CAIE FP1 (Further Pure Mathematics 1) 2003 November

\includegraphics{figure_1} The curve \(C\) has polar equation $$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$ where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]

Given that $$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$ find \(S_N = \sum_{n=N+1}^{2N} u_n\) in terms of \(N\). [3] Find a number \(M\) such that \(S_N < 10^{-20}\) for all \(N > M\). [3]

Three \(n \times 1\) column vectors are denoted by \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\), and \(\mathbf{M}\) is an \(n \times n\) matrix. Show that if \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\) are linearly dependent then the vectors \(\mathbf{Mx}_1\), \(\mathbf{Mx}_2\), \(\mathbf{Mx}_3\) are also linearly dependent. [2] The vectors \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) and the matrix \(\mathbf{P}\) are defined as follows: $$\mathbf{y}_1 = \begin{pmatrix} 1 \\ 5 \\ 7 \end{pmatrix}, \quad \mathbf{y}_2 = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \quad \mathbf{y}_3 = \begin{pmatrix} 5 \\ 51 \\ 55 \end{pmatrix},$$ $$\mathbf{P} = \begin{pmatrix} 1 & -4 & 3 \\ 0 & 2 & 5 \\ 0 & 0 & -7 \end{pmatrix}$$

1. Show that \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) are linearly dependent. [2]
2. Find a basis for the linear space spanned by the vectors \(\mathbf{Py}_1\), \(\mathbf{Py}_2\), \(\mathbf{Py}_3\). [2]

Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that $$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3] Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]

The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \, dx.$$ By considering \(\frac{d}{dx}(\tan x \sec^n x)\), or otherwise, show that $$(n + 1)I_{n+2} = 2^{\frac{4n}{n}} + nI_n.$$ [4] Find the value of \(I_6\). [4]

Find the sum of the squares of the roots of the equation $$x^3 + x + 12 = 0,$$ and deduce that only one of the roots is real. [4] The real root of the equation is denoted by \(\alpha\). Prove that \(-3 < \alpha < -2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt{6}\). [5]

Find the general solution of the differential equation $$\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = e^{-\alpha t},$$ where \(\alpha\) is a constant and \(\alpha \neq 2\). [7] Show that if \(\alpha < 2\) then, whatever the initial conditions, \(ye^{\alpha t} \to \frac{1}{(2-\alpha)^2}\) as \(t \to \infty\). [2]

Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express \(\sin^6 \theta\) in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4] Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]

The line \(l_1\) passes through the point \(A\) with position vector \(\mathbf{i} - \mathbf{j} - 2\mathbf{k}\) and is parallel to the vector \(3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}\). The variable line \(l_2\) passes through the point \((1 + 5 \cos t)\mathbf{i} - (1 + 5 \sin t)\mathbf{j} - 14\mathbf{k}\), where \(0 \leq t < 2\pi\), and is parallel to the vector \(15\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\). The points \(P\) and \(Q\) are on \(l_1\) and \(l_2\) respectively, and \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find the length of \(PQ\) in terms of \(t\). [4]
2. Hence show that the lines \(l_1\) and \(l_2\) do not intersect, and find the maximum length of \(PQ\) as \(t\) varies. [3]
3. The plane \(\Pi_1\) contains \(l_1\) and \(PQ\); the plane \(\Pi_2\) contains \(l_2\) and \(PQ\). Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), correct to the nearest tenth of a degree. [4]

Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 6 & 4 & 1 \\ -6 & -1 & 3 \\ 8 & 8 & 4 \end{pmatrix}.$$ [8] Hence find a non-singular matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A} + \mathbf{A}^2 + \mathbf{A}^3 = \mathbf{PDP}^{-1}\). [4]

Answer only one of the following two alternatives. EITHER The curve \(C\) has equation \(y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}\).

1. Express \(y\) in the form \(P + \frac{Q}{x-2} + \frac{R}{x+3}\). [3]
2. Show that \(\frac{dy}{dx} = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\). [4]
3. Write down the equations of all the asymptotes of \(C\). [3]
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\). [4]

OR A curve has equation \(y = \frac{5}{3}x^{\frac{3}{2}}\), for \(x \geq 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).

1. Find the length of \(R\). [4]
2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\). [5]
3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac{232\pi}{15}\). [5]