A LEVEL MATHEMATICS - (P3) 9709 · A LEVEL MATHEMATICS - (P3) – 9709 CAIE SYLLABUS Paper 3 1. Algebra 2. Logarithmic and exponential functions 3. Trigonometry 4. Differentiation

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A LEVEL MATHEMATICS - (P3) – 9709

CAIE SYLLABUS

Paper 3

1. Algebra 2. Logarithmic and exponential functions 3. Trigonometry 4. Differentiation 5. Integration 6. Numerical solution of equations 7. Vectors 8. Differential equations 9. Complex numbers

NOTE:

1. This checklist does not have the explanation of the concepts. 2. For explanation please refer to the “Revision Guidlines / Notes with solved examples” on the

website tkmaths.com. 3. Visit to the website tkmaths.com and the blog tkmathsisfun.blogspot.com

will be beneficial for you in your learning process.

Checklist for Exam Preparation

S.# Topic What you need to know

1 Algebra • Sketch of 𝒚 = |𝒂𝒙 + 𝒃| • Solving equation and inquality e.g.

- |𝟐𝒙 − 𝟓| < 𝟑 - |𝟐𝒙 − 𝟏| ≥ |𝟑 − 𝒙| - |𝟑 + 𝒙| > 𝟒 − 𝟐𝒙 - |𝟒𝒙 − 𝟑| = 𝟕 , - |𝟑𝒙 − 𝟓| = 𝒙 + 𝟏 , |𝒙 − 𝟓| = 𝟐|𝒙 + 𝟏|

• Long division of a polynomial (dgree not exceeding by 4) by a linear or quadratic polynomial

• Partial fractin with the given denominators.


• • Use the factor and reminder theorem. • Solving cubic equation. • Binomial Theorem i.ee use the expansion

of (𝟏 + 𝒙)𝒏 , where n is a rational number and |𝒙| < 𝟏.

2. Logarithmic and exponential functions

• Relationship between Logarithm and undices. • Laws of logarithm • understand the definition and properties of

ⅇ𝑥and lnx, including their relationship as inverse functions and their graphs

• Including knowledge of the graph 𝑦 = ⅇ𝑘𝑥of for both positive and negative values of k

• use logarithms to solve equations and inequalities in which the unknown appears in indices

• use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

3 Trigonometry • understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent,

• graphs of all six trigonometric functions for angles of any magnitude.

• trigonometrical identities • solving equations, • Harmonic Form • Double angle formula • Compound angle formula

- the expansions of sin(A ± B), cos(A ± B) and tan(A ± B)

- the formulae for sin 2A, cos 2A and tan2A - the expression of 𝑎 𝑠𝑖𝑛 𝜃 + 𝑏 𝑐𝑜𝑠 𝜃 in the

forms 𝑅 𝑠𝑖𝑛(𝜃 ± 𝛼) and 𝑅 𝑐𝑜𝑠(𝜃 ± 𝛼) .

4. Differentiation • The product rule • The quotient rule • Derivatives of exponential functions. • Derivative of natural logarithmic functions. • Derivative of trigonometric functions. i.e.


sinx, cos x, tanx, 𝑡𝑎𝑛−1 𝑥 • Implict differentiation. • Parametric differentiation.

5. Integration • integration of ⅇ𝑎𝑥+𝑏, 1

𝑎𝑥+𝑏 , sin(ax + b), cos(ax

+ b), 𝑠ⅇ𝑐2(𝑎𝑥 + 𝑏) and 1

𝑥2+𝑎2

• use of double-angle formulae to integrate 𝑠𝑖𝑛2 𝑥 or 𝑐𝑜𝑠2(2𝑥).

• integrate rational functions by means of decomposition into partial fractions

• recognise an integrand of the form 𝑘𝑓′(𝑥)

𝑓(𝑥), and

integrate such functions • use integration by parts e.g. integration of x

sin 2x, 𝑥2ⅇ−𝑥 , ln x, 𝑥 𝑡𝑎𝑛−1 𝑥. • use a given substitution to simplify and

evaluate either a definite or an indefinite integral. e.g. to integrate 𝑠𝑖𝑛.2 2𝑥 𝑐𝑜𝑠 𝑥 using the substitution u = sin x

6 Numerical Solution of equation

• finding a pair of consecutive integers between which a root lies.

• understand how a given simple iterative formula of the form 𝑥𝑛+1 = 𝐹(𝑥𝑛) relates to the equation being solved.

• use a given iteration to determine a root to a prescribed degree of accuracy

7. Vectors • Addition and subtraction of vectors. • Multiplication of a vector by a scalar. • calculate the magnitude of a vector, and use

unit vectors, displacement vectors and position vectors.

• finding the equation of a line given the position vector of a point on the line and a direction vector, or the position vectors of two points on the line.

• determine whether two lines are - parallel, - intersect or are - skew, and - find the point of intersection of two lines

when it exists • Calculation of the shortest distance between a

line and a point.


• use formulae to calculate the scalar product of two vectors.

• Use scalar products to find the angle between two lines

• finding the foot of the perpendicular from a point to a line.

• questions may involve 3D objects such as cuboids, tetrahedra (pyramids), etc

8. Differential Equations • First Order Differential Equation. • Forming a differential equation from a

problem. • The technique of separating the variables. • find by integration a general form of solution

for a first order differential equation in which the variables are separable.

• Use an initial condition to find a particular solution

• interpret the solution of a differential equation in the context of a problem being modelled by the equation

9. Complex number • understand the idea of a complex number, meaning of the terms - real part, - imaginary part, - modulus, - argument, - conjugate, and - the fact that two complex numbers are

equal if and only if both real and imaginary parts are equal

• Notations Re z, Imz, |z|, arg z, z* should be known.

• Operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy.

• Solving a cubic or quartic equation where one complex root is given.

• Represent complex numbers geometrically by means of an Argand diagram

• carry out operations of multiplication and division of two complex numbers expressed in

polar form 𝑟(𝑐𝑜𝑠 𝜃 + 𝑖 𝑠𝑖𝑛 𝜃) = 𝑟ⅇⅈ𝜗


-

-

• Find the two square roots of a complex number e.g. the square roots of 5 + 12i in exact Cartesian form.

• Illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram e.g. |z – a| < k, |z – a| = |z – b|, arg(z – a) = α.

The following is the list of the CAIE Past Paper questions which I would recommend you to

practice for concept building. . (A little hint of the question is given in a bracket in front of the

question below.)

1. Partial Fraction

i. J16/32/q7. (mix of partial fraction with Integration.)

ii. M16/32/q9. (improper algebraic fraction and then integration.)

iii. N17/33/q8 (improper algebraic fraction and integrate to prove the given answer.)

iv. J18/31/qs.9. (improper algebraic fraction then apply Binomial theorem.)

v. M19/32/q8 (proper algenraic fraction and then apply Binomial theorem.)

vi. J18/33/qs.6 (good mix of partial fraction and differential equation.)

2. Long Division

i. N17/33/q1. (find quotient and remainder applying long division.)


3. Remainder and Factor Theorem

i. M16/32/q4. (apply factor theorem, solve cubic equation and justify your answer.)

ii. J18/31/q4. (apply factor theorem to find the value of unknown constant.)

iii. N15/31/q6 (apply factor and remainderr theorem)

4. Binomial Theorem

i. J19/32/q1. (coefficient of x cube and expand the binomial.)

ii. J18/33/q1. ( in the ascending power of x expand)

iii. J17/33/q2 (Binomial expansion)

5. Modulus Function

i. N18/32/q.1 (solving modulus inequality)

ii. N18/33/q1 (solve modulus inequality with unknown constant both side modulus.)

iii. N15/31/q1 ( solve modulus inequality both side modulus)

iv. J15/33/q2 (solve modulus inequality with one side modulus and other side linear

equ.)

6. Logarithm, Exponential and Linear Law

EXPONENTIAL EQUATION

i. J16/32/q1 (exponential eq. use logarithms to solve.)

ii. N19/32/q2. (solve exponential equation, apply substitution.)

iii. J18/33/q.2 (solve exponential equation, apply substitution)

iv. N18/32/q4 (solve exponential equation.)

v. N18/33/q2 (solve exponential equation.)

vi. J15/33/q1 (sketch the exponential equation.)

LOGARITHMIC EQUATION


vii. March16/32/q1. (solve logarithmic equation)

viii. J18/31/q.1 (solve logarithmic equation.)

ix. M19/32/q1 (change logarithmic equation into quadratic then solve.)

x. J15/33/q1 (solve logarithmic equation)

LINEAR LAW

xi. N17/33/q2. (draw a suitable straight line using given equation and values.)

7. Trigonometry

i. J16/32/q5. (Prove the identity and hence solve the equation.)

ii. M16/32/q2. (Solve trigonometric equation, apply double angle formula.)

iii. N17/33/q4. (prove the identity, apply double angle formula and then sketch.)

iv. M19/32/q3. (sine and cosine double angle formula and solve the equatioin.)

v. J19/32/q10 (mix of double angle application, integration and differentiation in the

given diagram.)

vi. J18/33/q.5 (cubic expansion to proof then solve the equation.)

vii. J18/33/q7. (Harmonic form then apply integration.)

viii. N18/33/q6. (Harmonic form then solve trig equation.)

ix. J15/32/q5 (Harmonic form)

8. Differentiation

i. J16/32/q4. (Apply qutient or product rule to get exact value of Stationary pt.)

ii. M16/32/q6. (implicit differentiation, product rule )

iii. N17/33/q5. (Implicit equation, differentiate it, show tangent is parallel to the x-axis.)

iv. M19/32/q.5 (differentiate the implicit equation. Prove the given form.)

v. J18/33/q4. (good mix of quotient rule, sketch and iteration)

vi. J18/33/q.8 (implicit equation, tangent parallel to y-axis)

vii. N18/33/q4 (parametric equation, tangent is parallel to the y-axis)


viii. N15/33/q4 (good mix of paramatric equation and iteration)

ix. J15/33/q5 (parametric equation)

x. J17/32/q4 (parametric equation, find the equation of the normal.)

9. Integration

i. J16/32/q3. (apply Integration by parts and get the exact value).

ii. J18/31/q8. (apply integration by parts then apply iteration.)

iii. M19/32/q4. (apply integration by parts to prove the answer.)

iv. J18/33/q3. (apply integration by parts, give answer in terms of pi)

v. M16/32/q5. (apply Integration by substitution and find the exact value.)

vi. J18/31/q.5 (apply integration by substitution, trig eq. given.)

vii. N17/33/q9. (mix of differentiation and Integration. Diagram given with shaded

region, find stationary point and area.)

viii. N19/32/q10. (diagram given, shahed area given, apply integration by

substitution)

ix. J15/32/q5 (diagram and situation given. Proof the equation and apply iteration.

10. Numerical Solution of equaiton

i. J16/32/q8. (diagram given, apply differentiation then iteraiton)

ii. M16/32/q3. (Simple basic iteration question.)

iii. N17/33/q3. (two iterative equation given, show that one fails to converge.)

iv. N19/32/q2. (simple application of iterative concept.)

v. J19/32/q.6 (diagram with shaded region given, situation given proof and apply

iteration.)

vi. N18/33/q3. (Sketch suitable pair of graph to prove one intersection point then apply

iteration.)

11. Vectors


Note: Plane is not a part of syllabus from 2020

i. J16/32/q9(i) (position vector of quadrilateral given, verify it is rhombs)

ii. N17/33/q10(i, ii). (two vector equation of line are given, show lines do not intersect

and find angle between them.)

iii. J18/31/q.10 (i). (perpendicular distance from a point to the line.)

iv. J19/32/q9(i). (two position vector given, line given , show do not intersect.

12. Differential Equation.

i. J16/32/q6. (Solve the given differential equation.)

ii. M16/32/q7. (Solve the given differential equation of polynomial and exponential.)

iii. N17/33/6 (differential equation of trigonometrix function, find the value of x.)

iv. J18/31/qs6 (situation is given to solve differential equation.)

v. N19/32/q6. (solve differntial equation using given condition.)

vi. N18/32/qs.6 (from the given situation setup the diffferential equation.)

vii. M17/32/q7. (from the given situation form a differential equation.)

13. Complex Number

i. J16/32/q10 (sketch loci Argand diagram, find least distance, modulus and

argument)

ii. M16/32/q10 (conjugate, sketch of loci argand diagram, find the greatest and least

values of arg. Z)

iii. N17/33/q7. (square root of z, sketch the loci argand diagram.)

iv. J18/31/qs7. (Without using a calculator solve equation, sketch argand diagrm, find

angle.)

v. N19/32/q7 (Without calculator solve the equation, sketch the argand diagram and

shade the region)


vi. J18/33/qs.9 (find the complex number. Sketch loci argand diagram, calculate the

greatest value of arg z.)

vii. J19/32/q5. (complex root of the cubic equation is given, find the other two roots.

viii. N18/32/q9. (rationalizing the denominator, polar coordinate form, sketch loci

argand diagram and shade the regon.)

ix. N18/33/qs.8 (complex number in exponential form)

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A LEVEL MATHEMATICS - (P3) 9709 · A LEVEL MATHEMATICS - (P3) – 9709 CAIE SYLLABUS Paper 3 1. Algebra 2. Logarithmic and exponential functions 3. Trigonometry 4. Differentiation