Questions — Edexcel (10514 questions)

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Edexcel C4 Q7
14 marks Standard +0.3
A curve has parametric equations $$x = 3 \cos^2 t, \quad y = \sin 2t, \quad 0 \leq t < \pi.$$
  1. Show that \(\frac{dy}{dx} = -\frac{2}{3} \cot 2t\). [4]
  2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [3]
  3. Show that the tangent to the curve at the point where \(t = \frac{\pi}{6}\) has the equation $$2x + 3\sqrt{3} y = 9.$$ [3]
  4. Find a cartesian equation for the curve in the form \(y^2 = \text{f}(x)\). [4]
Edexcel C4 Q1
8 marks Standard +0.8
A curve has the equation $$2x^2 + xy - y^2 + 18 = 0.$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [8]
Edexcel C4 Q2
8 marks Standard +0.3
Use the substitution \(x = 2\tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]
Edexcel C4 Q3
9 marks Standard +0.3
  1. Show that \((1 + \frac{1}{24})^{-\frac{1}{2}} = k\sqrt{6}\), where \(k\) is rational. [2]
  2. Expand \((1 + \frac{1}{4}x)^{-\frac{1}{2}}\), \(|x| < 2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
  3. Use your answer to part \((b)\) with \(x = \frac{1}{6}\) to find an approximate value for \(\sqrt{6}\), giving your answer to 5 decimal places. [3]
Edexcel C4 Q4
9 marks Standard +0.3
Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.
  1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
  2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]
Edexcel C4 Q5
11 marks Standard +0.3
A curve has parametric equations $$x = \frac{t}{2-t}, \quad y = \frac{1}{1+t}, \quad -1 < t < 2.$$
  1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\left(\frac{2-t}{1+t}\right)^2\). [4]
  2. Find an equation for the normal to the curve at the point where \(t = 1\). [3]
  3. Show that the cartesian equation of the curve can be written in the form $$y = \frac{1+x}{1+3x}.$$ [4]
Edexcel C4 Q6
13 marks Standard +0.8
  1. Find \(\int \tan^2 x \, dx\). [3]
  2. Show that $$\int \tan x \, dx = \ln|\sec x| + c,$$ where \(c\) is an arbitrary constant. [4]
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = x^2 \tan x\). The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{3}\) is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Show that the volume of the solid formed is \(\frac{1}{18}\pi^2(6\sqrt{3} - \pi) - \pi \ln 2\). [6]
Edexcel C4 Q7
17 marks Standard +0.8
\includegraphics{figure_2} Figure 2 shows a hemispherical bowl of radius 5 cm. The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h\) cm and the volume of water in the bowl is \(V\) cm³, where $$V = \frac{1}{3}\pi h^2(15 - h).$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).
  1. Show that $$\frac{dh}{dt} = -\frac{kh(15-h)}{3(10-h)},$$ where \(k\) is a positive constant. [5]
  2. Express \(\frac{3(10-h)}{h(15-h)}\) in partial fractions. [3]
Given that when \(t = 0\), \(h = 5\),
  1. show that $$h^2(15-h) = 250e^{-kt}.$$ [6]
Given also that when \(t = 2\), \(h = 4\),
  1. find the value of \(k\) to 3 significant figures. [3]
Edexcel C4 Q1
5 marks Easy -1.2
  1. Expand \((1 + 4x)^5\) in ascending powers of \(x\) up to and including the term in \(x^5\), simplifying each coefficient. [4]
  2. State the set of values of \(x\) for which your expansion is valid. [1]
Edexcel C4 Q2
6 marks Moderate -0.3
Use the substitution \(u = 1 + \sin x\) to find the value of $$\int_0^{\frac{\pi}{4}} \cos x (1 + \sin x)^3 \, dx.$$ [6]
Edexcel C4 Q3
8 marks Moderate -0.3
  1. Express \(\frac{x+11}{(x+4)(x-3)}\) as a sum of partial fractions. [3]
  2. Evaluate $$\int_0^2 \frac{x+11}{(x+4)(x-3)} \, dx,$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction. [5]
Edexcel C4 Q4
8 marks Challenging +1.2
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 2\sin x + \cosec x\), \(0 < x < \pi\). The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\) is rotated through \(360°\) about the \(x\)-axis. Show that the volume of the solid formed is \(\frac{1}{2}\pi(4\pi + 3\sqrt{3})\). [8]
Edexcel C4 Q5
8 marks Standard +0.3
A curve has the equation $$x^2 - 3xy - y^2 = 12.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
  2. Find an equation for the tangent to the curve at the point \((2, -2)\). [3]
Edexcel C4 Q6
10 marks Standard +0.3
Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} 1 \\ 5 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) respectively. Find, in exact, simplified form,
  1. the cosine of \(\angle AOB\), [4]
  2. the area of triangle \(OAB\), [4]
  3. the shortest distance from \(A\) to the line \(OB\). [2]
Edexcel C4 Q7
14 marks Standard +0.8
A curve has parametric equations $$x = t(t - 1), \quad y = \frac{4t}{1-t}, \quad t \neq 1.$$
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [4]
The point \(P\) on the curve has parameter \(t = -1\).
  1. Show that the tangent to the curve at \(P\) has the equation $$x + 3y + 4 = 0.$$ [3]
The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
  1. Find the coordinates of \(Q\). [7]
Edexcel C4 Q8
16 marks Standard +0.3
An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.
  1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]
Given that after one week there are 360 insects in the colony,
  1. find the value of \(k\) to 3 significant figures. [2]
Given also that after two and three weeks there are 440 and 600 insects respectively,
  1. comment on suitability of the model. [2]
An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25\cos 0.5t).$$
  1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [4]
  2. Compare the suitability of the two models. [3]
Edexcel C4 Q1
6 marks Standard +0.3
The region bounded by the curve \(y = x^2 - 2x\) and the \(x\)-axis is rotated through \(2\pi\) radians about the \(x\)-axis. Find the volume of the solid formed, giving your answer in terms of \(\pi\). [6]
Edexcel C4 Q2
6 marks Standard +0.3
Use the substitution \(u = 1 - x^2\) to find $$\int \frac{1}{1-x^2} \, dx.$$ [6]
Edexcel C4 Q3
8 marks Standard +0.3
A curve has the equation $$2 \sin 2x - \tan y = 0.$$
  1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [5]
  2. Show that the tangent to the curve at the point \(\left(\frac{\pi}{6}, \frac{\pi}{3}\right)\) has the equation $$y = \frac{1}{2}x + \frac{\pi}{4}.$$ [3]
Edexcel C4 Q4
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
  1. Show that the area of triangle \(OAB\) is \(a^2\). [6]
Edexcel C4 Q5
9 marks Standard +0.3
The gradient at any point \((x, y)\) on a curve is proportional to \(\sqrt{y}\). Given that the curve passes through the point with coordinates \((0, 4)\),
  1. show that the equation of the curve can be written in the form $$2\sqrt{y} = kx + 4,$$ where \(k\) is a positive constant. [5]
Given also that the curve passes through the point with coordinates \((2, 9)\),
  1. find the equation of the curve in the form \(y = \text{f}(x)\). [4]
Edexcel C4 Q6
10 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a vertical cross-section of a vase. The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60°\). When the depth of water in the vase is \(h\) cm, the volume of water in the vase is \(V\) cm\(^3\).
  1. Show that \(V = \frac{1}{9}\pi h^3\). [3]
The vase is initially empty and water is poured in at a constant rate of 120 cm\(^3\) s\(^{-1}\).
  1. Find, to 2 decimal places, the rate at which \(h\) is increasing
    1. when \(h = 6\),
    2. after water has been poured in for 8 seconds. [7]
Edexcel C4 Q7
13 marks Standard +0.3
Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix}\) respectively.
  1. Find a vector equation for the line \(l_1\) which passes through \(A\) and \(B\). [2]
The line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -7 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.$$
  1. Show that lines \(l_1\) and \(l_2\) do not intersect. [5]
  2. Find the position vector of the point \(C\) on \(l_2\) such that \(\angle ABC = 90°\). [6]
Edexcel C4 Q8
14 marks Standard +0.3
$$\text{f}(x) = \frac{x(3x-7)}{(1-x)(1-3x)}, \quad |x| < \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = A + \frac{B}{1-x} + \frac{C}{1-3x}.$$ [4]
  2. Evaluate $$\int_0^{\frac{1}{4}} \text{f}(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational. [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
Edexcel C4 Q1
8 marks Moderate -0.3
The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$
  1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
  2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
  3. Explain why this model would not be appropriate for large values of \(t\). [1]