Course Content
Chapter-1. Functions
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Lesson-1.1.1- understand the terms: function, domain, range (image set), one-one function, inverse function and composition of functions.
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Lesson-1.1.2- use the notation f(x) = sin x, f: x ↦ lg x, (x > 0), f –1(x) and f 2 (x) [= f(f(x))]
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Lesson-1.1.3- understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic or trigonometric.
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Lesson-1.1.4- explain in words why a given function is a function or why it does not have an inverse.
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Lesson-1.1.5- find the inverse of a one-one function and form composite functions.
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Lesson-1.1.6- use sketch graphs to show the relationship between a function and its inverse.
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Chapter-2. Quadratic Functions
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Lesson-2.1.1- find the maximum or minimum value of the quadratic function f : x ↦ ax 2 + bx + c by any method.
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Lesson-2.1.2- use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain.
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Lesson-2.1.3- know the conditions for f(x) = 0 to have: (i) two real roots, (ii) two equal roots, (iii) no real roots.
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Lesson- 2.1.4- know the conditions for f(x) = 0 to have: and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve.
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Lesson-2.1.5- solve quadratic equations for real roots and find the solution set for quadratic inequalities.
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Chapter-3. Equations, Inequalities and Graphs
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Lesson-3.1.1- solve graphically or algebraically equations of the type |ax + b| = c (c ⩾ 0) and |ax + b| = |cx + d
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Lesson-3.1.2- solve graphically or algebraically inequalities of the type |ax + b| > c (c ⩾ 0), |ax + b| ⩽ c (c > 0) and |ax + b| ⩽ |cx + d|
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Lesson-3.1.3- use substitution to form and solve a quadratic equation in order to solve a related equation.
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Lesson-3.1.4- sketch the graphs of cubic polynomials and their moduli, when given in factorised form y = k(x – a)(x – b)(x – c).
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Lesson-3.1.5- solve cubic inequalities in the form k(x – a)(x – b)(x – c) ⩽ d graphically.
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Chapter-4. Indices and Surds
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Lesson-4.1.1- perform simple operations with indices and with surds, including rationalising the denominator.
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Chapter-5. Factors of Polynomials
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Lesson-5.1.1- know and use the remainder and factor theorems.
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Lesson-5.1.2- find factors of polynomials.
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Lesson-5.1.3- solve cubic equations
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Chapter-6. Simultaneous Equations
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Lesson-6.1.1- solve simple simultaneous equations in two unknowns by elimination or substitution.
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Chapter-7. Logarithmic and Exponential Functions
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Lesson-7.1.1- know simple properties and graphs of the logarithmic and exponential functions including l nx and e x (series expansions are not required) and graphs of ke nx + a and k ln(ax + b) where n, k, a and b are integers.
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Lesson-7.1.2- know and use the laws of logarithms (including change of base of logarithms).
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Lesson-7.1.3- solve equations of the form ax = b
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Chapter-8. Straight line Graphs
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Lesson-8.1.1- interpret the equation of a straight line graph in the form y = mx + c
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Lesson-8.1.2- transform given relationships, including y = axn and y = Abx , to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph.
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Lesson-8.1.3- solve questions involving mid-point and length of a line.
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Lesson-8.1.4- know and use the condition for two lines to be parallel or perpendicular, including finding the equation of perpendicular bisectors.
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Chapter-9. Circular Measure
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Lesson-9.1.1- solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure.
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Chapter-10. Trigonometry
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Lesson-10.1.1- know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent).
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Lesson-10.1.2- understand amplitude and periodicity and the relationship between graphs of related trigonometric functions, e.g. sin x and sin 2x
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Lesson-10.1.3- draw and use the graphs of y = a sin bx + c y = a cos bx + c y = a tan bx + c where a is a positive integer, b is a simple fraction or integer (fractions will have a denominator of 2, 3, 4, 6 or 8 only), and c is an integer.
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Lesson-10.1.4- solve simple trigonometric equations involving the six trigonometric functions and the above relationships (not including general solution of trigonometric equations)
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Lesson-10.1.5- prove simple trigonometric identities
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Chapter-11. Permutations and Combinations
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Lesson-11.1.1- recognise and distinguish between a permutation case and a combination case.
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Lesson-11.1.2- know and use the notation n! (with 0! = 1), and the expressions for permutations and combinations of n items taken r at a time.
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Lesson-11.1.3- answer simple problems on arrangement and selection (cases with repetition of objects, or with objects arranged in a circle, or involving both permutations and combinations, are excluded).
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Chapter-12. Series
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Lesson-12.1.1- use the Binomial Theorem for expansion of (a + b) n for positive integer n.
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Lesson-12.1.2- recognise arithmetic and geometric progressions
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Lesson-12.1.3- use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.
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Chapter-13. Vectors in Two Dimensions
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Lesson-13.1.1- know and use position vectors and unit vectors.
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Lesson-13.1.2- find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars.
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Lesson-13.1.3- compose and resolve velocities
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Chapter-14. Differentiation and Integration
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Lesson-14.1.1- understand the idea of a derived function
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Lesson-14.1.2- differentiate products and quotients of functions
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Lesson-14.1.3- apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems.
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Lesson-14.1.4- use the first and second derivative tests to discriminate between maxima and minima.
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Lesson-14.1.5- understand integration as the reverse process of differentiation.
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Lesson-14.1.6- evaluate definite integrals and apply integration to the evaluation of plane areas.
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Lesson-14.1.7- apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x–t and v–t graphs.
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