# 16 Further Pure 1

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1 and Core 2.

Candidates will also be expected to know for section 16.6 that the roots of an equation $\mbox{f}(x) = 0$ can be located by considering changes of sign of $\mbox{f}(x)$ in an interval of $x$ in which $\mbox{f}(x)$ is continuous.

Candidates may use relevant formulae included in the formulae booklet without proof.

 Graphs of rational functions of the form. $\frac{ax + b}{cx + d}, \space \frac{ax+b}{cx^2 + dx + e}$ or $\frac{x^2 + ax + b}{x^2 + cx + d}$ Sketching the graphs. Finding the equations of the asymptotes which will always be parallel to the coordinate axes. Finding points of intersection with the coordinate axes or other straight lines. Solving associated inequalities. Using quadratic theory (not calculus) to find the possible values of the function and the coordinates of the maximum or minimum points on the graph. Eg for $y = \frac{x^2 + 2}{x^2-4x},\space y = k \Rightarrow x^2 + 2 = kx^2 - 4kx$ which has real roots if $16k^2 + 8k - 8 \geq -0$, ie if $k \leq -1 \space or \space k \geq \frac{1}{2}$; stationary points are $(1, -1)$ and $(-2, \frac{1}{2})$ Graphs of parabolas, ellipses and hyperbolas with equations $y^2=4ax, \space \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $xy = c^2$ Sketching the graphs. Finding points of intersection with the coordinate axes or other straight lines. Candidates will be expected to interpret the geometrical implication of equal roots, distinct real roots or no real roots. Knowledge of the effects on these equations of single transformations of these graphs involving translations, stretches parallel to the $x$-axis or $y$-axis, and reflections in the line $y = x$. Including the use of the equations of the asymptotes of the hyperbolas given in the formulae booklet.
 Non-real roots of quadratic equations. Complex conjugates – awareness that non-real roots of quadratic equations with real coefficients occur in conjugate pairs. Sum, difference and product of complex numbers in the form $x + \mbox{i}y$ Comparing real and imaginary parts. Including solving equations eg $2z + z^{*} = 1 + \mbox{i}$ where $z^{ * }$ is theconjugate of $z$ .
 Manipulating expressions involving $\alpha + \beta$ and $\alpha \beta$ . Eg $\alpha^3 + \beta^3 = (\alpha+\beta)^3 = (\alpha + \beta)^3-3\alpha \beta(\alpha + \beta)$ Forming an equation with roots $\alpha^3, \beta^3,$ or $\frac{1}{\alpha},\frac{1}{\beta},\alpha+\frac{2}{\beta}, \beta+\frac{2}{\alpha}$ etc.
 Use of formulae for the sum of the squares and the sum of the cubes of the natural numbers. Eg to find a polynomial expression for $\displaystyle\sum_{r=1}^{n} r^2(r+2)$ or $\displaystyle\sum_{r=1}^{n} (r^2-r+1)$
 Finding the gradient of the tangent to a curve at a point, by taking the limit as $h$ tends to zero of the gradient of a chord joining two points whose $x$-coordinates differ by $h$. The equation will be given as $y=\mbox{f}(x)$, where $\mbox{f}(x)$ is a simple polynomial such as $x^2-2x \mbox{ or } x^4+3$. Evaluation of simple improper integrals. E.g. $\displaystyle\int_{1}^4 \frac{1}{\sqrt{x}} \mbox{d}x , \int_{4}^{\infty} x^{-\frac{3}{2}} \mbox{d}x$
 Finding roots of equations by interval bisection, linear interpolation and the Newton-Raphson method. Graphical illustration of these methods. Solving differential equations of the form $\displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x} = \mbox{f}(x)$ Using a step-by-step method based on the linear approximations $y_{n+1} \approx y_n + h \mbox{f}(x_n); x_{n+1} = x_n + h,$ with given values for $x_0, y_0$ and $h$. Reducing a relation to a linear law. E.g. $\frac{1}{x} + \frac{1}{y} = k; \space y^2 = ax^3+b; \space y = a x^n; \space y = ab^x$ Use of logarithms to base 10 where appropriate. Given numerical values of $(x, y)$, drawing a linear graph and using it to estimate the values of the unknown constants.
 General solutions of trigonometric equations including use of exact values for the sine, cosine and tangent of $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ Eg $\sin 2x = \frac{\sqrt{3}}{2}, \mbox{ } \cos\Big(x + \frac{\pi}{6}\Big) = -\frac{1}{\sqrt{2}}, \mbox{ } \tan \Big( \frac{\pi}{3}-2x \Big) = 1$, $\sin 2x=0.3, \mbox{ } \cos(3x-1) = -0.2$
 $2 \times 2$ and $2 \times 1$ matrices; addition and subtraction, multiplication by a scalar. Multiplying a $2 \times 2$ matrix by a $2 \times 2$ matrix or by a $2 \times 1$ matrix. The identity matrix $\mathbf{I}$ for a $2 \times 2$ matrix. Transformations of points in the $x - y$ plane represented by $2 \times 2$ matrices. Transformations will be restricted to rotations about the origin, reflections in a line through the origin, stretches parallel to the $\mbox{x}$-axis and $\mbox{y}$-axis, and enlargements with centre the origin. Use of the standard transformation matrices given in the formulae booklet. Combinations of these transformations e.g. $\Bigg[ \begin{matrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix} \Bigg], \mbox{ } \Bigg[ \begin{matrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{matrix} \Bigg], \mbox{ } \Big[ \begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix} \Big], \mbox{ } \Big[ \begin{matrix} 2 & 0 \\ 0 & 2 \end{matrix} \Big]$