Edexcel FP3 (Further Pure Mathematics 3) Specimen

Find the eigenvalues of the matrix \(\begin{pmatrix} 7 & 6 \\ 6 & 2 \end{pmatrix}\) (Total 4 marks)

Find the values of \(x\) for which $$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms. (Total 6 marks)

\includegraphics{figure_1} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a(t - \sin t), \quad y = a(1 - \cos t), \quad 0 \leq t \leq 2\pi$$ Find, by using integration, the length of \(C\). (Total 6 marks)

Find \(\int \sqrt{x^2 + 4} \, dx\). (Total 7 marks)

Given that \(y = \arcsin x\) prove that

1. \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\) [3]
2. \((1-x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} = 0\) [4]

(Total 7 marks)

$$I_n = \int_0^{\pi} x^n \sin x \, dx$$

1. Show that for \(n \geq 2\) $$I_n = n \left( \frac{\pi}{2} \right)^{n-1} - n(n-1)I_{n-2}$$ [4]
2. Hence obtain \(I_3\), giving your answers in terms of \(\pi\). [4]

(Total 8 marks)

$$\mathbf{A}(x) = \begin{pmatrix} 1 & x & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}, \quad x \neq \frac{5}{2}$$

1. Calculate the inverse of \(\mathbf{A}(x)\). $$\mathbf{B} = \begin{pmatrix} 1 & 3 & -1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{pmatrix}$$ [8] The image of the vector \(\begin{pmatrix} p \\ q \\ r \end{pmatrix}\) when transformed by \(\mathbf{B}\) is \(\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}\)
2. Find the values of \(p\), \(q\) and \(r\). [4]

(Total 14 marks)

The points \(A\), \(B\), \(C\), and \(D\) have position vectors $$\mathbf{a} = 2\mathbf{i} + \mathbf{k}, \quad \mathbf{b} = \mathbf{i} + 3\mathbf{j}, \quad \mathbf{c} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}, \quad \mathbf{d} = 4\mathbf{j} + \mathbf{k}$$ respectively.

1. Find \(\overrightarrow{AB} \times \overrightarrow{AC}\) and hence find the area of triangle \(ABC\). [7]
2. Find the volume of the tetrahedron \(ABCD\). [2]
3. Find the perpendicular distance of \(D\) from the plane containing \(A\), \(B\) and \(C\). [3]

(Total 12 marks)

The hyperbola \(C\) has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

1. Show that an equation of the normal to \(C\) at \(P(a \sec \theta, b \tan \theta)\) is $$by + ax \sin \theta = (a^2 + b^2)\tan \theta$$ [6] The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(AB\) is \(M\).
2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies. [7]

(Total 13 marks)