Pre-U Pre-U 9795 (Pre-U Further Mathematics) Specimen

The region \(R\) of an Argand diagram is defined by the inequalities $$0 \leqslant \arg(z + 4\mathrm{i}) \leqslant \frac{1}{4}\pi \quad \text{and} \quad |z| \leqslant 4.$$ Draw a clearly labelled diagram to illustrate \(R\). [4]

It is given that $$\mathrm{f}(n) = 7^n (6n + 1) - 1.$$ By considering \(\mathrm{f}(n + 1) - \mathrm{f}(n)\), prove by induction that \(\mathrm{f}(n)\) is divisible by 12 for all positive integers \(n\). [6]

Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]

Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]

Find the general solution of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 6\frac{\mathrm{d}y}{\mathrm{d}x} + 9y = 72\mathrm{e}^{3x}.$$ [7]

\includegraphics{figure_6} The diagram shows a sketch of the curve \(C\) with polar equation \(r = a \cos^2 \theta\), where \(a\) is a positive constant and \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\). [2]
2. The equation of \(C\) may be expressed in the form \(r = \frac{1}{2}a(1 + \cos 2\theta)\). Using this form, show that the area of the region enclosed by \(C\) is given by $$\frac{1}{16}a^2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (3 + 4 \cos 2\theta + \cos 4\theta) \, \mathrm{d}\theta,$$ and find this area. [6]

The equation $$8x^3 + 12x^2 + 4x - 1 = 0$$ has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(2\alpha + 1, 2\beta + 1, 2\gamma + 1\) is $$y^3 - y - 1 = 0.$$ [3] The sum \((2\alpha + 1)^n + (2\beta + 1)^n + (2\gamma + 1)^n\) is denoted by \(S_n\). Find the values of \(S_3\) and \(S_{-2}\). [5]

The curve \(C\) has equation $$y = \frac{x^2 - 2x - 3}{x + 2}.$$

1. Find the equations of the asymptotes of \(C\). [4]
2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]

Given that \(w_n = 3^{-n} \cos 2n\theta\) for \(n = 1, 2, 3, \ldots\), use de Moivre's theorem to show that $$1 + w_1 + w_2 + w_3 + \ldots + w_{N-1} = \frac{9 - 3\cos 2\theta + 3^{-N+1} \cos 2(N-1)\theta - 3^{-N+2} \cos 2N\theta}{10 - 6\cos 2\theta}.$$ [7] Hence show that the infinite series $$1 + w_1 + w_2 + w_3 + \ldots$$ is convergent for all values of \(\theta\), and find the sum to infinity. [2]

1. Find the inverse of the matrix \(\begin{pmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & 2 \end{pmatrix}\), and hence solve the set of equations \begin{align} x + 3y + 4z &= -5,
   2x + 5y - z &= 10,
   3x + 8y + 2z &= 8. \end{align} [5]
2. Find the value of \(k\) for which the set of equations \begin{align} x + 3y + 4z &= -5,
   2x + 5y - z &= 15,
   3x + 8y + 3z &= k, \end{align} is consistent. Find the solution in this case and interpret it geometrically. [5]

A group \(G\) has distinct elements \(e, a, b, c, \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation. Prove that if $$a \circ a = b, \quad b \circ b = a$$ then the set of elements \(\{e, a, b\}\) forms a subgroup of \(G\). [5] Prove that if $$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$ then the set of elements \(\{e, a, b, c\}\) does not form a subgroup of \(G\). [5]

With respect to an origin \(O\), the points \(A, B, C, D\) have position vectors $$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$ respectively. Find

1. a vector perpendicular to the plane \(OAB\), [2]
2. the acute angle between the planes \(OAB\) and \(OCD\), correct to the nearest \(0.1°\), [3]
3. the shortest distance between the line which passes through \(A\) and \(B\) and the line which passes through \(C\) and \(D\), [4]
4. the perpendicular distance from the point \(A\) to the line which passes through \(C\) and \(D\). [3]

Given that \(y = \cos\{\ln(1 + x)\}\), prove that

1. \((1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} = -\sin\{\ln(1 + x)\}\), [1]
2. \((1 + x)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\). [2]

Obtain an equation relating \(\frac{\mathrm{d}^3 y}{\mathrm{d}x^3}\), \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\). [2] Hence find Maclaurin's series for \(y\), up to and including the term in \(x^3\). [4] Verify that the same result is obtained if the standard series expansions for \(\ln(1 + x)\) and \(\cos x\) are used. [3]

Let \(J_n = \int_1^{\mathrm{e}} (\ln x)^n \, \mathrm{d}x\), where \(n\) is a positive integer. By considering \(\frac{\mathrm{d}}{\mathrm{d}x}(x(\ln x)^n)\), or otherwise, show that $$J_n = \mathrm{e} - nJ_{n-1}.$$ [4] Let \(J_n = \frac{J_n}{n!}\). Show that $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots + \frac{1}{10!} = \frac{1}{\mathrm{e}}(1 + J_{10}).$$ [6] It can be shown that $$\sum_{r=2}^{n} \frac{(-1)^r}{r!} = \frac{1}{\mathrm{e}}(1 + (-1)^n J_n)$$ for all positive integers \(n\). Deduce the sum to infinity of the series $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots,$$ justifying your conclusion carefully. [3]