Edexcel FP3 (Further Pure Mathematics 3) 2014 June

The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find

1. a vector equation of the line \(l\), [2]
2. the position vector of the point where \(l\) meets \(\Pi\). [4]
3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]

$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$

1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]

The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).

1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]

Using calculus, find the exact value of

1. \(\int_1^2 \frac{1}{\sqrt{x^2 - 2x + 3}} \, dx\) [4]
2. \(\int_0^1 e^{-x} \sinh x \, dx\) [4]

Using the definitions of hyperbolic functions in terms of exponentials,

1. show that $$\operatorname{sech}^2 x = 1 - \tanh^2 x$$ [3]
2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$ [4]

Given that \(y = \arctan \frac{x}{\sqrt{1 + x^2}}\) show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2}}\) [4]

[In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.] The points \(P(3\cos \alpha, 2\sin \alpha)\) and \(Q(3\cos \beta, 2\sin \beta)\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$

1. Show the equation of the chord \(PQ\) is $$\frac{x}{3}\cos\frac{(\alpha + \beta)}{2} + \frac{y}{2}\sin\frac{(\alpha + \beta)}{2} = \cos\frac{(\alpha - \beta)}{2}$$ [4]
2. Write down the coordinates of the mid-point of \(PQ\). [1]

Given that the gradient, \(m\), of the chord \(PQ\) is a constant,

1. show that the centre of the chord lies on a line $$y = -kx$$ expressing \(k\) in terms of \(m\). [5]

A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x^2 + y^2 = r^2\) where \(r\) is a constant.

1. Show that \(1 + \left(\frac{dy}{dx}\right)^2 = \frac{r^2}{r^2 - x^2}\) [3]
2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4\pi r^2\). [5]
3. Write down the length of the arc of the curve \(y = \sqrt{1 - x^2}\) from \(x = 0\) to \(x = 1\) [1]

The position vectors of the points \(A\), \(B\) and \(C\) from a fixed origin \(O\) are $$\mathbf{a} = \mathbf{i} - \mathbf{j}, \quad \mathbf{b} = \mathbf{i} + \mathbf{j} + \mathbf{k}, \quad \mathbf{c} = 2\mathbf{j} + \mathbf{k}$$ respectively.

1. Using vector products, find the area of the triangle \(ABC\). [4]
2. Show that \(\frac{1}{6}\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0\) [3]
3. Hence or otherwise, state what can be deduced about the vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\). [1]

$$I_n = \int (x^2 + 1)^{-n} dx, \quad n > 0$$

1. Show that, for \(n > 0\) $$I_{n+1} = \frac{x(x^2 + 1)^{-n}}{2n} + \frac{2n - 1}{2n}I_n$$ [5]
2. Find \(I_2\) [3]