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OCR MEI C4 2014 June Q4
8 marks Moderate -0.3
  1. Show that \(\cos(\alpha + \beta) = \frac{1 - \tan \alpha \tan \beta}{\sec \alpha \sec \beta}\). [3]
  2. Hence show that \(\cos 2\alpha = \frac{1 - \tan^2 \alpha}{1 + \tan^2 \alpha}\). [2]
  3. Hence or otherwise solve the equation \(\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \frac{1}{2}\) for \(0° \leqslant \theta \leqslant 180°\). [3]
OCR MEI C4 2014 June Q5
7 marks Standard +0.3
A curve has parametric equations \(x = e^{2t}, y = te^{2t}\).
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\). [4]
  2. Find the cartesian equation of the curve in the form \(y = ax^b \ln x\), where \(a\) and \(b\) are constants to be determined. [3]
OCR MEI C4 2014 June Q6
6 marks Standard +0.8
Fig. 6 shows the region enclosed by the curve \(y = (1 + 2x^2)^{\frac{1}{2}}\) and the line \(y = 2\). \includegraphics{figure_6} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\). [6]
OCR MEI C4 2014 June Q7
18 marks Standard +0.3
Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}
  1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
    1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
    2. Hence find the equation of this plane. [2]
  3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]
The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).
  1. Find the volume of the tetrahedron ABCD. [2]
OCR MEI C4 2014 June Q8
18 marks Standard +0.8
Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel. \includegraphics{figure_8.1} The height of the water surface above the hole \(t\) seconds after opening the hole is \(h\) metres, where $$\frac{dh}{dt} = -A\sqrt{h}$$ and where \(A\) is a positive constant. Initially the water surface is 1 metre above the hole.
  1. Verify that the solution to this differential equation is $$h = \left(1 - \frac{1}{2}At\right)^2.$$ [3]
The water stops leaking when \(h = 0\). This occurs after 20 seconds.
  1. Find the value of \(A\), and the time when the height of the water surface above the hole is 0.5 m. [4]
Fig. 8.2 shows a similar situation with a different barrel; \(h\) is in metres. \includegraphics{figure_8.2} For this barrel, $$\frac{dh}{dt} = -B\frac{\sqrt{h}}{(1+h)^2},$$ where \(B\) is a positive constant. When \(t = 0\), \(h = 1\).
  1. Solve this differential equation, and hence show that $$h^{\frac{1}{2}}(30 + 20h + 6h^2) = 56 - 15Bt.$$ [7]
  2. Given that \(h = 0\) when \(t = 20\), find \(B\). Find also the time when the height of the water surface above the hole is 0.5 m. [4]
Edexcel C4 Q1
6 marks Moderate -0.3
  1. Find the binomial expansion of \((2 - 3x)^{-3}\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
  2. State the set of values of \(x\) for which your expansion is valid. [1]
Edexcel C4 Q2
8 marks Standard +0.3
A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
  2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]
Edexcel C4 Q3
10 marks Standard +0.3
  1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that $$\frac{2x^3 - 5x^2 + 6}{x^2 - 3x} \equiv Ax + B + \frac{C}{x} + \frac{D}{x-3}.$$ [5]
  2. Evaluate $$\int_1^2 \frac{2x^3 - 5x^2 + 6}{x^2 - 3x} \, dx,$$ giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers. [5]
Edexcel C4 Q4
12 marks Standard +0.3
A mathematician is selling goods at a car boot sale. She believes that the rate at which she makes sales depends on the length of time since the start of the sale, \(t\) hours, and the total value of sales she has made up to that time, £\(x\). She uses the model $$\frac{dx}{dt} = \frac{k(5-t)}{x},$$ where \(k\) is a constant. Given that after two hours she has made sales of £96 in total,
  1. solve the differential equation and show that she made £72 in the first hour of the sale. [8]
The mathematician believes that is it not worth staying at the sale once she is making sales at a rate of less than £10 per hour.
  1. Verify that at 3 hours and 5 minutes after the start of the sale, she should have already left. [4]
Edexcel C4 Q5
12 marks Standard +0.3
Relative to a fixed origin, two lines have the equations $$\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}$$ and $$\mathbf{r} = \begin{pmatrix} -3 \\ 1 \\ -6 \end{pmatrix} + t \begin{pmatrix} 3 \\ a \\ b \end{pmatrix},$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters. Given that the two lines are perpendicular,
  1. find a linear relationship between \(a\) and \(b\). [2]
Given also that the two lines intersect,
  1. find the values of \(a\) and \(b\), [8]
  2. find the coordinates of the point where they intersect. [2]
Edexcel C4 Q6
13 marks Standard +0.8
\includegraphics{figure_6} Figure 1 shows the curve with equation \(y = x\sqrt{1-x}\), \(0 \leq x \leq 1\).
  1. Use the substitution \(u^2 = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac{8}{15}\). [8]
  2. Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360°\) about the \(x\)-axis. [5]
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]