CAIE FP1 (Further Pure Mathematics 1) 2019 November

The curve \(C\) has equation \(y = x^a\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis. [6]

It is given that \(y = \ln(ax + 1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac{d^n y}{dx^n} = (-1)^{n-1} \frac{(n-1)!a^n}{(ax+1)^n}.$$ [6]

The integral \(I_n\), where \(n\) is a positive integer, is defined by $$I_n = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^{-n} \sin \pi x \, dx.$$

1. Show that $$n(n+1)I_{n+2} = 2^{n+1} n + \pi - \pi^2 I_n.$$ [5]
2. Find \(I_5\) in terms of \(\pi\) and \(I_1\). [2]

The line \(y = 2x + 1\) is an asymptote of the curve \(C\) with equation $$y = \frac{x^2 + 1}{ax + b}.$$

1. Find the values of the constants \(a\) and \(b\). [3]
2. State the equation of the other asymptote of \(C\). [1]
3. Sketch \(C\). [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.] [3]

Let \(S_N = \sum_{r=1}^{N} (5r + 1)(5r + 6)\) and \(T_N = \sum_{r=1}^{N} \frac{1}{(5r + 1)(5r + 6)}\).

1. Use standard results from the List of Formulae (MF10) to show that $$S_N = \frac{1}{3}N(25N^2 + 90N + 83).$$ [3]
2. Use the method of differences to express \(T_N\) in terms of \(N\). [4]
3. Find \(\lim_{N \to \infty} (N^{-3} S_N T_N)\). [2]

With \(O\) as the origin, the points \(A\), \(B\), \(C\) have position vectors $$\mathbf{i} - \mathbf{j}, \quad 2\mathbf{i} + \mathbf{j} + 7\mathbf{k}, \quad \mathbf{i} - \mathbf{j} + \mathbf{k}$$ respectively.

1. Find the shortest distance between the lines \(OC\) and \(AB\). [5]
2. Find the cartesian equation of the plane containing the line \(OC\) and the common perpendicular of the lines \(OC\) and \(AB\). [4]

The equation \(x^3 + 2x^2 + x + 7 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\).

1. Use the relation \(x^2 = -7y\) to show that the equation $$49y^3 + 14y^2 - 27y + 7 = 0$$ has roots \(\frac{\alpha}{\beta \gamma}\), \(\frac{\beta}{\gamma \alpha}\), \(\frac{\gamma}{\alpha \beta}\). [4]
2. Show that \(\frac{\alpha^2}{\beta^2 \gamma^2} + \frac{\beta^2}{\gamma^2 \alpha^2} + \frac{\gamma^2}{\alpha^2 \beta^2} = \frac{58}{49}\). [3]
3. Find the exact value of \(\frac{\alpha^2}{\beta^3 \gamma^3} + \frac{\beta^2}{\gamma^3 \alpha^3} + \frac{\gamma^2}{\alpha^3 \beta^3}\). [2]

The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).

1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
2. Find \(\mathbf{M}^T \mathbf{P}\). [3]

1. Use de Moivre's theorem to show that $$\sec 6\theta = \frac{\sec^6 \theta}{32 - 48 \sec^2 \theta + 18 \sec^4 \theta - \sec^6 \theta}.$$ [6]
2. Hence obtain the roots of the equation $$3x^6 - 36x^4 + 96x^2 - 64 = 0$$ in the form \(\sec q\pi\), where \(q\) is rational. [5]

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$

1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]

Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}

1. Solve the system of equations when \(\theta \neq -1\). [3]
2. Find the general solution when \(\theta = -1\). [3]
3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]

Answer only one of the following two alternatives. **EITHER** It is given that \(w = \cos y\) and $$\tan y \frac{d^2 y}{dx^2} + \left( \frac{dy}{dx} \right)^2 + 2 \tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$

1. Show that $$\frac{d^2 w}{dx^2} + 2 \frac{dw}{dx} + w = -e^{-2x}.$$ [4]
2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0\), \(y = \frac{1}{4}\pi\) and \(\frac{dy}{dx} = \frac{1}{\sqrt{3}}\). [10]

**OR** The curves \(C_1\) and \(C_2\) have polar equations, for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), as follows: \begin{align} C_1 : r &= 2(e^\theta + e^{-\theta}),
C_2 : r &= e^{2\theta} - e^{-2\theta}. \end{align} The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(e^{2\alpha} - 2e^\alpha - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt{2}\). [6]
2. Sketch \(C_1\) and \(C_2\) on the same diagram. [3]
3. Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures. [5]