Mathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman

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1 03 04 Mathematics syllabus for Grade and For Bilingual Schools in the Sultanate of Oman Prepared By: A Stevens (Qurum Private School) M Katira (Qurum Private School) M Hawthorn (Al Sahwa Schools) In Conjunction with Ministry of Education: Ahmed Al-Thuhli Head of International schools Programs section Moosa Hadi Math Supervisor Based on:. Tetbooks: Advanced Maths AS Core for Edecel (C C) Pearson Longman Advanced Maths A Core for Edecel (C3 C4) Pearson Longman. Week comprising 6 lessons 3. Lessons 40min each

2 Eponents & Logs Equations ALGEBRA Inequalities Quadratic Equations & Functions Topic No. of Weeks Area of maths covered GRADE Semester Chapter in book PEARSON LONGMAN C:3 (pg 4-63) 0/03 Academic Year Components of topic to be covered Solving quadratic equations by: o Factorizing o Quadratic formula o Completing the square Sketching quadratic graphs o Ma/Min; o Shape o Turning Point (verte) & Ais of Symmetry Nature of roots(working with the discriminate) Objectives Completing the square. Solution of quadratic equations. Solution of quadratic equations by factorisation, using the formula and completing the square. Quadratic functions and their graphs Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations. C:4 (pg 65-70) Solving inequalities o Linear o Quadratic 0.5 The discriminate of a quadratic function Solution of linear and quadratic inequalities. For eample, a + b > c + d, p + q + r > 0, p + q + r < a + b C:5 (pg 7-87) Solving simultaneous equations o Linear, o One linear & one quadratic (algebraically & graphically) Intersection of linear & quadratic functions (3 cases of discriminate) Simultaneous equations: analytical solution by substitution. For eample, where one equation is linear and one equation is quadratic. Graphs of functions; sketching curves. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations. C:8 (Pg 3-33) Logarithms: o Definition.- log ep. o Rules: logcab ; logc(a b), logc(a n ) o Special cases ; logaa, loga, loga( a). Eponential function: o Graphs. o Relationship between (log and ep). Sketch ab y a and translations y a c Laws of logarithms a To include: log c ab ; log c b ; log a n c Special cases log a a ; log a ; log ( ) a a

3 Trig functions and angles in all TRIGONOMETRY Solving triangles, Radians and applications GEOMETRY Co-ordinate geometry C:6 (pg 9-0) Revision: o Coordinates, Midpoint, Gradient(value,+,-) o y = m + c (Drawing and writing the equations if two points are given) o The length of a line segment joining (,y) to (,y) Straight line : o Gradient as tan. o Special cases for gradient ( 0,, -, ) o Parallel and perpendicular lines (m=m, m.m= -) Equation of a line: y y = m ( ) or y y = y y ( ) Forms of equations of straight lines: o y=m + c, y=m, y=+c o y=c, y=o, =c, =0 o a+by+c=0 Sketching Applications: o Advanced use of the previous knowledge 3.5 Equation of a straight line, including the forms y y = m( ) and a + by + c = 0. To include: the equation of a line through two given points the equation of a line parallel (or perpendicular) to a given line through a given point. For eample, the line perpendicular to the line 3 + 4y = 8 through the point (, 3) has equation 4 y 3 3 Conditions for two straight lines to be parallel or perpendicular to each other. EX: ask for an equation for a line that is parallel to a line cuts a given equation for a curve in two points. To include equation of circle in the form ( a) + (y b) = r C:7 C:6. (Pages 5,53 & 8-30) C: (pg 54-68) Solutions of triangle (sin, cos, area rule) Radians: o Definition. o Radians Degrees. o Angles and Quadrants (0 360 ). o Area of sector and length of arc. (A=0.5r, s=r) o Area of a triangle. (A=0.5absinC) o Area of segment. (A= A- A) o Special triangles Trigonometric functions for any angle: o Sign. o Magnitude. o Special cases. Graphs of trigonometric functions: o y=sin, cos, tan. o Transformations of the graphs : y = ± af( ± ±A)±B 3 The sine and cosine rules, and the area of a triangle in the form ½ ab sin C. Radian measure, including use for arc length and area of sector. Use of the formulae s r and A r Sine, cosine and tangent functions. Their graphs, symmetries and periodicity. Knowledge of graphs of curves with equations such as y 3sin, y sin, y sin is epected 6

4 Geometric series SEQUENCE & SERIES Arithmetic series Binomial epansion ALGEBRA Algebra and functions Grade : Semester Identities C: (pg 87-00) C:4 (pg 8-7) Long division o Revision the concepts ; (Quotient, Divisor, Dividend, and Remainder) o Dividing a polynomial by ( a+b). o Dividing a polynomial by ( a +b+c). o A simpler method of division (e. a b` B A ) c d C D Remainder and Factor theorem Factorising polynomials Binomial epansion using Pascal s triangle. Notation n! and Formula for binomial epansion Algebraic division; use of the Factor Theorem and the Remainder Theorem. Students should know that if f() = 0 when = a, then ( a) is a factor of f(). Students may be required to factorise cubic epressions such as and Students should be familiar with the terms quotient and remainder and be able to determine the remainder when the polynomial f() is divided by (a + b). Should use a known factor to determine another factor. The students should be able to use Pascal s triangle to epand small positive integers n. Familiar with notations n! and able to use the binomial epansion formula to epand for all positive integers n for C:8.3 (pg 36-45) Arithmetic Series o Definition. o The concepts; common difference, progression. o Formula for the nth term arithmetic series. o Advanced applications. o Formula for the sum of n term(s) of arithmetic series. o Advanced applications. Arithmetic series, including the formula for the term & sum of the first n natural numbers. The general term and the sum to n terms of the series are required. The proof of the term & the sum to n terms formula should be known. C:0 (pg ) Geometric Series o Definition. o The concepts; common ratio, progression. o Formula for the nth term in the arithmetic sequences. o Advanced applications. o Formula for the sum of n term(s) of arithmetic sequences. o Advanced applications..5 The sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of r <. The general term and the sum to n terms are required. The proof of the sum formula should be known.

5 TRIGONOMETRY Identities & Equations ALGEBRA Standard functions and curve sketching C:7 (pg 3-8) C: (pg 68-77) Sketching and interpreting the curves of standard functions: o y=a, y=a+b, y=a +b+c o y =, y = 3 o y =, y= 3, y=, y, y n n Introducing the concepts ; Continuity, Discontinuity, Asymptote Geometrical Interpretation of the solution of equations o Solving two equations graphically. o Points of intersection between an equation and a given line (intersection- being tangent- etc) o Eplaining if two points intersect or don't. o Other advanced applications. Trigonometric Identities: o tan = sin cos. o sin + cos =. o sin = cos(90 -), cos = sin(90 -) Solution of trigonometric equations: o Simple. (E; Asin(a±b)=B) o Quadratic. o Advanced equations (required the above knowledge).5 Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations. Functions to include simple cubic functions and k the reciprocal function y with 0 Knowledge of the term asymptote is epected. Knowledge and use of tan and sin cos sin cos Proving various identities Solution of simple trigonometric equations in a given interval. Students should be able to solve equations such as

6 ALGEBRA Transformations of graphs C3: (pg - 4) Even and Odd functions: o Definitions. o Determining of a given function is an odd or even function. o Relationship between the graphs of odd and even functions. Modulus functions: o Definition of. o Equations with modulus signs. o Inequalities with modulus signs. o Sketching functions involving modulus signs. 4 o Comparison: f(), f(). Transformation of graph of f() to : o y = f()±a o y = f(±a)±b o y = -f() o y = f(-) o y = a f() o y = f(a) Sketching a graph of a function using the previous transformations rules. TOTAL NO. OF WEEKS 8 Definition of a function. a y f (), f ( ) with a odd or even. The modulus function. a b c d and a b 3 Combinations of the transformations y = f() as represented by y = af(), y = f() + a, y = f( + a), y = f(a). Students should be able to sketch the graph of, for eample, y = f(3), y = f(-) +, given the graph of y = f() or the graph of, for eample, y 3sin, y cos 4 The graph of y = f(a + b) will not be required.

7 Differentiation CALCULUS Differentiation ALGEBRA Partial fractions Topic No. of Weeks Area of maths covered Grade Semester Chapter in book PEARSON LONGMAN 0/03 Academic Year Components of topic to be covered Objectives C4:8 (pg 79-87) C:9 (pg 48-64) Distinct linear factors. Repeated linear factors. Improper fractions. Rates of change Tangent to a curve Gradient of a curve Differentiation The notation Function notation Vocabulary Differentiating from first principles Differentiation of polynomials Tangents and normals Rational functions. Partial fractions to include denominators such as (a+b)(c+d)(e+f) and a bc d and ( a b).use other resources for quadratic factor that cannot be factorised. The derivative of f() as the gradient of the tangent to the graph of y = f () at a point; the gradient of the tangent as a limit; interpretation as a rate of change; second order derivatives. dy For eample, knowledge that is the rate of change of y with d respect to. Knowledge of the chain rule is not required. The n notation f () may be used. Differentiation of, and related sums and differences. For eample, for n, the ability to differentiate epressions such as and is epected. 3 Applications of differentiation to gradients, tangents and normals. Use of differentiation to Find equations of tangents and normals at specific points on a curve. Using the st principle to find simple differentiations. C:5 (pg 30-43) Increasing and decreasing functions Stationary points Identifying the type of a stationary point Maimum and minimum problems Applications of differentiation to maima and minima and stationary points, increasing and decreasing functions. The notation f () may be used for the second order derivative. To include applications to curve sketching. Maima and minima problems may be set in the contet of a practical problem.

8 Integration INTEGRATION Integration TRIGONOMETRY Trig involving all trig ratio s in all quadrants C3:3 (pg 46-80) C:0 (pg 65-73) C:9 (pg ) Reciprocal Functions: o Definition. o Identities. o Graphs. o Comparing the si trigonometric functions. o Identities (involved all the si trigonometric functions). o Equations (involved all the si trigonometric functions). Addition formulae: o sin(a±b), cos(a±b), tan(a±b). o Advanced applications of all of the previous formulae. Double angle formulae: o sina, cosa, tana. o Advanced applications of all of the previous formulae. Half-angle formulae: o Sin0.5A, cos0.5a, tan0.5a. o Advanced applications of all of the previous formulae The reverse of differentiation Finding the constant C Using the integral sign Rules for integrating n Integration of a polynomial Applying integration Indefinite and definite integrals Area under a curve To find an area using integration Area between a curve and a straight line Area between two curves The Trapezium rule Formula for the trapezium rule 3.5 Knowledge of secant, cosecant and cotangent. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains. Angles measured in both degrees and radians. Knowledge and use of sec θ=+tan θ and cosec θ = + cot θ. Knowledge and use of double angle formulae; use of formulae for sin(a ± B), cos (A ± B) and tan (A ± B) and of epressions for acos θ + bsinθ in the equivalent forms of r cos (θ ± a) or r sin (θ ± a). To include application to half angles. Knowledge of the (tan ½ θ ) formulae will not be required. Students should be able to solve equations such as a cos θ + b sin θ = c in a given interval, and to prove simple identities such as cos cos + sin sin cos. Note: Inverse trigonometrical functions and the factor formulae are not required. Indefinite integration as the reverse of differentiation. Students should know that a constant of integration is required. Integration of n For eample, the ability to integrate epressions such as ½ 3 -½ and ( + ) is epected ½ Given f () and a point on the curve, students should be able to find an equation of the curve in the form y = f(). Evaluation of definite integrals. Interpretation of the definite integral as the area under a curve. Students will be epected to be able to evaluate the area of a region bounded by a curve and given straight lines. Eg find the finite area bounded by the curve y = 6 and the line y =. dy will not be required. Approimation of area under a curve using the trapezium rule. For eample, evaluate using the values of ( + ) at = 0, 0.5, 0.5, 0.75 and.

9 EXPONENTS & LOGS The functions and ln Probability Elementary probability The terminology of probability Sample space Addition rule Multiplication rule Tree diagrams Independent and mutually eclusive events Number of arrangements Elementary probability. Sample space. Eclusive and complementary events. Conditional probability. Understanding and use of P(A ) = - P(A), P(A U B) = P(A) + P(B) - P(A B), P(A B) = P(A) P(B A). Independence of two events. P(B A)=P(B), P(A B) = P(A), P(A B) = P(A)P(B). Sum and product laws. Use of tree diagrams and Venn diagrams. Sampling with and without replacement Grade Semester C3:4 (pg 90-96) Logarithms: o Relationship and graph of e and ln. o ln e ln o e N N o ln=loge e and its inverse ln: o Graphs of y af ( a) b if f ( ) e where a and b are positive or negative o Graphs of y af ( a) b if f ( ) ln where a and b are positive or negative o Solving equations involving e and ln The function e and its graph. To include the graph of y = ae b +c + d. The function ln and its graph; ln as the inverse function of e. Solution of equations of the form e a + b = p and ln (a + b) = q is epected.

10 INTEGRATION Integration Further Differentiation CALCULUS Differentiation C3:5 (pg 98-36) The chain rule Differentiation of powers of f() The use of Differentiation of the eponential function and natural logarithms Differentiation of functions of the type e f() Differentiation of ln Differentiation of ln k Differentiation of ln (f()) Simplifying a logarithmic function before differentiating Differentiation of products and quotients The product rule The quotient rule Differentiation of trigonometric functions Differentiation of sin and cos The derivatives of sin f() and cos f() Angle in degrees The derivatives of powers of sin and cos Trigonometric differentiation involving the product and quotient rules Differentiation of tan, cot, sec and cosec 3 Differentiation of e, ln, sin, cos, tan and their sums and differences. Differentiation using the product rule, the quotient rule and the chain rule. Differentiation of cosec, cot and sec are required. Skill will be epected in the differentiation of functions generated from standard forms using products, quotients and composition, such as Sin4, and tan. The use of Eg finding dy d dy d e 3 for d dy sin 3y C4:0 (pg 0-3) Implicit differentiation Second derivatives of implicit functions stationary points Connected rates of change Differentiation of simple functions defined implicitly. The finding of equations of tangents and normals to curves given implicitly is required. C4: (.-.4) (pg 44-70) Integration as the limit of a sum Solids of revolution Volume of revolution Integration of e and Limits of the definite integral b a Integration by recognition k f d n ' Integrals of the type f ' f Integrals of the type kf e d d 3 Integration of e, Students should recognize integrals of the form f ( ) d ln f ( ) f ( ) Evaluation of volume of revolution. y d is required, These methods as the reverse processes of the chain and product rules respectively. NOTE: Trigonometric Integration is NOT required.

11 STATISTICS Normal Distribution? The normal distribution The standard normal distribution Area under the curve of the normal distribution curve The Normal distribution including the mean, variance and use of tables of the cumulative distribution Knowledge of the shape and the symmetry of the distribution is required. Knowledge of the probability density function is not required. Derivation of the mean, variance and cumulative distribution function is not required. Interpolation is not necessary. Questions may involve the solution of simultaneous equations TOTAL NUMBER OF WEEKS 5

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