# 17 Further Pure 2

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof except where proof is required in this module and requested in a question.

 The relations between the roots and the coefficients of a polynomial equation; the occurrence of the non-real roots in conjugate pairs when the coefficients of the polynomial are real.
 The Cartesian and polar coordinate forms of a complex number, its modulus, argument and conjugate. $x+\mbox{i}y$ and $r(\cos \theta + \mbox{i} \sin \theta)$. The sum, difference, product and quotient of two complex numbers. The parts of this topic also included in module Further Pure 1 will be examined only in the context of the content of this module. The representation of a complex number by a point on an Argand diagram; geometrical illustrations. Simple loci in the complex plane. For example, $|z - 2 - \mbox{i} | \leqslant 5, \space \space \mbox{arg}(z - 2)= \frac{ \pi }{3}$ Maximum level of difficulty $| z - a | = | z - b |$ where $a$ and $b$ are complex numbers.
 De Moivre's theorem for integral $n$. Use of $z+\frac{1}{z} = 2 \cos \theta$ and $z-\frac{1}{z} = 2 \mbox{i} \sin \theta$, leading to, for example, expressing $\sin^5 \theta$ in terms of multiple angles and $\tan 5 \theta$ in term of powers of $\tan \theta$. Applications in evaluating integrals, for example,$\int \sin^5 \theta \mbox{d}\theta$. De Moivre's theorem; the $n \text{th}$ roots of unity, the exponential form of a complex number. The use, without justification, of the identity $e^{ix}= \cos x + \mbox{i} \sin x$ Solutions of equations of the form $z^n = a + \mbox{i}b$ To include geometric interpretation and use, for example, in expressing $\cos \frac{5 \pi}{12}$ in surd form.
 Applications to sequences and series, and other problems. Eg proving that $7^n + 4^n+1$is divisible by 6, or $(\cos \theta + \mbox{i} \sin \theta)^n = \cos n \theta + \mbox{i} \sin n \theta$ where n is a positive integer.
 Summation of a finite series by any method such as induction, partial fractions or differencing. Eg $\displaystyle\sum_{r=1}^n r.r! = \displaystyle\sum_{r=1}^n \big[(r+1)! - r! \big]$

 Use of the derivatives of $\sin^{-1}x, \space \cos^{-1}x, \space \tan^{-1}x$ as given in the formulae booklet. To include the use of the standard integrals $\int \frac{1}{a^2 + x^2} \mbox{d}x; \space \int \frac{1}{\sqrt{a^2-x^2}}\mbox{d}x$ given in the formulae booklet.
 Hyperbolic and inverse hyperbolic functions and their derivatives; applications to integration. The proofs mentioned below require expressing hyperbolic functions in terms of exponential functions. To include solution of equations of the form $a \sinh x + b \cosh x = c$. Use of basic definitions in proving simple identities. Maximum level of difficulty: $\sinh(x+y) \equiv \sinh x \cosh y + \cosh x \sinh y$. The logarithmic forms of the inverse functions, given in the formulae booklet, may be required. Proofs of these results may also be required. Proofs of the results of differentiation of the hyperbolic functions, given in the formula booklet, are included. Knowledge, proof and use of: $\cosh^2 x - \sinh^2 x = 1$ $1 - \tanh^2 x = \mbox{sech}^2 x$ $\coth^2 x - 1 = \mbox{cosech}^2 x$ Familiarity with the graphs of $\sinh x, \space \cosh x, \space \tanh x, \space \sinh^{-1} x, \space \cosh^{-1} x, \space \tanh^{-1} x$.
 Calculation of the arc length of a curve and the area of a surface of revolution using Cartesian or parametric coordinates. Use of the following formulae will be expected: $s = \displaystyle\int_{x_1}^{x_2} \Bigg[ 1 + \bigg({\operatorname{d}\!y\over\operatorname{d}\!x} \bigg)^2 \Bigg]^{\frac{1}{2}} \mbox{d}x = \int_{t_1}^{t_2} \Bigg[ \bigg( {\operatorname{d}\!x\over\operatorname{d}\!t} \bigg)^2 + \bigg( {\operatorname{d}\!y\over\operatorname{d}\!t} \bigg)^2 \Bigg] ^ {\frac{1}{2}} \mbox{d}t$ $S = 2 \pi \displaystyle\int_{x_1}^{x_2} y \Bigg[ 1 + \bigg( {\operatorname{d}\!y\over\operatorname{d}\!x} \bigg)^2 \Bigg]^{\frac{1}{2}} \mbox{d}x = 2 \pi \int_{t_1}^{t_2} y \Bigg[ \bigg( {\operatorname{d}\!x\over\operatorname{d}\!t} \bigg)^2 + \bigg( {\operatorname{d}\!y\over\operatorname{d}\!t} \bigg)^2 \Bigg]^{\frac{1}{2}} \mbox{d}t$