)\). The radius of the base of the tower is 1. The string has length \(\pi\) and you may ignore the size of the goat. The curve \(C\) represents the edge of the region that the goat can reach as shown in Figure 2.
  1. Write down the equation of \(C\) for \(y < 0\). [1] When the goat is at the point \(G(x, y)\), with \(x > 0\) and \(y > 0\), as shown in Figure 2, the string lies along \(OAG\) where \(OA\) is an arc of the circle with angle \(OTA = \theta\) radians and \(AG\) is a tangent to the circle at \(A\).
  2. With the aid of a suitable diagram show that $$x = \sin \theta + (\pi - \theta) \cos \theta$$ $$y = 1 - \cos \theta + (\pi - \theta) \sin \theta$$ [5]
  3. By considering \(\int y \frac{dx}{d\theta} d\theta\), show that the area between \(C\), the positive \(x\)-axis and the positive \(y\)-axis can be expressed in the form $$\int_0^{\pi} u \sin u \, du + \int_0^{\pi} u^2 \sin^2 u \, du + \int_0^{\pi} u \sin u \cos u \, du$$ [5]
  4. Show that \(\int_0^{\pi} u^2 \sin^2 u \, du = \frac{\pi^3}{6} + \int_0^{\pi} u \sin u \cos u \, du\) [4]
  5. Hence find the area of grass that can be reached by the goat. [8]
Edexcel AEA 2011 June Q1
Standard +0.3
Solve for \(0 \leq \theta \leq 180°\) $$\tan(\theta + 35°) = \cot(\theta - 53°)$$ [Total 4 marks]
Edexcel AEA 2011 June Q2
Challenging +1.8
Given that $$\int_0^{\frac{\pi}{2}} (1 + \tan\left[\frac{1}{2}x\right])^2 \, dx = a + \ln b$$ find the value of \(a\) and the value of \(b\). [Total 7 marks]
Edexcel AEA 2011 June Q3
17 marks Challenging +1.8
A sequence \(\{u_n\}\) is given by $$u_1 = k$$ $$u_{2n} = u_{2n-1} \times p \qquad n \geq 1$$ $$u_{2n+1} = u_{2n} \times q \qquad n \geq 1$$ where \(k\), \(p\) and \(q\) are positive constants with \(pq \neq 1\)
  1. Write down the first 6 terms of this sequence. [3]
  2. Show that \(\sum_{r=1}^{2n} u_r = \frac{k(1+p)(1-(pq)^n)}{1-pq}\) [6]
In part (c) \([x]\) means the integer part of \(x\), so for example \([2.73] = 2\), \([4] = 4\) and \([0] = 0\)
  1. Find \(\sum_{r=1}^{\infty} 6 \times \left(\frac{4}{3}\right)^{\left[\frac{r}{2}\right]} \times \left(\frac{3}{5}\right)^{\left[\frac{r-1}{2}\right]}\) [4]
[Total 13 marks]
Edexcel AEA 2011 June Q4
13 marks Challenging +1.2
The curve \(C\) has parametric equations $$x = \cos^2 t$$ $$y = \cos t \sin t$$ where \(0 \leq t < \pi\)
  1. Show that \(C\) is a circle and find its centre and its radius. [5]
% Figure 1 shows a sketch of C with point P, rectangle R with diagonal OP \includegraphics{figure_1} Figure 1 Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \((\cos^2 \alpha, \cos\alpha \sin \alpha)\), \(0 < \alpha < \frac{\pi}{2}\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(OP\) as a diagonal, where \(O\) is the origin.
  1. Show that the area of \(R\) is \(\sin\alpha \cos^3 \alpha\) [1]
  2. Find the maximum area of \(R\), as \(\alpha\) varies. [7]
[Total 13 marks]
Edexcel AEA 2011 June Q5
17 marks Challenging +1.8
% Figure 2 shows curve with vertical asymptotes at x = -2 and x = 2, horizontal asymptote at y = 1, with U-shaped region between asymptotes \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{x^2 - 2}{x^2 - 4}\) and \(x \neq \pm 2\). The curve cuts the \(y\)-axis at \(U\).
  1. Write down the coordinates of the point \(U\). [1]
The point \(P\) with \(x\)-coordinate \(a\) (\(a \neq 0\)) lies on \(C\).
  1. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left(0, \frac{a^2 - 2}{a^2 - 4} - \frac{(a^2 - 4)^2}{4}\right)$$ [6]
The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).
    1. Show that $$\frac{a^2}{2(a^2-4)} - \frac{(a^2-4)^2}{4} = a^2 + \frac{(a^2-4)^4}{16}$$
    2. Hence, show that $$(a^2 - 4)^2 = 1$$
    3. Find the centre and radius of \(E\).
    [10]
[Total 17 marks]