H2 Further Mathematics 9649 Pure Math Notes

H2 Further Mathematics 9649 Pure Mathematics Zhuo Zhuzhen Hwa Chong Institution

Preface One of my most vivid memory from JC is how I was con- templating my life decisions after finding out that my friend who dropped FM for computing was getting easy As while I was stuck with my Ds no matter how hard I tried. Indeed, I don’t think many can deny the fact that FM is a difficult subject, and I would argue its one of the most difficult A level subject there is. Perhaps you are feeling the same struggle - and maybe even a bit of helplessness - as I once did. Pure mathematics can feel really demoralising at times, es- pecially given how creative examiners can be. However, just remember that the very fact that you are reading this means that you have the ability to do well in this subject (since you passed the selection test). The power of spamming papers and questions should not be understated - remember to never give up no matter how difficult it seems. I created this set of notes as a convenient reference for myself for everything that I needed to know about FM and as a tool for memorising the dozens of formulas. Now that it has served its purpose, I hope it will be as useful to you in achieving your desired goals, just as I was able to achieve mine. All the best for all your future exams! I

Acknowledgements This entire set of notes is an adapted and summarised ver- sion of the Further Mathematics notes created by HCI Math Department. Most figures are taken directly from the source. Special thanks must also be given to my FM teachers, Mr Ng Say Tiong and Miss Ho Jia Yuan, for working so hard and being so selfless. II

Table of Contents 1 Recurrence Relations and Mathematical Induction 1 1.1 Convergence of Infinite Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Solutions to RR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3.1 First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3.2 Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Numerical Methods 3 2.1 Determining the Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Analytical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Approximate Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.1 Linear Interpolation Method . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.2 Fixed Point Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.3 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Approximate Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 Trapezium Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.2 Simpson’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Polar Curves and Conic Sections 8 3.1 Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.1 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.2 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 III

3.2.4 Directrix-Eccentricity-Focus Definitions . . . . . . . . . . . . . . . . . . . 11 3.2.5 Polar Equations of Conic Sections . . . . . . . . . . . . . . . . . . . . . . 11 4 Application of Definite Integrals 13 4.1 Area of Curve in Polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Arc Length of Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Volume of Revolution of Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Surface Area of Revolution of Curve . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Matrices 15 5.1 Terminologies and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Properties of Matrix Addition and Scalar Multiplication . . . . . . . . . . . . . 15 5.3 Properties of Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4 Transpose of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4.1 Properties of Transpose Matrices . . . . . . . . . . . . . . . . . . . . . . 16 5.5 Augmented Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.6 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.7 Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.8 Geometrical Interpretation of Solution to System of Linear Equations. . . . . . . 18 5.8.1 2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.8.2 3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.9 Determinant of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.9.1 2 × 2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.9.2 3 × 3 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.9.3 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.10 Non-singular Matrices and its Inverse . . . . . . . . . . . . . . . . . . . . . . . . 21 5.10.1 Computing Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 Vector Space 23 6.1 Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.1.1 Find Subspace of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2 Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 IV

6.2.1 Null Space and Range Space . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2.2 Rank and Nullity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.3.1 Finding Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.4 Diagonalisation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7 Differentiation Equation 28 7.1 Solving D.E. Analytically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.1.1 First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.1.2 Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.2 Approximation to Solutions of First Order D.E. . . . . . . . . . . . . . . . . . . 29 7.3 Equilibria and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.4 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8 Complex Numbers 32 8.1 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.1.1 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.2 De Moivre’s Theorem and its Application . . . . . . . . . . . . . . . . . . . . . . 34 8.2.1 nth root of a Complex Number . . . . . . . . . . . . . . . . . . . . . . . 34 8.2.2 Deriving Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . 34 V

1 RECURRENCE RELATIONS AND MATHEMATICAL INDUCTION CHAPTER 1 Recurrence Relations and Mathematical Induction 1.1 Convergence of Infinite Sequence When given a recurrence relation, if un → L when n → ∞, then un+1 → L. Hence, a recurrence relation can be reduced to an expression with one algebraic term to find the limit of the equatio