Edexcel F3 (Further Pure Mathematics 3) 2018 Specimen

The curve \(C\) has equation $$y = 9 \cosh x + 3 \sinh x + 7x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm. [6]

An ellipse has equation $$\frac{x^2}{25} + \frac{y^2}{4} = 1$$ The point \(P\) lies on the ellipse and has coordinates \((5\cos \theta, 2\sin \theta)\), \(0 < \theta < \frac{\pi}{2}\) The line \(L\) is a normal to the ellipse at the point \(P\).

1. Show that an equation for \(L\) is $$5x \sin \theta - 2y \cos \theta = 21 \sin \theta \cos \theta$$ [5]

Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(PQ\),

1. find the exact area of triangle \(OPM\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2\theta\) [6]

Without using a calculator, find

1. \(\int_{-2}^{1} \frac{1}{x^2 + 4x + 13} \, dx\), giving your answer as a multiple of \(\pi\), [5]
2. \(\int_{-1}^{4} \frac{1}{\sqrt{4x^2 - 12x + 34}} \, dx\), giving your answer in the form \(p \ln\left(q + r\sqrt{2}\right)\), [7]

where \(p\), \(q\) and \(r\) are rational numbers to be found.

$$\mathbf{M} = \begin{pmatrix} 1 & k & 0 \\ -1 & 1 & 1 \\ 1 & k & 3 \end{pmatrix}, \text{ where } k \text{ is a constant}$$

1. Find \(\mathbf{M}^{-1}\) in terms of \(k\). [5]

Hence, given that \(k = 0\)

1. find the matrix \(\mathbf{N}\) such that $$\mathbf{MN} = \begin{pmatrix} 3 & 5 & 6 \\ 4 & -1 & 1 \\ 3 & 2 & -3 \end{pmatrix}$$ [4]

Given that \(y = \text{artanh}(\cos x)\)

1. show that $$\frac{dy}{dx} = -\text{cosec } x$$ [2]
2. Hence find the exact value of $$\int_{0}^{\frac{\pi}{4}} \cos x \, \text{artanh}(\cos x) \, dx$$ giving your answer in the form \(a \ln\left(b + c\sqrt{3}\right) + d\pi\), where \(a\), \(b\), \(c\) and \(d\) are rational numbers to be found. [5]

The coordinates of the points \(A\), \(B\) and \(C\) relative to a fixed origin \(O\) are \((1, 2, 3)\), \((-1, 3, 4)\) and \((2, 1, 6)\) respectively. The plane \(\Pi\) contains the points \(A\), \(B\) and \(C\).

1. Find a cartesian equation of the plane \(\Pi\). [5]

The point \(D\) has coordinates \((k, 4, 14)\) where \(k\) is a positive constant. Given that the volume of the tetrahedron \(ABCD\) is 6 cubic units,

1. find the value of \(k\). [4]

The curve \(C\) has parametric equations $$x = 3t^4, \quad y = 4t^3, \quad 0 \leq t \leq 1$$ The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).

1. Show that $$S = k\pi \int_{0}^{1} t^2(t^2 + 1)^{\frac{1}{2}} dt$$ where \(k\) is a constant to be found. [4]
2. Use the substitution \(u^2 = t^2 + 1\) to find the value of \(S\), giving your answer in the form \(p\pi\left(11\sqrt{2} - 4\right)\) where \(p\) is a rational number to be found. [7]

$$I_n = \int_{0}^{\ln 2} \tanh^{2n} x \, dx, \quad n \geq 0$$

1. Show that, for \(n \geq 1\) $$I_n = I_{n-1} - \frac{1}{2n-1}\left(\frac{3}{5}\right)^{2n-1}$$ [5]
2. Hence show that $$\int_{0}^{\ln 2} \tanh^{-1} x \, dx = p + \ln 2$$ where \(p\) is a rational number to be found. [5]