Questions — Edexcel (10514 questions)

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Edexcel C4 2015 June Q6
8 marks Standard +0.8
\includegraphics{figure_2} Figure 2 shows a sketch of the curve with equation \(y = \sqrt{(3-x)(x+1)}\), \(0 \leqslant x \leqslant 3\) The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis, and the \(y\)-axis.
  1. Use the substitution \(x = 1 + 2\sin\theta\) to show that $$\int_0^3 \sqrt{(3-x)(x+1)} dx = k \int_{-\frac{\pi}{6}}^{\frac{\pi}{2}} \cos^2\theta d\theta$$ where \(k\) is a constant to be determined. [5]
  2. Hence find, by integration, the exact area of \(R\). [3]
Edexcel C4 2015 June Q7
13 marks Standard +0.8
  1. Express \(\frac{2}{P(P-2)}\) in partial fractions. [3]
A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{2}P(P-2)\cos 2t, \quad t \geqslant 0$$ where \(P\) is the population in thousands, and \(t\) is the time measured in years since the start of the study. Given that \(P = 3\) when \(t = 0\),
  1. solve this differential equation to show that $$P = \frac{6}{3 - e^{\frac{1}{2}\sin 2t}}$$ [7]
  2. find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures. [3]
Edexcel C4 2015 June Q8
10 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3^x$$ The point \(P\) lies on \(C\) and has coordinates \((2, 9)\). The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).
  1. Find the exact value of the \(x\) coordinate of \(Q\). [4]
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360°\) about the \(x\)-axis.
  1. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are exact constants. [You may assume the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] [6]
Edexcel C4 Q1
6 marks Standard +0.3
Use the substitution \(u = 4 + 3x^2\) to find the exact value of $$\int_0^2 \frac{2x}{(4 + 3x^2)^2} \, dx .$$ [6]
Edexcel C4 Q2
8 marks Standard +0.3
A curve has equation $$x^3 - 2xy - 4x + y^3 - 51 = 0.$$ Find an equation of the normal to the curve at the point \((4, 3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]
Edexcel C4 Q3
13 marks Standard +0.3
$$f(x) = \frac{1 + 14x}{(1 - x)(1 + 2x)}, \quad |x| < \frac{1}{2}.$$
  1. Express \(f(x)\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_{-\frac{1}{6}}^{\frac{1}{4}} f(x) \, dx\), giving your answer in the form \(\ln p\), where \(p\) is rational. [5]
  3. Use the binomial theorem to expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\), simplifying each term. [5]
Edexcel C4 Q4
10 marks Standard +0.3
The line \(l_1\) has vector equation \(\mathbf{r} = \begin{pmatrix} 11 \\ 5 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix}\), where \(\lambda\) is a parameter. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 24 \\ 4 \\ 13 \end{pmatrix} + \mu \begin{pmatrix} 7 \\ 1 \\ 5 \end{pmatrix}\), where \(\mu\) is a parameter.
  1. Show that the lines \(l_1\) and \(l_2\) intersect. [4]
  2. Find the coordinates of their point of intersection. [2]
Given that \(\theta\) is the acute angle between \(l_1\) and \(l_2\),
  1. find the value of \(\cos \theta\). Give your answer in the form \(k\sqrt{3}\), where \(k\) is a simplified fraction. [4]
Edexcel C4 Q5
12 marks Standard +0.3
\includegraphics{figure_1} The curve shown in Fig. 1 has parametric equations $$x = \cos t, \quad y = \sin 2t, \quad 0 \leq t < 2\pi.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [3]
  2. Find the values of the parameter \(t\) at the points where \(\frac{dy}{dx} = 0\). [3]
  3. Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the \(x\)-axis. [2]
  4. Show that a cartesian equation for the part of the curve where \(0 \leq t < \pi\) is $$y = 2x\sqrt{(1 - x^2)}.$$ [3]
  5. Write down a cartesian equation for the part of the curve where \(\pi \leq t < 2\pi\). [1]
Edexcel C4 Q6
13 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the curve with equation $$y = x^2 \sin\left(\frac{1}{2}x\right), \quad 0 < x \leq 2\pi.$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.
  1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). [7]
The table shows corresponding values of \(x\) and \(y\).
\(x\)\(\pi\)\(\frac{5\pi}{4}\)\(\frac{3\pi}{2}\)\(\frac{7\pi}{4}\)\(2\pi\)
\(y\)\(9.8696\)\(14.247\)\(15.702\)\(G\)\(0\)
  1. Find the value of \(G\). [1]
  2. Use the trapezium rule with values of \(x^2 \sin\left(\frac{1}{2}x\right)\)
    1. at \(x = \pi\), \(x = \frac{3\pi}{2}\) and \(x = 2\pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
    2. at \(x = \pi\), \(x = \frac{5\pi}{4}\), \(x = \frac{3\pi}{2}\), \(x = \frac{7\pi}{4}\) and \(x = 2\pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
    [5]
Edexcel C4 Q7
13 marks Standard +0.3
In an experiment a scientist considered the loss of mass of a collection of picked leaves. The mass \(M\) grams of a single leaf was measured at times \(t\) days after the leaf was picked. The scientist attempted to find a relationship between \(M\) and \(t\). In a preliminary model she assumed that the rate of loss of mass was proportional to the mass \(M\) grams of the leaf.
  1. Write down a differential equation for the rate of change of mass of the leaf, using this model. [2]
  2. Show, by differentiation, that \(M = 10(0.98)^t\) satisfies this differential equation. [2]
Further studies implied that the mass \(M\) grams of a certain leaf satisfied a modified differential equation $$10 \frac{dM}{dt} = -k(10M - 1), \quad (1)$$ where \(k\) is a positive constant and \(t \geq 0\). Given that the mass of this leaf at time \(t = 0\) is 10 grams, and that its mass at time \(t = 10\) is 8.5 grams,
  1. solve the modified differential equation (1) to find the mass of this leaf at time \(t = 15\). [9]
Edexcel C4 Q1
6 marks Moderate -0.8
A measure of the effective voltage, \(M\) volts, in an electrical circuit is given by $$M^2 = \int_0^1 V^2 \, dt$$ where \(V\) volts is the voltage at time \(t\) seconds. Pairs of values of \(V\) and \(t\) are given in the following table.
\(t\)00.250.50.751
\(V\)-4820737-161-29
\(V^2\)
Use the trapezium rule with five values of \(V^2\) to estimate the value of \(M\). [6]
Edexcel C4 Q2
7 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = x^2 + 3\). The shaded region is bounded by \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 3\). The shaded region is rotated \(360°\) about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer as an exact multiple of \(\pi\). [7]
Edexcel C4 Q3
6 marks Moderate -0.8
  1. Given that \(a^x = e^{kx}\), where \(a\) and \(k\) are constants, \(a > 0\) and \(x \in \mathbb{R}\), prove that \(k = \ln a\). [2]
  2. Hence, using the derivative of \(e^{kx}\), prove that when \(y = 2^x\), $$\frac{dy}{dx} = 2^x \ln 2.$$ [2]
  3. Hence deduce that the gradient of the curve with equation \(y = 2^x\) at the point \((2, 4)\) is \(\ln 16\). [2]
Edexcel C4 Q4
8 marks Moderate -0.3
$$f(x) = (1 + 3x)^{-1}, \quad |x| < \frac{1}{3}.$$
  1. Expand \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\). [3]
  2. Hence show that, for small \(x\), $$\frac{1 + x}{1 + 3x} \approx 1 - 2x + 6x^2 - 18x^3.$$ [2]
  3. Taking a suitable value for \(x\), which should be stated, use the series expansion in part \((b)\) to find an approximate value for \(\frac{101}{103}\), giving your answer to 5 decimal places. [3]
Edexcel C4 Q5
11 marks Standard +0.3
  1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2}\ln 2.$$ [6]
\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Find the volume of the solid of revolution generated. [2]
  2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]
Edexcel C4 Q6
12 marks Standard +0.3
Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are $$\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})$$ and \(\mathbf{r} = 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k})\), where \(\lambda\) and \(\mu\) are scalars.
  1. Show that the submarines are moving in perpendicular directions. [2]
  2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]
The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).
  1. Show that only one of the submarines passes through the point \(B\). [3]
  2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]
Edexcel C4 Q7
13 marks Standard +0.3
In a chemical reaction two substances combine to form a third substance. At time \(t\), \(t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation $$\frac{dx}{dt} = k(1 - 2x)(1 - 4x), \text{ where } k \text{ is a positive constant.}$$
  1. Solve this differential equation and hence show that $$\ln\left|\frac{1 - 2x}{1 - 4x}\right| = 2kt + c, \text{ where } c \text{ is an arbitrary constant.}$$ [7]
  2. Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\). [4]
  3. Find the limiting value of the concentration \(x\) as \(t\) becomes very large. [2]
Edexcel C4 Q8
15 marks Standard +0.3
\includegraphics{figure_2} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(ABCD\) is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t, \quad y = 9 \sin 2t, \quad 0 \leq t < 2\pi.$$
  1. Find the cartesian equation of the curve in the form \(y^2 = f(x)\). [4]
  2. Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by $$\int_0^{\frac{\pi}{2}} A \sin 2t \sin t \, dt, \text{ stating the value of the constant } A.$$ [3]
  3. Find the value of this integral. [4]
The sides of the rectangle \(ABCD\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm,
  1. find the total area of the red glass. [4]
Edexcel C4 Q9
6 marks Moderate -0.3
The following is a table of values for \(y = \sqrt{1 + \sin x}\), where \(x\) is in radians.
\(x\)00.511.52
\(y\)11.216\(p\)1.413\(q\)
  1. Find the value of \(p\) and the value of \(q\). [2]
  2. Use the trapezium rule and all the values of \(y\) in the completed table to obtain an estimate of \(I\), where $$I = \int_0^2 \sqrt{1 + \sin x} \, dx.$$ [4]
Edexcel C4 Q10
8 marks Standard +0.3
\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]
Edexcel C4 Q11
8 marks Moderate -0.3
Given that $$\frac{10(2 - 3x)}{(1 - 2x)(2 + x)} \equiv \frac{A}{1 - 2x} + \frac{B}{2 + x},$$
  1. find the values of the constants \(A\) and \(B\). [3]
  2. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x^3\), of \(\frac{10(2 - 3x)}{(1 - 2x)(2 + x)}\), for \(|x| < \frac{1}{2}\). [5]
Edexcel C4 Q12
12 marks Moderate -0.8
A radioactive isotope decays in such a way that the rate of change of the number \(N\) of radioactive atoms present after \(t\) days, is proportional to \(N\).
  1. Write down a differential equation relating \(N\) and \(t\). [2]
  2. Show that the general solution may be written as \(N = Ae^{-kt}\), where \(A\) and \(k\) are positive constants. [5]
Initially the number of radioactive atoms present is \(7 \times 10^{18}\) and 8 days later the number present is \(3 \times 10^{17}\).
  1. Find the value of \(k\). [3]
  2. Find the number of radioactive atoms present after a further 8 days. [2]
Edexcel C4 Q13
12 marks Standard +0.3
Relative to a fixed origin \(O\), the point \(A\) has position vector \(4\mathbf{i} + 8\mathbf{j} - \mathbf{k}\), and the point \(B\) has position vector \(7\mathbf{i} + 14\mathbf{j} + 5\mathbf{k}\).
  1. Find the vector \(\overrightarrow{AB}\). [1]
  2. Calculate the cosine of \(\angle OAB\). [3]
  3. Show that, for all values of \(\lambda\), the point \(P\) with position vector $$\lambda\mathbf{i} + 2\lambda\mathbf{j} + (2\lambda - 9)\mathbf{k}$$ lies on the line through \(A\) and \(B\). [3]
  4. Find the value of \(\lambda\) for which \(OP\) is perpendicular to \(AB\). [3]
  5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(AB\). [2]
Edexcel C4 Q14
12 marks Standard +0.3
  1. Use integration by parts to find the exact value of \(\int_1^3 x^2 \ln x \, dx\). [6]
  2. Use the substitution \(x = \sin \theta\) to show that, for \(|x| \leq 1\), $$\int \frac{1}{(1 - x^2)^{\frac{3}{2}}} \, dx = \frac{x}{(1 - x^2)^{\frac{1}{2}}} + c, \text{ where } c \text{ is an arbitrary constant.}$$ [6]
Edexcel C4 Q15
15 marks Challenging +1.2
\includegraphics{figure_1} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta, \quad y = 4 \sin \theta, \quad 0 \leq \theta < 2\pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where \(\theta = \alpha\), \(\theta = -\alpha\), \(\theta = \pi - \alpha\), \(\theta = -\pi + \alpha\).
  1. Find an equation of the tangent to the ellipse at \((5 \cos \alpha, 4 \sin \alpha)\), and show that it can be written in the form $$5y \sin \alpha + 4x \cos \alpha = 20.$$ [4]
  2. Find by integration the area enclosed by the ellipse. [4]
  3. Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac{80}{\sin 2\alpha} - 20\pi.$$ [4]
  4. Given that \(0 < \alpha < \frac{\pi}{4}\), find the value of \(\alpha\) for which the areas of two types of wood are equal. [3]