Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2018 June

1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
2. Using the method of differences, prove that \(\sum_{r=1}^{n} \frac{3}{(3r-1)(3r+2)} = \frac{1}{2} - \frac{1}{3n+2}\). [2]
3. Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(3r-1)(3r+2)}\). [1]

1. Determine the asymptotes and turning points of the curve with equation \(y = \frac{x^2+3}{x+1}\). [7]
2. Sketch the curve. [3]

The complex numbers \(z_1\) and \(z_2\) are such that \(|z_1| = 2\), \(\arg(z_1) = \frac{7}{12}\pi\), \(|z_2| = \sqrt{2}\) and \(\arg(z_2) = -\frac{1}{8}\pi\).

1. Find, in exact form, the modulus and argument of \(\frac{z_1}{z_2}\). [3]
2. Let \(z_3 = \left(\frac{z_1}{z_2}\right)^n\). It is given that \(n\) is the least positive integer for which \(z_3\) is a positive real number. Find this value of \(n\) and the exact value of \(z_3\). [4]

A curve has polar equation \(r = \frac{3}{10}e^{3\theta}\) for \(\theta \geq 0\). The length of the arc of this curve between \(\theta = 0\) and \(\theta = \alpha\) is denoted by \(L(\alpha)\).

1. Show that \(L(\alpha) = \frac{1}{3}(e^{3\alpha} - 1)\). [5]
2. The point \(P\) on the curve corresponding to \(\theta = \beta\) is such that \(L(\beta) = OP\), where \(O\) is the pole. Find the value of \(\beta\). [2]

Find, in the form \(y = f(x)\), the solution of the differential equation \(\frac{dy}{dx} + y\tanh x = 2\cosh x\), given that \(y = \frac{3}{4}\) when \(x = \ln 2\). [8]

The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).

1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]

The function \(y\) satisfies \(\frac{d^2y}{dx^2} + x^2y = x\), and is such that \(y = 1\) and \(\frac{dy}{dx} = 1\) when \(x = 1\).

1. Using the given differential equation
   1. state the value of \(\frac{d^2y}{dx^2}\) when \(x = 1\), [1]
   2. find, by differentiation, the value of \(\frac{d^3y}{dx^3}\) when \(x = 1\). [2]
2. Hence determine the Taylor series for \(y\) about \(x = 1\) up to and including the term in \((x-1)^3\) and deduce, correct to 4 decimal places, an approximation for \(y\) when \(x = 1.1\). [3]

1. Write down the values of the constants \(a\) and \(b\) for which \(m^3 = \frac{1}{6}m^3(am^2 + 2) - \frac{1}{12}m^2(bm)\). [1]
2. Prove by induction that \(\sum_{r=1}^{n} r^5 = \frac{1}{6}n^3(n+1)^3 - \frac{1}{12}n^2(n+1)^2\) for all positive integers \(n\). [7]

1. Use de Moivre's theorem to prove that \(\cos 3\theta = 4c^3 - 3c\), where \(c = \cos\theta\). [3]
2. Solve the equation \(2\cos 3\theta - \sqrt{3} = 0\) for \(0 < \theta < \pi\), giving each answer in an exact form. [2]
3. Deduce, in trigonometric form, the three roots of the equation \(x^3 - 3x - \sqrt{3} = 0\). [3]

1. Let \(G\) be a group of order 10. Write down the possible orders of the elements of \(G\) and justify your answer. [2]
2. Let \(G_1\) be the cyclic group of order 10 and let \(g\) be a generator of \(G_1\) (that is, an element of order 10). List the ten elements of \(G_1\) in terms of \(g\) and state the order of each element. [4]
3. The group \(G_2\) is defined as the set of ordered pairs \((x, y)\), where \(x \in \{0, 1\}\) and \(y \in \{0, 1, 2, 3, 4\}\), together with the binary operation \(\oplus\) defined by $$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$ where \(x_3 = x_1 + x_2\) modulo 2 and \(y_3 = y_1 + y_2\) modulo 5.
   1. List the elements of \(G_2\) and state the order of each element. [3]
   2. State, with justification, whether \(G_1\) and \(G_2\) are isomorphic. [1]

Let \(\mathbf{A}\) be the matrix \(\begin{pmatrix} 17 & 12 \\ 12 & 10 \end{pmatrix}\).

1. 1. Determine the integer \(n\) for which \(27\mathbf{A} - \mathbf{A}^2 = n\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [2]
   2. Hence find \(\mathbf{A}^{-1}\) in the form \(p\mathbf{A} + q\mathbf{I}\) for rational numbers \(p\) and \(q\). [2]
2. The plane transformation \(T\) is defined by \(T: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \mathbf{A} \begin{pmatrix} x \\ y \end{pmatrix}\). It is given that \(T\) is a stretch, with scale factor \(k\), parallel to the line \(y = mx\), where \(m > 0\).
   1. Find the value of \(k\). [2]
   2. By considering \(\mathbf{A} \begin{pmatrix} x \\ mx \end{pmatrix}\), or otherwise, determine the value of \(m\). [4]

The curve \(C\) is given by \(y = \frac{1}{4}x^2 - \frac{1}{2}\ln x\) for \(2 \leq x \leq 8\).

1. Find, in its simplest exact form, the length of \(C\). [5]
2. When \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed. Show that the area of this surface is \(\pi(270 - 47\ln 2 - 2(\ln 2)^2)\). [10]

The planes \(\Pi_1\) and \(\Pi_2\) are both perpendicular to \(\mathbf{n}\), where \(\mathbf{n} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). The points \(A(0, -9, 13)\) and \(B(8, 7, -3)\) lie in \(\Pi_1\) and \(\Pi_2\) respectively.

1. Find the equations of \(\Pi_1\) and \(\Pi_2\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\) and show that \(\overrightarrow{AB}\) is parallel to \(\mathbf{n}\). [4]
2. Calculate the perpendicular distance between \(\Pi_1\) and \(\Pi_2\). [2]
3. Write down two vectors which are perpendicular to \(\mathbf{n}\) and hence find, in the form $$\mathbf{r} = \mathbf{u} + \lambda\mathbf{v} + \mu\mathbf{w},$$ an equation for the plane \(\Pi_3\) which is parallel to \(\Pi_1\) and \(\Pi_2\) and exactly half-way between them. [4]
4. The locus of all points \(P\) such that \(AP = BP = 12\sqrt{2}\) is denoted by \(L\).
   1. Give a full geometrical description of \(L\). [4]
   2. Using the result of part (iii), or otherwise, find a point on \(L\) which has integer coordinates. [4]