Questions — Edexcel (10514 questions)

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Edexcel P4 2022 October Q8
4 marks Standard +0.8
A student was asked to prove by contradiction that "there are no positive integers \(x\) and \(y\) such that \(3x^2 + 2xy - y^2 = 25\)" The start of the student's proof is shown in the box below.
\fbox{\begin{minipage}{0.8\textwidth} Assume that integers \(x\) and \(y\) exist such that \(3x^2 + 2xy - y^2 = 25\) \(\Rightarrow (3x - y)(x + y) = 25\) If \((3x - y) = 1\) and \((x + y) = 25\) $3x - y = 1
x + y = 25\( \)\Rightarrow 4x = 26 \Rightarrow x = 6.5, y = 18.5$ Not integers \end{minipage}}
Show the calculations and statements that are needed to complete the proof. [4]
Edexcel P4 2022 October Q9
5 marks Standard +0.3
With respect to a fixed origin \(O\), the equations of lines \(l_1\) and \(l_2\) are given by $$l_1: \mathbf{r} = \begin{pmatrix} 2 \\ 8 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} -4 \\ -1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 4 \\ 8 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters. Prove that lines \(l_1\) and \(l_2\) are skew. [5]
Edexcel P4 2022 October Q10
8 marks Standard +0.3
A spherical ball of ice of radius 12 cm is placed in a bucket of water. In a model of the situation, • the ball remains spherical as it melts • \(t\) minutes after the ball of ice is placed in the bucket, its radius is \(r\) cm • the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius • the radius of the ball of ice is 6 cm after 15 minutes Using the model and the information given,
  1. find an equation linking \(r\) and \(t\), [5]
  2. find the time taken for the ball of ice to melt completely, [2]
  3. On Diagram 1 on page 27, sketch a graph of \(r\) against \(t\). [1]
Edexcel P4 2022 October Q11
9 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a sketch of the closed curve with equation $$(x + y)^3 + 10y^2 = 108x$$
  1. Show that $$\frac{dy}{dx} = \frac{108 - 3(x + y)^2}{20y + 3(x + y)^2}$$ [5]
The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km. The points \(P\) and \(Q\) represent points that are furthest north and furthest south of the origin \(O\), as shown in Figure 4. Using the result given in part (a),
  1. find how far the point \(Q\) is south of \(O\). Give your answer to the nearest 100 m. [4]
Edexcel C4 Q1
5 marks Moderate -0.3
Use the binomial theorem to expand $$\sqrt{(4-9x)}, \quad |x| < \frac{4}{9},$$ in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying each term. [5]
Edexcel C4 Q2
7 marks Standard +0.3
A curve has equation $$x^2 + 2xy - 3y^2 + 16 = 0.$$ Find the coordinates of the points on the curve where \(\frac{dy}{dx} = 0\). [7]
Edexcel C4 Q3
8 marks Moderate -0.3
  1. Express \(\frac{5x + 3}{(2x - 3)(x + 2)}\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_0^1 \frac{5x + 3}{(2x - 3)(x + 2)} dx\), giving your answer as a single logarithm. [5]
Edexcel C4 Q4
7 marks Challenging +1.2
Use the substitution \(x = \sin \theta\) to find the exact value of $$\int_0^1 \frac{1}{(1-x^2)^{3/2}} dx.$$ [7]
Edexcel C4 Q5
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the graph of the curve with equation $$y = xe^x, \quad x \geq 0.$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
  1. Use integration to find the exact value of the area for \(R\). [5]
  2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and \(0.8\).
    \(x\)00.20.40.60.8
    \(y = xe^x\)00.298361.99207
    [1]
  3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures. [4]
Edexcel C4 Q6
12 marks Standard +0.3
A curve has parametric equations $$x = 2\cot t, \quad y = 2\sin^2 t, \quad 0 < t \leq \frac{\pi}{2}.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [4]
  2. Find an equation of the tangent to the curve at the point where \(t = \frac{\pi}{4}\). [4]
  3. Find a cartesian equation of the curve in the form \(y = f(x)\). State the domain on which the curve is defined. [4]
Edexcel C4 Q7
13 marks Standard +0.3
The line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$$ and the line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},$$ where \(\lambda\) and \(\mu\) are parameters. The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).
  1. Find the coordinates of \(B\). [4]
  2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. [4]
The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\). The point \(D\) is such that \(ABCD\) is a parallelogram.
  1. Show that \(|\overrightarrow{AB}| = |\overrightarrow{BC}|\). [3]
  2. Find the position vector of the point \(D\). [2]
Edexcel C4 Q8
13 marks Standard +0.3
Liquid is pouring into a container at a constant rate of \(20\text{ cm}^3\text{s}^{-1}\) and is leaking out at a rate proportional to the volume of the liquid already in the container.
  1. Explain why, at time \(t\) seconds, the volume, \(V\text{ cm}^3\), of liquid in the container satisfies the differential equation $$\frac{dV}{dt} = 20 - kV,$$ where \(k\) is a positive constant. [2]
The container is initially empty.
  1. By solving the differential equation, show that $$V = A + Be^{-kt},$$ giving the values of \(A\) and \(B\) in terms of \(k\). [6]
Given also that \(\frac{dV}{dt} = 10\) when \(t = 5\),
  1. find the volume of liquid in the container at 10 s after the start. [5]
Edexcel C4 2013 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$\sqrt{(9 + 8x)}, \quad |x| < \frac{9}{8}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [5]
  2. Use your expansion to estimate the value of \(\sqrt{11}\), giving your answer as a single fraction. [3]
Edexcel C4 2013 June Q2
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = xe^{-\frac{1}{2}x}\), \(x > 0\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = xe^{-\frac{1}{2}x}\).
\(x\)01234
\(y\)0\(e^{-\frac{1}{2}}\)\(3e^{-\frac{3}{2}}\)\(4e^{-2}\)
  1. Complete the table with the value of \(y\) corresponding to \(x = 2\) [1]
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places. [4]
    1. Find \(\int xe^{-\frac{1}{2}x} \, dx\).
    2. Hence find the exact area of \(R\), giving your answer in the form \(a + be^{-2}\), where \(a\) and \(b\) are integers. [6]
Edexcel C4 2013 June Q3
7 marks Moderate -0.3
A curve \(C\) has parametric equations $$x = 2t + 5, \quad y = 3 + \frac{4}{t}, \quad t \neq 0$$
  1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) with coordinates \((9, 5)\). [4]
  2. Find a cartesian equation of the curve in the form $$y = \frac{ax + b}{cx + d}$$ where \(a\), \(b\), \(c\) and \(d\) are integers. [3]
Edexcel C4 2013 June Q4
10 marks Moderate -0.3
With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).
  1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
  2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
  3. Find a vector equation of \(l_2\). [2]
  4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
  5. find the length of \(OX\), giving your answer to 3 significant figures. [3]
Edexcel C4 2013 June Q5
9 marks Standard +0.3
The curve \(C\) has the equation $$\sin(\pi y) - y - x^2 y = -5, \quad x > 0$$
  1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5] The point \(P\) with coordinates \((2, 1)\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
  2. Find the exact value of the \(x\)-coordinate of \(A\). [4]
Edexcel C4 2013 June Q6
11 marks Moderate -0.3
    1. Express \(\frac{7x}{(x + 3)(2x - 1)}\) in partial fractions. [3]
    2. Given that \(x > \frac{1}{2}\), find $$\int \frac{7x}{(x + 3)(2x - 1)} \, dx$$ [3]
  2. Using the substitution \(u^3 = x\), or otherwise, find $$\int \frac{1}{x + x^3} \, dx, \quad x > 0$$ [5]
Edexcel C4 2013 June Q7
10 marks Challenging +1.2
\includegraphics{figure_2} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan \theta, \quad y = 1 + 2\cos 2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The curve \(C\) crosses the \(x\)-axis at \((\sqrt{3}, 0)\). The finite shaded region \(S\) shown in Figure 2 is bounded by \(C\), the line \(x = 1\) and the \(x\)-axis. This shaded region is rotated through \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. Show that the volume of the solid of revolution formed is given by the integral $$k \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (16 \cos^2 \theta - 8 + \sec^2 \theta) \, d\theta$$ where \(k\) is a constant. [5]
  2. Hence, use integration to find the exact value for this volume. [5]
Edexcel C4 2013 June Q8
9 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm. At time \(t\) minutes, the depth of liquid in the tank is \(h\) centimetres. The liquid leaks from a hole \(P\) at the bottom of the tank. The liquid leaks from the tank at a rate of \(32\pi \sqrt{h}\) cm\(^3\) min\(^{-1}\).
  1. Show that at time \(t\) minutes, the height \(h\) cm of liquid in the tank satisfies the differential equation $$\frac{dh}{dt} = -0.02\sqrt{h}$$ [4]
  2. Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm. [5]
Edexcel C4 2015 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$(4 + 5x)^{\frac{1}{2}}, \quad |x| < \frac{4}{5}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient in its simplest form. [5]
  2. Find the exact value of \((4 + 5x)^{\frac{1}{2}}\) when \(x = \frac{1}{10}\) Give your answer in the form \(k\sqrt{2}\), where \(k\) is a constant to be determined. [1]
  3. Substitute \(x = \frac{1}{10}\) into your binomial expansion from part (a) and hence find an approximate value for \(\sqrt{2}\) Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers. [2]
Edexcel C4 2015 June Q2
11 marks Standard +0.3
The curve \(C\) has equation $$x^2 - 3xy - 4y^2 + 64 = 0$$
  1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
  2. Find the coordinates of the points on \(C\) where \(\frac{dy}{dx} = 0\) (Solutions based entirely on graphical or numerical methods are not acceptable.) [6]
Edexcel C4 2015 June Q3
8 marks Challenging +1.2
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = 4x - xe^{\frac{1}{x}}, x \geqslant 0\) The curve meets the \(x\)-axis at the origin \(O\) and cuts the \(x\)-axis at the point \(A\).
  1. Find, in terms of \(\ln 2\), the \(x\) coordinate of the point \(A\). [2]
  2. Find $$\int xe^{\frac{1}{x}} dx$$ [3]
  3. Find, by integration, the exact value for the area of \(R\). Give your answer in terms of \(\ln 2\) [3]
The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis and the curve with equation $$y = 4x - xe^{\frac{1}{x}}, x \geqslant 0$$
Edexcel C4 2015 June Q4
11 marks Standard +0.3
With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = \begin{pmatrix} 5 \\ -3 \\ p \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ -3 \end{pmatrix}, \quad l_2: \mathbf{r} = \begin{pmatrix} 8 \\ 5 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l_1\) and \(l_2\) intersect at the point \(A\).
  1. Find the coordinates of \(A\). [2]
  2. Find the value of the constant \(p\). [3]
  3. Find the acute angle between \(l_1\) and \(l_2\), giving your answer in degrees to 2 decimal places. [3]
The point \(B\) lies on \(l_2\) where \(\mu = 1\)
  1. Find the shortest distance from the point \(B\) to the line \(l_1\), giving your answer to 3 significant figures. [3]
Edexcel C4 2015 June Q5
6 marks Moderate -0.3
A curve \(C\) has parametric equations $$x = 4t + 3, \quad y = 4t + 8 + \frac{5}{2t}, \quad t \neq 0$$
  1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) where \(t = 2\), giving your answer as a fraction in its simplest form. [3]
  2. Show that the cartesian equation of the curve \(C\) can be written in the form $$y = \frac{x^2 + ax + b}{x - 3}, \quad x \neq 3$$ where \(a\) and \(b\) are integers to be determined. [3]