Questions — Edexcel (10514 questions)

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Edexcel C4 Q16
8 marks Moderate -0.3
The speed, \(v\) m s\(^{-1}\), of a lorry at time \(t\) seconds is modelled by $$v = 5(e^{0.1t} - 1) \sin (0.1t), \quad 0 \leq t \leq 30.$$
  1. Copy and complete the following table, showing the speed of the lorry at 5 second intervals. Use radian measure for \(0.1t\) and give your values of \(v\) to 2 decimal places where appropriate.
    \(t\)0510152025
    \(v\)1.567.2317.36
    [3]
  2. Verify that, according to this model, the lorry is moving more slowly at \(t = 25\) than at \(t = 24.5\). [1]
The distance, \(s\) metres, travelled by the lorry during the first 25 seconds is given by $$s = \int_0^{25} v \, dt.$$
  1. Estimate \(s\) by using the trapezium rule with all the values from your table. [4]
Edexcel C4 Q17
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{4}{x - 3}\), \(x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates \(3.25\) and \(5\) respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [1]
The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through \(360°\) about the \(y\)-axis to form a solid shape \(S\).
  1. Find the volume of \(S\), giving your answer in the form \(\pi (a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]
The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
  1. show that the volume of the tower is approximately \(15\,500\) m\(^3\). [2]
Edexcel C4 Q18
7 marks Moderate -0.3
  1. Use integration by parts to find $$\int x \cos 2x \, dx.$$ [4]
  2. Prove that the answer to part \((a)\) may be expressed as $$\frac{1}{2} \sin x (2x \cos x - \sin x) + C,$$ where \(C\) is an arbitrary constant. [3]
Edexcel C4 Q19
8 marks Moderate -0.3
The circle \(C\) has equation \(x^2 + y^2 - 8x - 16y - 209 = 0\).
  1. Find the coordinates of the centre of \(C\) and the radius of \(C\). [3]
The point \(P(x, y)\) lies on \(C\).
  1. Find, in terms of \(x\) and \(y\), the gradient of the tangent to \(C\) at \(P\). [3]
  2. Hence or otherwise, find an equation of the tangent to \(C\) at the point \((21, 8)\). [2]
Edexcel C4 Q20
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$
  1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]
The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
  1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]
Edexcel C4 Q21
10 marks Standard +0.3
  1. Prove that, when \(x = \frac{1}{12}\), the value of \((1 + 5x)^{-\frac{1}{2}}\) is exactly equal to \(\sin 60°\). [3]
  2. Expand \((1 + 5x)^{-\frac{1}{2}}\), \(|x| < 0.2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  3. Use your answer to part \((b)\) to find an approximation for \(\sin 60°\). [2]
  4. Find the difference between the exact value of \(\sin 60°\) and the approximation in part \((c)\). [1]
Edexcel C4 Q22
10 marks Moderate -0.3
A curve is given parametrically by the equations $$x = 5 \cos t, \quad y = -2 + 4 \sin t, \quad 0 \leq t < 2\pi$$
  1. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate. [4]
  2. Sketch the graph at \(C\). [2]
\(P\) is the point on \(C\) where \(t = \frac{1}{2}\pi\).
  1. Show that the normal to \(C\) at \(P\) has equation $$8\sqrt{3}y = 10x - 25\sqrt{3}.$$ [4]
Edexcel C4 Q23
11 marks Moderate -0.3
A Pancho car has value \(£V\) at time \(t\) years. A model for \(V\) assumes that the rate of decrease of \(V\) at time \(t\) is proportional to \(V\).
  1. By forming and solving an appropriate differential equation, show that \(V = Ae^{-kt}\), where \(A\) and \(k\) are positive constants. [3]
The value of a new Pancho car is \(£20\,000\), and when it is 3 years old its value is \(£11\,000\).
  1. Find, to the nearest \(£100\), an estimate for the value of the Pancho when it is 10 years old. [5]
A Pancho car is regarded as 'scrap' when its value falls below \(£500\).
  1. Find the approximate age of the Pancho when it becomes 'scrap'. [3]
Edexcel C4 Q24
13 marks Standard +0.3
Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((9\mathbf{i} - 2\mathbf{j} + \mathbf{k})\), \((6\mathbf{i} + 2\mathbf{j} + 6\mathbf{k})\) and \((3\mathbf{i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.
  1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]
Given that \(C\) lies on \(l\),
  1. find the value of \(p\) and the value of \(q\), [2]
  2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]
The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).
  1. Find the position vector of \(D\). [6]
Edexcel C4 Q25
12 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows part of the curve with equation \(y = x^2 + 2\). The finite region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
  1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [5]
  2. State, with a reason, whether your answer in part \((a)\) is an under-estimate or over-estimate of the area of \(R\). [1]
  3. Using integration, find the volume of the solid generated when \(R\) is rotated through \(360°\) about the \(x\)-axis, giving your answer in terms of \(\pi\). [6]
Edexcel C4 Q26
8 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = 1 + \frac{1}{2\sqrt{x}}\). The shaded region \(R\), bounded by the curve, that \(x\)-axis and the lines \(x = 1\) and \(x = 4\), is rotated through \(360°\) about the \(x\)-axis. Using integration, show that the volume of the solid generated is \(\pi (5 + \frac{1}{2} \ln 2)\). [8]
Edexcel C4 Q27
9 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8\sqrt{\sin \frac{\pi x}{10}}\), in the interval \(0 \leq x \leq 10\). The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = -2\), \(x = 12\) and \(y = 10\). The units on both axes are metres.
  1. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)0246810
    \(y\)06.130
    [2]
The area of the cross-section of the tunnel is given by \(\int_0^{10} y \, dx\).
  1. Estimate this area, using the trapezium rule with all the values from your table. [4]
  2. Deduce an estimate of the cross-sectional area of the concrete surround. [1]
  3. State, with a reason, whether your answer in part \((c)\) over-estimates or under-estimates the true value. [2]
Edexcel C4 Q28
6 marks Standard +0.3
The function f is given by $$f(x) = \frac{3(x + 1)}{(x + 2)(x - 1)}, \quad x \in \mathbb{R}, x \neq -2, x \neq 1.$$
  1. Express \(f(x)\) in partial fractions. [3]
  2. Hence, or otherwise, prove that \(f'(x) < 0\) for all values of \(x\) in the domain. [3]
Edexcel C4 Q29
8 marks Moderate -0.3
  1. Expand \((1 + 3x)^{-2}\), \(|x| < \frac{1}{3}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  2. Hence, or otherwise, find the first three terms in the expansion of \(\frac{x + 4}{(1 + 3x)^2}\) as a series in ascending powers of \(x\). [4]
Edexcel C4 Q30
11 marks Standard +0.3
Liquid is poured into a container at a constant rate of 30 cm\(^3\) s\(^{-1}\). At time \(t\) seconds liquid is leaking from the container at a rate of \(\frac{1}{5}V\) cm\(^3\) s\(^{-1}\), where \(V\) cm\(^3\) is the volume of liquid in the container at that time.
  1. Show that $$-15 \frac{dV}{dt} = 2V - 450.$$ [3]
Given that \(V = 1000\) when \(t = 0\),
  1. find the solution of the differential equation, in the form \(V = f(t)\). [7]
  2. Find the limiting value of \(V\) as \(t \to \infty\). [1]
Edexcel F1 2022 January Q1
5 marks Moderate -0.3
$$\mathbf{M} = \begin{pmatrix} 3x & 7 \\ 4x + 1 & 2 - x \end{pmatrix}$$ Find the range of values of \(x\) for which the determinant of the matrix \(\mathbf{M}\) is positive. [5]
Edexcel F1 2022 January Q2
8 marks Moderate -0.8
The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$
  1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
  2. Without using your calculator and showing all stages of your working,
    1. determine the value of \(|z_1|\) [1]
    2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
  3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]
Edexcel F1 2022 January Q3
5 marks Standard +0.3
The parabola \(C\) has equation \(y^2 = 18x\) The point \(S\) is the focus of \(C\)
  1. Write down the coordinates of \(S\) [1]
The point \(P\), with \(y > 0\), lies on \(C\) The shortest distance from \(P\) to the directrix of \(C\) is 9 units.
  1. Determine the exact perimeter of the triangle \(OPS\), where \(O\) is the origin. Give your answer in simplest form. [4]
Edexcel F1 2022 January Q4
8 marks Standard +0.8
The equation $$x^4 + Ax^3 + Bx^2 + Cx + 225 = 0$$ where \(A\), \(B\) and \(C\) are real constants, has
  • a complex root \(4 + 3\text{i}\)
  • a repeated positive real root
  1. Write down the other complex root of this equation. [1]
  2. Hence determine a quadratic factor of \(x^4 + Ax^3 + Bx^2 + Cx + 225\) [2]
  3. Deduce the real root of the equation. [2]
  4. Hence determine the value of each of the constants \(A\), \(B\) and \(C\) [3]
Edexcel F1 2022 January Q5
8 marks Standard +0.3
$$\mathbf{P} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$$ The matrix \(\mathbf{P}\) represents the transformation \(U\)
  1. Give a full description of \(U\) as a single geometrical transformation. [2]
The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf{Q}\), is a reflection in the line \(y = -x\)
  1. Write down the matrix \(\mathbf{Q}\) [1]
The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf{R}\)
  1. Determine the matrix \(\mathbf{R}\) [2]
The transformation \(W\) is represented by the matrix \(3\mathbf{R}\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T'\) The transformation \(W'\) maps the triangle \(T'\) back to the original triangle \(T\)
  1. Determine the matrix that represents \(W'\) [3]
Edexcel F1 2022 January Q6
8 marks Standard +0.8
The quadratic equation $$Ax^2 + 5x - 12 = 0$$ where \(A\) is a constant, has roots \(\alpha\) and \(\beta\)
  1. Write down an expression in terms of \(A\) for
    1. \(\alpha + \beta\)
    2. \(\alpha\beta\)
    [2]
The equation $$4x^2 - 5x + B = 0$$ where \(B\) is a constant, has roots \(\alpha - \frac{3}{\beta}\) and \(\beta - \frac{3}{\alpha}\)
  1. Determine the value of \(A\) [3]
  2. Determine the value of \(B\) [3]
Edexcel F1 2022 January Q7
9 marks Standard +0.8
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The rectangular hyperbola \(H\) has equation \(xy = 36\) The point \(P(4, 9)\) lies on \(H\)
  1. Show, using calculus, that the normal to \(H\) at \(P\) has equation $$4x - 9y + 65 = 0$$ [4]
The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)
  1. Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are rational constants. [5]
Edexcel F1 2022 January Q8
10 marks Standard +0.3
$$f(x) = 2x^{-\frac{2}{3}} + \frac{1}{2}x - \frac{1}{3x - 5} - \frac{5}{2} \quad x \neq \frac{5}{3}$$ The table below shows values of \(f(x)\) for some values of \(x\), with values of \(f(x)\) given to 4 decimal places where appropriate.
\(x\)12345
\(f(x)\)0.5\(-0.2885\)0.5834
  1. Complete the table giving the values to 4 decimal places. [2]
The equation \(f(x) = 0\) has exactly one positive root, \(\alpha\). Using the values in the completed table and explaining your reasoning,
  1. determine an interval of width one that contains \(\alpha\). [2]
  2. Hence use interval bisection twice to obtain an interval of width 0.25 that contains \(\alpha\). [3]
Given also that the equation \(f(x) = 0\) has a negative root, \(\beta\), in the interval \([-1, -0.5]\)
  1. use linear interpolation once on this interval to find an approximation for \(\beta\). Give your answer to 3 significant figures. [3]
Edexcel F1 2022 January Q9
14 marks Standard +0.8
  1. Prove by induction that, for \(n \in \mathbb{N}\) $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$$ [5]
  2. Using the standard summation formulae, show that $$\sum_{r=1}^{n} r(r+1)(r-1) = \frac{1}{4}n(n+A)(n+B)(n+C)$$ where \(A\), \(B\) and \(C\) are constants to be determined. [4]
  3. Determine the value of \(n\) for which $$3\sum_{r=1}^{n} r(r+1)(r-1) = 17\sum_{r=n}^{2n} r^2$$ [5]
Edexcel FP1 Q1
5 marks Moderate -0.3
$$\text{f}(x) = 2x^3 - 8x^2 + 7x - 3.$$ Given that \(x = 3\) is a solution of the equation f\((x) = 0\), solve f\((x) = 0\) completely. [5]