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JAMES RUSE AGRICULTURAL HIGH SCHOOL MATHEMATICS PROGRAMME - PRELIMINARY MATHEMATICS EXTENSION 1 – 2006 Preliminary: Year 11 Terms 1,2,3 PREAMBLE This program contains three major sections A. Revision of year 9 and 10 work, B. Preliminary Course to be completed in Year 11. C. HSC Course to be completed in Year 12 Section A contains work on coordinate geometry, algebra, trigonometry, geometry, probability, logarithms and use of the calculator. This should not be taught as a block at the commencement of the year but should be revision progressively through the use of short questions at the start of periods, short diagnostic tests and revision assignments which could be given as homework exercises. The strand on functions and calculus has been designed so that a variety of functions have been introduced before concepts and techniques involved in differentiation are met. This has been done to ensure that initial practice of rules for differentiation is made more meaningful for pupils. Teachers need to read carefully the appropriate pages of both syllabus and notes, to consult the support notes for this program, and to peruse previous Higher School Certificate Examination Papers so that the spirit and intentions of the syllabus are interpreted correctly. Syllabus Summary – Mathematics (Ext 1) Preliminary JRAHS Ext I (16.12.05) - Prelim 2006 HSC 1 TOPIC SUMMARY – PRELIMINARY COURSE Revision (1) Algebra (2) Absolute Value (3) Co-ordinate Geometry (4) Trigonometry (5) Geometry Proofs (6) Graphs (7) Logarithms Preliminary Course Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12 Unit 13 Unit 14 Unit 15 Unit 16 Further Graphs and Locus Quadratics Trigonometry Expansions Radian Measure Limits and Calculus Products, Quotients and Function of Function Calculus of Trigonometric, Exponential and Logarithmic Functions Applications of Differentiation Geometry of the Circle Inequalities Further Trigonometry Polynomials Trigonometric Equations Permutations and Combinations Locus and The Parabola Geometry of the Parabola JRAHS Ext I (16.12.05) - Prelim 2006 2 A. Revision of Year 9 and 10 work. Revision 1. Algebra Student outcomes (i) (ii) Student is able to: Simplify expressions involving removing parenthesis Substitute: Evaluate expressions involving the 4 operations, powers and roots. Number substitution may involve integers, fractions, decimals or surds Implications, considerations and implementations Example: Expand and simplify: 4 x 2 5x 7 3 x 2 7 x 1 , 2 x 3 x 2 5x 2 . Example: Find value of t 4 t 2 1 when t2 3 or 2 4 A4C 2 4 evaluate when A , B , 4 B 3 3 7 8 C . 3 Substitute into formulae Find h if v = 10, r = 2 and v r 2 h or find t 1 when a = 4, u = – 4, s = 6 and s ut at 2 . 2 (iii) Factorise expressions containing Common factors. Differences of squares. Differences and sum of cubes. Trinomials. Four terms by grouping in pairs. e.g. Factor/factorise: 5 x 2 10 x , 16 x 2 1 , 3 x 2 4 x 7 , 2 x 3 16 , x 2 y 2 3x 3 y . (iv) Simplify Algebraic Expressions: (v) Solve linear equations: 5t 3 21 t , (vi) Solve linear inequalities: 3x 4 2 12 , 3 2x 1 . (vii) Solve quadratic equations: 5 x 2 11x 2 0 , 8t 2 1 10t , y 2 6 y , 3 a3 x 2 5x 6 2 , , a2 a 4 x2 2m n m 3n 2 3 , . 3 6 x x x 2 v 22 (viii) Solve equations reducible to quadratics: JRAHS Ext I (16.12.05) - Prelim 2006 3x 4 3x 1 5 x 2 2 , . x 3x 1 5 x 2 16 . 2 x 4 9 x 2 8 0 , x 2 3x x 2 3x 3 0 . 3 (ix) Solve simultaneous equations: Standard of difficulty only to the extent required on later topics. (x) Perform arithmetic operations on Surds: Simplification: Solve: 3x 2 y 7 and 5x y 2 18 . 50 , 4 18 Addition and subtraction: 2 3 27 , 3 2 18 3 . Multiplication: 3 1 3 1. Rationalising denominators: 3 2 (xi) 2 2 3 3 6 , 2 3 1 3 3 1 , 2 3 1 , 2 , 3 1 , 3 2 2 3 2 Simplify Indices: Revision of index laws for positive indices. Use in simplifying algebraic expressions e.g: 3x Meaning of negative indices. Evaluation of numeric expressions. 2 9 , 3 Simplification of algebraic expressions. 3 x 1 2 x .4 2 x 1 a 1 b 1 , , ab (2 x ) 3 y 2 Meaning of fractional index. Evaluation of numeric expressions. Writing expressions in index form Writing expressions in surd form. Solve Indicial equations: Simultaneous indicial equations: (xiii) Polynomials: 4 operations on polynomials. Remainder and factor theorems. Factoring cubic or quartic polynomials. JRAHS Ext I (16.12.05) - Prelim 2006 1 2 2 3 x, x , x 1 2 3 2 8 , 4 . x , x (xii) 2 x .4 x , (2 x ) 3 2 3 2 3 1 , x2 , 2x 3 x2 1 2 4 3 Solve: 2 32 , y 81 , 9 n 27 . n Solve: 5 x y 125 and 7 x y 1 . Exercises such as finding b and c if x 3 bx c is divisible by x 2 and x 3 . 4 Revision 2. Absolute Values Students will have been introduced to this topic in the junior school. Therefore a large amount of time should not be spent on this topic. Pre-testing should be used to identify areas of weakness that can be addressed. Student outcomes (i) Implications, considerations and implementations Student is able to: Give definition for absolute values: a for a 0 (a) a a for a 0 (b) a a 2 (c) a considered in terms of the distance of a from zero on a number line. (d) a b = the distance between a and b on a number line (ii) Evaluate expressions containing absolute values. Evaluate: 4 5 13 , 9 3 5 (iii) Graph simple functions involving absolute values. Graph: y x 2 , y 3x 2 , y 3 x (iv) Solve equations containing absolute values. Solve: x 2 5 , 3x 2 1 , 3x 2 5 x 4 , 3x 2 5 x 4 (v) Solve inequalities containing absolute values. Solve: 3x 1 7 , 2 x 5 , 5x 1 8 , x 1 x 1 , 2 x 1 x 2 3 (Note: The first 3 could be solved by considering distances on a number line) (vi) Sketch harder absolute value graphs JRAHS Ext I (16.12.05) - Prelim 2006 Sketch: x y 1 , x y a , y 2 x 1 5 Revision 3. Coordinate Geometry Student outcomes (i) (ii) Student is able to: Find for a Straight line: Slope of the line. Intercepts. Draw its graph. Condition for a point to be on a line. Distance between point. Midpoint of interval. Standard form for equations of lines. Point–gradient form Angle between line and positive direction of the x-axis (Angle of inclination) For Parallel and perpendicular lines: (a) Find gradient of a given line. Know condition for lines to be parallel ( m1 m2 ). Implications, considerations and implementations e.g: Find k if A(2,3) lies on line 3 x 4 y k . e.g: y = 0 for x–axis, x = 0, for y–axis. y = c for line parallel to x–axis, x = k for line parallel to y–axis. m tan e.g: Find k if 3x 4 y 7 and kx 5 y 10 are parallel. Find the value of k which makes lines parallel. (b) Use the properties for parallel and perpendicular lines to prove that a particular figure is a parallelogram e.g: Prove that the figure bounded by the lines 3x 2 y 8 , 4 x 3 y 7 , 3x 2 y 12 and 4 x 3 y 11 is a parallelogram. (c) Find the equation of a line through a given point parallel to a given line. e.g: Find equation of line through A(2, 3) and parallel to 3x 4 y 10 . Derivation to be based on fact that equation is of form 3x 4 y c and A(2,3) lies on it. An alternative equation for a line parallel to ax by c and passing through x0 , y 0 is ax x0 b y y0 0 . (d) Know the condition for lines to be perpendicular and prove that m1m2 1 Students should be able to find the slope of a line perpendicular to a given line (ie use 1 m2 ) and also prove that lines are m1 perpendicular (ie show that m1m2 1) (e) Use in proving that a particular figure is a rectangle (opposite sides equal, one angle a right angle). JRAHS Ext I (16.12.05) - Prelim 2006 6 (f) Find values of k which make a pair of lines perpendicular e.g.: Find k if 3x 4 y 7 is perpendicular to kx 5 y 10 . (g) Find the equation of a line through a given point which is perpendicular to a given line. e.g.: Find equation of the line through A(4,–2) and perpendicular to 3x 7 y 10 . Derivation to be based on fact that equation is of form 7 x 3 y c and A(4,–2) lies on it. An alternative equation for a line perpendicular to ax by c and passing through x0 , y 0 is bx x0 a y y0 0 . (iii) Find the equation of a line through the intersection of two given lines. Given lines l1 0 and l2 0 then the equation of a line through the intersection point of these lines has the form l1 kl2 0 . Reference: HSC Course in Mathematics Year 11 – 3 unit Chapter 4 - Jones and Couchman (iv) Find perpendicular distance from a point to a line. (v) Circles: Find equation of a circle from its locus definition. Write down the equation of a circle given its centre and radius. Find the centre and the radius of a circle from an equation given in central form or general form. (vi) e.g.1: Write down the centre and radius of the circle with equation x 32 y 42 25 e.g.2: Find the centre and radius of circle with equation x 2 y 2 6 x 8 y 1 0 . Test if a line is a tangent or a chord to a circle through (a) solution of simultaneous equations and (b) use of the perpendicular distance formula. e.g. Find the point(s) of intersection of the line x y 2 and the circle x 2 y 2 5 . e.g. Find R if the line x y 4 is tangent to the circle x 1 y 2 R 2 . 2 (vii) 3U Divide a line into a particular ratio m:n, both internally and externally. JRAHS Ext I (16.12.05) - Prelim 2006 2 e.g. Use similar triangles to derive the ratio formula 7 Revision 4. Trigonometry Students will have been introduced to this topic in the junior school. Therefore a large amount of time should not be spent on this topic. Pre-testing should be used to identify areas of weakness that can be addressed. Student outcomes (i) Student is able to: Redefine trig. ratios in terms of unit circle incorporating exact values. Evaluate trig. Ratios for angles in all quadrants (ASTC). Implications, considerations and implementations Value of expressions such as sin 210 , cos 225 , tan 240 . Simplifying expressions such as sin 180 , cos180 , tan 360 . Find exact values for trig. Ratios of angles associated with 30, 45, 60 and also 0, 90, 180, 270, 360 (ii) Sketch graphs of y sin x , y cos x , y tan x . (iii) Solve simple equations of form sin x c , cos x c , tan x c . (iv) Solve Problems involving right angled triangles including those in which: (a) a diagram is not given e.g, A ladder 20m long rests against a vertical wall. The ladder makes an angle of 60o with the ground. Find the distance that the ladder will reach up the wall. (b) angles of elevation and depression are involved (c) simple bearings are used (v) Use the sine rule to calculate the length of a Application to problems involving angles of side given 2 angles and a side or calculate an elevation and depression, bearings and angle given 2 sides and an angle. geometrical figures. (See Year 10 program and syllabus for guidance on standards). (vi) Use the cosine rule to obtain sides and angles and apply to bearings and geometrical figures. JRAHS Ext I (16.12.05) - Prelim 2006 8 (vii) Solve cases where two triangles have a common side and do not overlap and cases where triangles overlap. Questions should include problems on angles of elevation and depression as well as problems with bearings. (Note: The method to be adopted at all times requires students to obtain a general expression for the length and angle required and then to carry out a single calculation, or find calculation as a final step). JRAHS Ext I (16.12.05) - Prelim 2006 9 Revision 5. Geometry Revision of Years 7–10 geometry. The syllabus notes and sample questions should be closely examined in order to decide on the standard and explanations sequence. Practice should be given in choosing correct strategies and techniques for solutions of problems. Student outcomes (i) Implications, considerations and implementations Student is able to: Define triangles and special quadrilaterals. (ii) Define properties of angles on a straight line, vertically opposite angles, angles at a point and use these properties in a variety of problems. (iii) Identify parallel lines and angles on parallel lines and use tests for parallel lines. (iv) Identify and calculate: (a) exterior angles angle sum of a triangle. (b) angle sum of quadrilaterals and general polygons. (c) sum of external angles of general polygon. (v) Test for congruent triangles (vi) (a) Recognise properties of parallelograms, rhombus, rectangle and square and (b) use sufficiency conditions to test for these quadrilaterals. (vii) Test for similar triangles and calculate sides in similar triangles. (viii) State and use Pythagoras' Theorem and its converse. (ix) Derive and use area formulae for parallelogram, triangle, trapezium and rhombus. JRAHS Ext I (16.12.05) - Prelim 2006 10 (x) Prove the properties: (a) A line parallel to one side of a triangle divides the other two sides in the same proportion. (b) A line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. (c) Parallel lines preserve ratios of intercepts on transversals. (xi) Complete simple numerical exercises of a deductive nature, on the above properties. e.g. (a) In the figure AE = 15cm , EC = 21cm, DB = 24 cm , and DE||BC. Find the length of AD. C 21 B E 15 x D A (b) D, E, F are the midpoints of the sides of a triangle ABC. Prove that triangles ABC and DEF are similar. (xii) Solve problems involving all of the above properties with or without diagrams, using numerical or general cases. JRAHS Ext I (16.12.05) - Prelim 2006 For examples see Syllabus pages 17 - 20 11 Revision 6. Graphs Student outcomes (i) Implications, considerations and implementations Student is able to: Graph straight lines using intercepts. (ii) Graph parabola by using axes of symmetry/vertex approach with points of intersection with axes where appropriate. (iii) Graph polynomial functions: (iv) Graph circles of form x a 2 y b 2 R 2 and semi-circles of Graph: y xx 2x 3 , y x 1x 1 , y x 3 3x 2 4 x , y x 3 6 x 2 11x 6 . 2 the form y a 2 x 2 or y a 2 x 2 . (v) Graph hyperbolae: (vi) Graph examples involving asymptotes: E JRAHS Ext I (16.12.05) - Prelim 2006 1 2 ( x 2) , y , y , x3 x ( x 3) 1 1 y 2 , y 1 . x x Graph: y y x x ( x 1) , y , y 2 . ( x 2)( x 2) x 1 ( x 1) 2 12 Revision 7. Logarithmic and Exponential Functions Students will have been introduced to this topic in the junior school. Therefore a large amount of time should not be spent on this topic. Pre-testing should be used to identify areas of weakness that can be addressed. Student outcomes Implications, considerations and implementations Student is able to: (i) 12.3 Find values of 2 x ,3 x ,4 x , .... using a hand calculator. Draw graphs of y 2 x ,3 x ,4 x , .... Graph y e x . (ii) Define log a x, log e x . 12.1 N a x x log a N , N 0, a 0, x (iii) Evaluate simple logarithmic expressions 12.2 from this definition. 1 eg: log28, log93, log 2 . 4 (iv) Sketch log. graphs Sketch y log a x, a 2, 3, 4,, 10, e (v) Use index laws to prove rules for logarithms: (for a, x, y > 0) Verify the results: log a a x x and a loga x x . 12.2 (a) log a a 1 (b) log a 1 0 (c) log x log y log xy (d) log x n n log x x (e) log x log y log y 12.3 (f) log a b log c b log c a (vi) Simplify numerical expressions 12.3 log 2 20 log 2 5 , (vii) Solve simple equations. eg: log 7 x 2 , 3 x 10 , log 2 x 8 . log 2 8 . log 2 4 12.2 Practice in writing a logarithm statement as in index form and vice versa. JRAHS Ext I (16.12.05) - Prelim 2006 13 (viii) Simplify algebraic expressions 12.3 JRAHS Ext I (16.12.05) - Prelim 2006 x2 y in terms of a, b and c if Express log z log x a , log y b and log z c . 14 Prelim. UNIT 1. FURTHER GRAPHS and LOCUS Student outcomes (i) (ii) Implications, considerations and implementations Student is able to: Understand the concept of a function. Interpret function notation. State domain and range. State the natural domain of a function, as well as the range corresponding to this domain, as each function is graphed. x : x 3, x : 2 x 2, x : x 4, y : y 2 is to be Appropriate set notation e.g. introduced. For a real domain that excludes a point (say x 3 ) use x : x , x 3 or x : x 3 or x 3 but not x 3. Discussion of a curve such as x 2 y 2 r 2 as two functions y r 2 x 2 and y r 2 x 2 Also consider the semi-circles derived from the 2 2 circle x a y b r 2 . i.e. y b r 2 x a 2 (iii) Graph simple cases where the function rule varies for different parts of the domain 0 if x 0 e.g. f ( x) 2 x if x 0 and functions where domain is restricted e.g. graph y 4 x 2 for x 1 . (iv) Identify and use the symmetry properties of odd and even functions. Define even functions as functions such that f(x) = f(–x) and give a geometric interpretation ( = graph is invariant under a reflection in y-axis) Define odd functions as functions such that f(–x) = –f(x) and give a geometric interpretation ( = graph is invariant under a 180 rotation about the origin) JRAHS Ext I (16.12.05) - Prelim 2006 Use of symmetry properties of odd and even functions in graphing functions such as: y 2x 3 , y x2 1 . x2 1 Find the axis of symmetry of graphs of even functions. Understand point symmetry of graphs of odd functions. 15 (v) Determine the set of points which satisfy a Examples: given set of conditions either algebraically or (1) The locus whose equation is x 2 y 2 0 geometrically. consists of the points lying on either of the straight lines y x , and y x . (2) The locus of points P equidistant from two distinct points A and B. ie PA=PB [Locus is the perpendicular bisector of the segment AB]. (3) The locus of a point P such that its distance from a point A is k times its distance from point B. ie AP=kBP [Locus is a circle] (4) Locus of a point P and two fixed points A and B such that PA+PB = constant. [Locus is an ellipse] (5) Locus of points equidistant from a fixed point and a fixed line. [Locus is a parabola – will be treated in more depth in Unit 15] Circle already covered in Revision Unit 3 (vi) Graph inequalities with at most one nonlinear inequality. Examples: (1) Indicate the region determined by the inequality: 2 x 3 y 6 . (2) Indicate the region determined by the 2 inequality: x 3 y 2 1 . (3) Shade the region that satisfies all three inequalities: x 2 y 2 1 , y 2 x , x 0 . (vii) Sketch graphs by addition or subtraction of ordinates Graphs could include: y x 2 1 , y x 2 , x x E y x 2 x , y sin x cos x (viii) Sketch further graphs by: deciding if the function is even or odd, finding points of intersection with the axes, E finding asymptotes, plotting additional points. Examples could include: x2 2 x2 , y y x4 , y 2 ( x 1)( x 3) x 2 JRAHS Ext I (16.12.05) - Prelim 2006 16 Prelim. UNIT 2 QUADRATICS Student outcomes (i) Implications, considerations and implementations Student is able to: Solve quadratic equations using (a) Factorising (b) Completion of square Solve equations of the type: (a) x 2 7 x 12 0 , 4 x 2 7 x 3 0 , x 2 4x . (b) 4 x 2 7 x 3 0 , x 2 6 x 4 0 . (ii) Finding roots of quadratic equation using the Prove the quadratic formula by completing the quadratic formula. square. (iii) Solve quadratic inequalities by the use of an appropriate graph. (iv) Establish the relation between roots and coefficients. (v) Find the discriminant. Use the discriminant to determine types of roots. Solve x 2 4 x 3 0 , x 2 16 , x 2 3x . (1) Finding values of k that give (a) real, (b) unequal and (c) unreal roots for equations such as x 2 kx 4 0 . (2) Find the value(s) of m so that the line y mx m is a tangent to the parabola y x2 . (vi) Identify types of quadratic expressions: Find value(s) of k for which an expression of positive definite, negative definite, indefinite. the form kx2 3x 4 is positive definite or negative definite. (vii) Write identical quadratic expressions. Examples should include the expression of a quadratic polynomial ax 2 bx c in the form Axx 1 Bx C . (viii) Equations reducible to quadratic equations: Equations should be of the order of difficulty: R (a) x 4 9 x 2 20 0 . (a) From the 2 unit syllabus 3U 2 (b) From the 3 unit syllabus JRAHS Ext I (16.12.05) - Prelim 2006 1 1 (b) x 3 x 2 0 , x x 2n n 2 9.2 8 0 , 9 n 7.3 n 18 0 17 Prelim. UNIT 3 TRIGONOMETRY EXPANSIONS This topic is revised and extended in Prelim. Unit 10 Student outcomes (i) (ii) Student is able to: Write down basic identities: Implications, considerations and implementations sin 2 A cos A , cos2 A sin A , sin A sin A , cos A cos A , tan A tan A Prove that cos A B cos A cos B sin A sin B . Derive results for cos A B, sin A B , tan A B . (iii) Use formulae in both "directions". (iv) Derive the results for sin 2 A , cos 2 A , tan 2 A JRAHS Ext I (16.12.05) - Prelim 2006 e.g. Write an equivalent expression for sin A Bsin A B cos A Bcos A B and an expression for tan 2 X Y . 18 Prelim. UNIT 4 RADIAN MEASURE This topic is encountered again in the HSC course , Unit 6 Student outcomes (i) Implications, considerations and implementations Student is able to: (a) Define a radian as an angle subtended at the centre of a circle by an arc equal to a radius. Convert degrees radians using a calculator and the relationship 180 0 . (b) Evaluate expressions involving radians (ii) (iii) (b) sin 4 , sin 2 Find the length of an arc: l R . Calculate arc length given R and and calculating given l and R. 1 2 R. 2 Find the area of a segment as the difference between the areas of a triangle and sector. 1 A R 2 sin . 2 Find the area of sector: A (iv) Apply the formula for the length of an arc and the area of a sector to solving problems. e.g. Finding the volume of a cone formed from a sector of given dimensions. (v) Graph trigonometric functions using radians e.g. y 3 cos 2 x , y sin x , y 1 cos x (vi) Solve simple equations using graphs e.g. Graph y sin 2 x and y 12 x , hence find an approximation to the solution of sin 2 x 12 x . (vii) Prove that sin x < x < tan x for 0 x 2 . This can be done by comparing areas T B rtan r O r A area OAB < area sector OAB < area OAT 1 2 1 1 r sin r 2 r 2 tan 2 2 2 JRAHS Ext I (16.12.05) - Prelim 2006 19 (viii) Prove that for small values of x, sin x x tan x . (ix) lim sin x , x0 x lim tan x lim 1 cos x , x0 x x0 x Deduce results for JRAHS Ext I (16.12.05) - Prelim 2006 20 Prelim. UNIT 5 LIMITS AND CALCULUS Student outcomes (i) 8.3 Student is able to: Develop the notion of a gradient function by measurement and tabulation of gradient at a number of points on curves. Implications, considerations and implementations e.g. y x 2 , y x 3 Predict a gradient function for these functions proposed from tabulated values. (ii) Develop an informal idea of a limit. 8.4 Limits of expressions such as x2 a2 as x a xa 2 xh h 2 and as h0. h No extensive work with evaluating limits is required in the Mathematics (2u) syllabus Not to be examined (iii) Define continuity 8.2 (a) A function is continuous at a point x c if f x is defined at x c the limit of f x as x c (from above and below) exists and f c equals this limit No formal work is required. Not to be examined (b) A function is continuous for a x b if it is continuous for all point within the domain a x b . (iv) Define differentiability (a) A function is differentiable at a point x c if f x is defined at x c the limit of f x as x c (from above and below) exists and f (c) equals this limit Graphs such a those involving absolute values could be used to demonstrate that the gradients before and after a point may not be the same. Not to be examined (b) A function is differentiable for a x b if it is differentiable for all point within the domain a x b . JRAHS Ext I (16.12.05) - Prelim 2006 21 (v) 8.5 State the formal definition of derivative as: f ' x lim x 0 (vi) f ( x x) f ( x) x Practise finding derivatives of quadratic and cubic expressions from first principles. 8.7 Find derivative of xn for positive integer n. Alternative notation could include f ( x h) f ( x ) f ' x lim or h 0 h f ( x ) f (c ) f ' c lim x c xc Use of various notations should be introduced: dy d 2 x 3x , y , f x , f 3 , dx dx (vii) Find, by first principles, the derivative of 1 8.8 y x and y x . (viii) Prove from the definition of the rules, the 8.8 derivatives for cf(x), f(x)±g(x). (ix) Complete simple exercises. Examples: (a) Differentiate: x 3 2 x 1 , 3 x 8.8 x x (b) Differentiate expressions that have been 2 simplified by expanding: y 2 x 3 , y 4x 53x 2 . (c) If f x x 3 4 x 2 6 x 2 find f 2 . (x) Use correct notation: Derivatives of q f p , s g t . Derivatives with variables other than x and y: ds dV s 3t 2 4t 5 find , V 4r 2 find . dt dr (xi) Find equation of tangent and normal. e.g. Find the area of the triangles bounded by the coordinate axis and the tangent to y x 2 x at A(1,2). JRAHS Ext I (16.12.05) - Prelim 2006 22 Prelim. UNIT 6 PRODUCTS, FUNCTION OF FUNCTIONS, QUOTIENTS Student outcomes (i) Implications, considerations and implementations Student is able to: Prove that if y = u(x).v(x) then dy dv du dy u v uv u v or dx dx dx dx Simple exercises such as finding derivatives of 2x 13x 4. Further practise can be included with exercises using the function of function rule Find equations of tangent and normals. See Fitzpatrick Ch 14 (ii) Understand the concept of a function of a function introduced through calculators. Derive function of function rule. If y F (u ) where u f ( x) then dy dy du = . dx du dx (iii) Complete exercises using the function of function rule. Differentiate: 3 x 47 , 6 x2 1 , x 5 2 4 3 , etc. Exercises could include questions that reinforce the product rule Example: Find the derivative of differentiate x 2 4 x 1 (iv) Find the derivative of y y v x . 4 x 1 hence 1 considered as v( x) 1 Prove the quotient rule: du dv v u d u dx dx or dy vu uv = 2 dx dx v v2 v (v) Apply these rules to finding equations of tangents and normals. Find derivatives of functions such as: 3 x 1 , , etc. 2x 1 x 1 The first may be more easily done using the function of function rule. e.g. (i) Find the equation of the tangent to x 1 1 y at 2, . x 1 3 (ii) Find the equation of the normal to y 2 x 3 at the point where x 6 . JRAHS Ext I (16.12.05) - Prelim 2006 23 Prelim. UNIT 7 CALCULUS of TRIGONOMETRIC, EXPONENTIAL and LOGARITHMIC FUNCTIONS Student outcomes (i) Implications, considerations and implementations Student is able to: Prove the derivative of sin x . Find the derivatives of the other trig. functions and their reciprocals using trig. identities and rules of differentiation. (ii) Find simple derivatives of expressions containing trig. functions. (iii) Complete exercises using the product, quotient and function of function rules for expressions involving trigonometric expressions. (iv) Find the equation of the tangent or normal to Find the equation of the normal to y sin 2 x at a curve the point where x 8 (v) Find derivative of 10x from 10 x x 10 x . x 0 x lim 5x sin x , tan x x 2 sin x , lim 1 , cos x x . x sin x , cos3x 4 . sin x 1 10 x 1 (= ln10) can be estimated by x 0 x considering successively smaller values for x. Generalise for y = ax. Find derivative of ex, e f x , a x . (vi) Investigate the relation dy dx 1. dx dy This could be demonstrated by considering the functions such as y x 3 and y 3 x . (vii) Find the derivative of log e x and log e f x . Differentiate expressions in bases other than “e” Use of change of base theorem. (viii) Find simple derivatives of functions containing log. functions. (ix) Solve simple problems involving log. differentiation. JRAHS Ext I (16.12.05) - Prelim 2006 Differentiate: 3 log e x , x 2 ln x 2 x e.g. Find the equation of the tangent at the point where y log e x cuts the x–axis 24 Prelim. UNIT 8 APPLICATIONS OF CALCULUS This topic, even though it contains material from the HSC course, is included here to give a better overall view of differentiation and its uses and also to further reinforce the various methods of differentiation. The various sections in the topic will be revised and extended in the appropriate units of the HSC course. Student outcomes Implications, considerations and implementations Student is able to: CURVE SKETCHING (i) (a) State the significance of the sign of the first derivative. (b) Identify monotonic increasing and decreasing functions. (ii) (a) Find stationary points on a curve. dy (A definition can be found in the syllabus For a stationary point at x0 , y 0 , dx 0 . For note 10.2). a max/min turning point, finding values of x dy 0 is sufficient for sketching for which (b) Identify local maxima and minima. dx dy (c) Distinguish between local maximum or 0 does not always most curves. However dx minimum and absolute maximum or imply that there is a turning point; but in all minimum value of a function over a dy given domain. cases of a turning point must change sign dx for points before and after x0 , y 0 , while for dy some curves may not exist at : x0 , y 0 yet dx the curve changes direction. Consider curves such as y 3 x 2 and y x for which the gradient functions do not exist at their max/min turning points. y 5 y 4 5 3 4 2 3 1 2 x -10 -5 1 5 x -1 -10 -5 5 -1 (iii) Define the second derivative JRAHS Ext I (16.12.05) - Prelim 2006 introduce appropriate notation – d dy d 2 y , y , f x , g 4 , etc , dx dx dx 2 25 (iv) (a) Find second derivative and use it to determine concavity – concave up and concave down. NOTE: Stress that a point of inflexion is a point about which the concavity of the curve changes. 2 d y 0 is not a sufficient test for inflexion (b) Investigate the geometrical significance dx 2 of the sign of the second derivative. d2y points, since 0 when x = 0 for both dx 2 (c) Examine inflexional tangents – these may be horizontal, oblique or vertical. y x 2 and y x 3 but y x 2 has a minimum turning point at 0,0 while y x 3 has a (d) Find points of inflexion and horizontal horizontal inflexion point at 0,0 . points of inflexion. Curves may also have inflexion points where the tangent is vertical – consider y 3 x . There is still a change in y" before and after 0,0 . 1 x3 hx = 2 y 1.5 1 0.5 -3 -2 -1 x 1 2 3 -0.5 -1 -1.5 -2 d2y for dx 2 values before and after x0 , y 0 . In all cases students must test Students should be made aware of the geometrical significance of the inflection point it is the only point where the tangent at a point crosses the curve at that point. (v) Sketch simple polynomial curves and rational functions using calculus techniques combined with work from Years 9 and 10. (Polynomials of degree 3 and higher.) JRAHS Ext I (16.12.05) - Prelim 2006 e.g. sketch y x 3 3x 2 9 x 2 , y x 2 2 . x 26 MAXIMUM and MINIMUM VALUE PROBLEMS (vi) Construct the function to be analysed from data given in words or on a diagram. Geometrical and practical problems are to be stressed. e.g.1: Constructing various containers or enclosures to maximise/minimise areas, volumes, costs etc. given fixed perimeters, surface areas etc. Prove that a closed cylinder of fixed surface area has maximum volume when its diameter equals its height. e.g.2: Given the hourly running cost of a ship as a function of its velocity, find the most economical running speed. Students need to pay particular attention to restrictions on variables and their explanation of why there is a local as well as an absolute max./min. for the values under consideration. * Consider problems in which more than one case needs to be analysed – HSC 1988 Q7b ** Consider problems in which the required solution is at an endpoint – Fitzpatrick (2u) Ex15e Q5 JRAHS Ext I (16.12.05) - Prelim 2006 27 Prelim. UNIT 9 GEOMETRY OF CIRCLE Except for the assumption "equal arcs subtend equal angles at the circumference or the centre in circles of equal radius ", students should be reasonably familiar with this work from the junior school. The emphasis in this topic is to be on: (i) a logical development of the theorems based on the definitions and assumption, (ii) solution of problems giving reasons for each step. Students should be asked to prove the preliminary theorems and the theorems for which proof may be examined themselves. The chief thrust of the work in exercises should be centred on developing the ability to choose the appropriate facts for the solution of problems. All circle geometry proofs are 3 unit work only. Student outcomes (i) Implications, considerations and implementations Student is able to: Understand and use definitions, assumptions and theorems within circle geometry. ALL 3U (a) Define a circle, centre, radius, diameter, arc, sector, segment, chord, tangent, concyclic points, cyclic quadrilateral, an angle subtended by an arc or chord at the centre and at the circumference, and of an arc subtended by an angle should be given. Assumptions: (1) Two circles touch if they have a common tangent at the point of contact. (2) The tangent to a circle is perpendicular to the radius drawn to the point of contact. Converse. (ii) Discuss and prove the following results. Reproduction of memorised proofs will not be required. Prove any of the following results using properties obtained (a) Equal angles at the centre stand on equal chords. Converse. (b) The perpendicular from the centre of a circle to a chord bisects the chord. Converse JRAHS Ext I (16.12.05) - Prelim 2006 28 (c) Equal chords in equal circles are equidistant from the centres. Converse (d) Any 3 non-collinear points lie on a unique circle whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining pairs of non-collinear points. (e) The angle at the centre is twice the angle at the circumference subtended by the same or equal arcs. (f) Angles at the circumference in the same segment are equal. (g) The angle in a semi-circle is a right angle. (h) Opposite angles of a cyclic quadrilateral are supplementary. (i) The exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite angle. (j) If the opposite angles in a quadrilateral are supplementary then the quadrilateral is cyclic (also a test for 4 points to be concyclic). (k) If the exterior angle of a quadrilateral equals it opposite interior angle then the quadrilateral is cyclic. (l) If an interval subtends equal angles at 2 points on the same side of it then the endpoints of the interval and the 2 points are concyclic. (m) The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. (n) Tangents to a circle from an external point are equal. (o) When circles touch, the line of centres passes through their point of contact. (p) The products of the intercepts of 2 intersecting chords/secants are equal. JRAHS Ext I (16.12.05) - Prelim 2006 29 (q) The square of the length of the tangent from an external point is equal to the product of the intercepts on the secant. (iii) Solve problems. The syllabus supplies a number of numerical and deductive exercises that can be used as a guide to the standard to be encouraged. (a) Find the value of x. xo o 100 (b) A B >> 30 130 o o O >> C E D AB||CD. O is centre of circle. Find size of angle BDE. (c) D C O 150 75 o o B A O is the centre. Prove that AC||BD JRAHS Ext I (16.12.05) - Prelim 2006 30 (d) T P 6 cm 10 cm O TP is a tangent. O is the centre. Find the length of TP (e) B >> A ^ ^ >> D C ABCD is a parallelogram inscribed in a circle. Prove that ABCD is also a rectangle. (e) A Q P B AP and AQ are diameters of two circles that intersect at A and B. Prove that P, B and Q are collinear. JRAHS Ext I (16.12.05) - Prelim 2006 31 A (f) D E O B C AB is a chord of lenght 8 cm. CD is a diameter of length 10 cm. AB and CD intersect at E and AE = 3 cm. Find the length of OE. (iv) Problem Areas: Have adequate practice in the type of situations that invariably cause problems. These include: (a) 126 (a) failure to discern between cyclic quadrilaterals and quadrilaterals which are not cyclic. o o 126 xo xo Find the value of x. (b) (b) failure to recognise all relationships existing in a figure associated with angles between a tangent and a chord and the angles in the alternate segments. o 63 156 o x o B T A ATB is a tangent. Find the value of x. JRAHS Ext I (16.12.05) - Prelim 2006 32 (c) failure to use the fact that the external angle of a cyclic quadrilateral is equal to a remote interior angle. (c) D A F B E C G ABCD and ABEF are cyclic quadrilaterals. AB is a common chord. DC and FE intersect at G. (i) Prove that GCBE is a cyclic quadrilateral. (ii) Hence prove that GCE = GBE. (d) C (d) failure to recognise that a quadrilateral is cyclic. D A E B ABC is an acute angled triangle. BD and CE are altitudes. Prove that AED = ACB. Sources of additional exercises are: JENKS "Geometrical Exercises" FORDER "School Geometry" HALL & STEPHENS "Geometry" JRAHS Ext I (16.12.05) - Prelim 2006 33 Prelim. UNIT 10 INEQUALITIES Student outcomes (i) 3U (ii) 3U (iii) 3U Student is able to: Define a > b , a < b. Implications, considerations and implementations a > b if a–b > 0 Understand the theorems on inequalities : If a > b then (a) a+c > b+c (b) ac > bc if c > 0 (c) ac < bc if c < 0 Know the special rule for inequalities involving only positive numbers. (a) If a b , c d ac bd (b) If a b a 2 b 2 1 1 (c) If a b a b (iv) 3U Solve simple inequalities. (v) 3U Graph inequalities on the number plane: Intersections and unions of regions. e.g ax by c 0 , x a y b r 2 , (vi) Prove simple inequalities e.g. x 2 y 2 2 xy for all x and y with equality when x = y. (Consider x 2 y 2 2 xy ) or x 4 y 4 2 x 2 y 2 for all x and y with equality when x = y 3U (Stress ax b x b when a 0 ). a 2 2 y x 2 etc. Exercises relying on harder AM/GM relationships are covered in Ext 2 course (vii) 3U Solve inequalities JRAHS Ext I (16.12.05) - Prelim 2006 e.g. 2t 1 x2 1 1 0 ; t 1 x 34 Prelim. UNIT 11 FURTHER TRIGONOMETRY Student outcomes (i) Student is able to: (a) Write down basic identities: Complete simple proofs of identities. (b) Find the solution of simple linear equations (ii) 3U Implications, considerations and implementations sin 2 A cos A , cos2 A sin A , sin A sin A , cos A cos A , sin A , tan A tan A , tan A cos A sin 2 A cos 2 A 1 , 1 tan 2 A sec 2 A , cos sin 2 cos sin 2 2 , 1 1 2 sec 2 B , etc. 1 sin B 1 sin B 2 sin A 1, 3 cos A 1 0 , sin A 2 Write down expansions for cos A B, sin A B , tan A B . (iii) 3U Find the values of sin105o, tan15o etc. expressed as surds. (iv) Practice using formulae in both "directions". 3U (v) Find the acute angle between two lines. 3U Treat case where one line is parallel to the y-axis as a special case. (vi) (a) Write down the expansions for sin 2 A , cos 2 A , tan 2 A e.g. Write an equivalent expression for sin A Bsin A B cos A Bcos A B and an expression for cos3x 5 y . 3U (b) Simplify expressions involving the above results. ( c) Calculate exact values using the above results. (vii) State the sine and cosine rule. Complete calculations in triangles with common sides. JRAHS Ext I (16.12.05) - Prelim 2006 35 (viii) Complete more difficult (a) The elevation of a hill at a place P due East of it is 48o, and questions involving simple 3at a place Q due South of P the elevation is 30o. If the dimensional problems and/or 3U distance from P to Q is 500 metres, find the height of the the necessity to select a hill. particular triangle (or triangles) from a figure: (b) From a point A the bearings of two points B and C are found to be 333oT and 013oT respectively. From a point D, 5km due north of A, the bearings are 301oT and 021oT respectively. By considering the triangle ABC, show that if the distance between B and C is d km, then 2 2 sin 210 sin 59 0 sin 210 sin 59 0 2 0 d 25 2 cos 40 sin 8 0 sin 32 0 sin 32 0 sin 8 0 JRAHS Ext I (16.12.05) - Prelim 2006 36 Prelim. UNIT 12 POLYNOMIALS Student outcomes (i) 3U (ii) Implications, considerations and implementations Student is able to: Define polynomial of degree n. Coefficient of polynomials. Monic polynomials. Zeros of polynomials. Graph polynomial: Linear, quadratic, cubic, quartic polynomials. 3U Graph of functions y x 2 x 4 , 2 2 y x 1 x 5 . 3 Know the significance of multiple roots of a polynomial. (It should be noted that a polynomial of odd degree always has at least one zero.) 2 Prove that Px x a Qx is tangent to x-axis at x a . (iii) 3U Divide polynomials. Understand the division transformations: Px Qxx a r . (iv) 3U Graph rational functions. Discuss natural domain. JRAHS Ext I (16.12.05) - Prelim 2006 Graph of y x 1 x2 4 , y x 1 x2 4 37 (v) Prove and use Examples: 3U (a) The Remainder Theorem. Find k if x 3 kx 4 is divisible by x 2 . (b) The Factor Theorem. Factor polynomials such as x 3 6 x 2 11x 6 . Solve polynomial equations. Solve equations such as x 3 x 2 5 x 2 0 . Graph polynomial equations Px x 3 6 x 2 11x 6 developed from the graphs of their linear factors. Deduce from the factor theorem: (a) If Px has zeros a1 , a2 , a3 ,, an then x a1 x a2 x a3 x an is factor of Px . (b) If Px is of degree n and has n distinct zeros a1 , a2 , a3 ,, an then Px P0 x a1 x a2 x a3 x an . (c) A polynomial of degree n cannot have more that n zeros. (d) If Px vanishes for more than n values of x it vanishes for all values of x. If x 3 9 x 8 px 2 qx 1 x 2 r find p, q, r. 3 2 (e) If Ax Bx for more than n values of x they are equal for all values of x. (vi) (a) Consider a polynomial as a product of linear and quadratic factors. e.g. a polynomial which is monic, of degree 4 and with double root at x 3 is of form: x 32 x 2 bx c . (b) Construct polynomials from given set of data. e.g. P(x) has degree 3, with zeros at x 0, 1 2 , 1 2 and P1 3 . What is P(x)? 3U JRAHS Ext I (16.12.05) - Prelim 2006 38 (vii) 3U Recognize the relationship between roots and Finding expressions such as 2 + 2 + 2 coefficients for quadratic , cubic and quartic polynomials. (a) By use of algebraic identities: (++)2 = 2+2+2 + 2(++) (Note: x 3 px q 0 is often used in H.S.C. exams as all cubic equations are reducible to this form by a translation of the origin to the inflexion point.) (b) By substitution into the polynomial. If , , are roots of x 3 px q 0 , show that 3 p q and 2 p p 3q 3q and 1 1 1 2 2 2 3 p q 2 p 3 JRAHS Ext I (16.12.05) - Prelim 2006 q 3 3 39 Prelim. UNIT 13 TRIGONOMETRIC EQUATIONS and IDENTITIES This unit deals with solutions expressed in degrees and minutes. HSC Unit 6 and HSC Unit 10 extend the solutions using radians and inverse trigonometry Student outcomes (i) 3U Implications, considerations and implementations Student is able to: (a) Write down the expansions for sin 2 A , cos 2 A , tan 2 A . Numerical exercises involving 2A , 3A , t . (b) Prove the results for sin 3A , cos 3 A , tan 3A . (c) (ii) Know the t result for sin A , cos A , tan A . 2t . 1 t2 hence deduce that tan 150 2 3 . e.g prove that if t = tan then tan 2 (d) Simplfy identities involving sin x y , 2A , 3A , t. e.g. cot Z tan Z 2cosec2Z , sin 3B cos 3B 2 sin 5B , sin 2 B cos 2 B sin 4 B 1 cos 2 A tan A for 0 A . 1 cos 2 A 2 (e) Eliminate parameters for simple cases. e.g. find y in terms of x if x sec , y cos 2 (Note restrictions) Solve equations (expessing the solution in a general formula with degrees): 3U Derive and use the general solution: tan x c x 180m , where tan c , 90 0 90 0 and m is an arbitrary integer. cos x c x 360m , where cos c , 0 180 0 and m is an arbitrary integer. e.g. cos x 0.5 x 360m 120 and m is an arbitrary integer. sin x c 180m 1 , where NOTE: Equations of the form tan Ax tan Bx etc. will be covered in the HSC section involving Radian Measure m sin c , 90 0 90 0 and m is an arbitrary integer. (iii) Application to solving equations such as sin x 0.2 , 3 cos 2x 1 , tan 4x 3 0 in cases where the general solution and/or solutions from a given finite domain are required. Solve equations of the type Asin x B cos x B which can be rewritten as tan x . A JRAHS Ext I (16.12.05) - Prelim 2006 e.g. 2 sin x 3 cos x . 40 (iv) (a) Solve quadratic equations involving one trig. function. (a) e.g. Solve 2 sin 2 x 3 sin x 2 0 , 2 sin 2 x 3 sin x 3 0 . (b) Solve equations in which trig. identities are used. (b) e.g. sec x 2 cos x , 3 sin 2 x 2 cos 2 x , 2 cos 2 A 7 cos A . 3U (c) Solve quadratic equations of the type: A sin 2 x B sin x cos x C cos 2 x 0 . (v) 3U (c) e.g. 2 sin 2 x 3 sin x cos x 2 cos 2 x 0 . Dividing by cos 2 x equation in tan 2 x . (a) Use auxilliary angle to write expressions of the form Asin x B cos x in the form Rsin x or R cosx (b) Solve equations of the form Asin x B cos x C . (vi) Solving equations of the form Asin x B cos x C using the t method. 3U Discuss the problem of the failure to find the solution x 180 which arises when the original equation reduces to a linear equation, rather than a quadratic equation, after substitution of the t results. JRAHS Ext I (16.12.05) - Prelim 2006 41 Prelim. UNIT 14 PERMUTATIONS AND COMBINATIONS Student outcomes (i) Student is able to: determine the number of arrangements of unlike elements in a line. 3U Implications, considerations and implementations e.g. (a) Find the way of arranging 6 cards each of which is a different colour. Factional notations (b) Find the number of 4 letter words that can be formed from the letters of PROBLEM. (c) Find the number of ways of sitting 8 people at a rectangular table with 4 seats on each side if one side faces a painting and the other side does not. (The technique to be used in each case is simply by filling spaces. However students are to be instructed that they must write a DETAILED explanation of the reason for each numerical entry into a space). (ii) determine the number of arrangements of unlike elements in a line and involving special conditions. e.g. Find the number of ways of positioning A, B, C, D, E, F in 6 seats in line if A and B are to fill the end positions. e.g. Find the number of ways of positioning A, B, C, D, E, F if A, B, C are to sit together. 3U determine the number of arrangements of unlike elements in a line and involving groups. (iv) use complementary events in space filling. e.g. Find the number of ways of positioning A, B, C, D, E, F in 6 seats in line if A and B are together but E and F are not. determine the number of arrangements of unlike elements in a circle. e.g. Find the number of arrangements of 7 people around a circular table. (= 6!) Since we can place any person in any position at the start and then there are 6 people to place in the remaining 6 positions = 6 5 4 3 2 1 . 3U (iii) 3U (v) 3U In general n objects can be arranged in n 1! ways around a circle. (vi) 3U determine the number of arrangements of unlike elements in a circle and involving special conditions. JRAHS Ext I (16.12.05) - Prelim 2006 42 (vii) 3U determine the number of arrangements of unlike elements in a circle and involving groups. (viii) Arrangements of objects some of which are to be separated . 3U e.g. Find the number of 7 letter words which can be formed from the letters of PROBING if no 2 vowels are to be together. (Note: Decide on the number of ways of arranging the consonants P R B N G. There are 5! such arrangements. There are 6 possible places for the first vowel positioned and 5 places for the second vowel. There are 3600 such words). (ix) Arrangment of elements some of which are alike. e.g. Find the number of ways of arranging the letter of the word CALL. 3U (Note: The student should be led by carefully considering suitable examples to a general method for arranging n objects of which p are of one type, q are of a second type and r are of a third type.) Examples using this method: e.g. Find the number of ways of arranging the letters of WOOLLOOMOOLOO if all letters are used in the arrangement. e.g. 3 green, 4 blue and 5 red cards are placed alongside one another. Find the number of colour patterns that can be formed. (x) 3U (xi) 3U Introduction of the symbol for placement: n! n Pr = (n–r)! Selection of r objects from a group developed by realising that order of choice is irrelevant and that the r objects can be considered to be identical. Proof that the number of selections is n! n r!(n–r)! , which is given the symbol r or n C r (Read as "n choose r") JRAHS Ext I (16.12.05) - Prelim 2006 The result for nr should be initially developed through a series of exercises of the following type: Find the number of ways of selecting 3 boys from 7 boys to form a committee. Students should also see the relation between n and the number of branches on a tree r diagram. 43 (xii) Exercises involving use of n C r result. 3U e.g. (a) Find the number of different committees of 4 that can be selected from 10 people. (b) Find the number of different committees containing 2 boys and 3 girls that can be selected from 5 boys and 6 girls. (c) Find the number of different committees of 5 that can be selected from 5 boys and 5 girls if a definite boy Tom and a girl Maree are to be included. (xiii) Selection of groups of identical sizes. 3U JRAHS Ext I (16.12.05) - Prelim 2006 e.g. 22 students are divided into 2 hockey teams each containing 11 players? How many team combinations are possible? 22 C1111C11 Note this result is 2 ! 44 Prelim. UNIT 15 LOCUS and THE PARABOLA. Student outcomes (i) Student is able to: Find the perpendicular distance from a point to a line. (ii) Define the parabola as a locus (iii) Use the locus definition to obtain the equation of a parabola. Implications, considerations and implementations e.g. Find the equation of the locus of a point equidistant from (3,4) and the line y 1 . e.g. Find the equation of the locus of a point equidistant from (5,12) and the line 3 x 4 y 12 0 . (iv) Sketch a parabola given in one of the following forms: x 2 4ay , y 2 4ax , x h 2 4a y k or y k 2 4ax h (v) Write down the equation of a parabola given two of the following: focus, vertex or directrix. (vi) Sketch parabolas R (vii) Complete the square to find the vertex, the focus and the directrix from the equation of a parabola. (viii) Solve inequalities involving quadratic E functions. JRAHS Ext I (16.12.05) - Prelim 2006 Note: A sketch in these types of questions is essential. y x 2 4 , y x 2x 3 , y 2 x 2 5x 2 y 2 x 2 6 x 3 x 1 12 418 y 34 2 Note: A sketch in these types of questions is essential. e.g. Find the inequality in x and y such that the point P(x,y) is closer to A( –1, –2) than it is to the line y = 2. 45 Prelim. UNIT 16 GEOMETRY OF THE PARABOLA. Student outcomes (i) 3U Student is able to: Convert parametric equations to a cartesian equation. (ii) Find the gradient of a curve for curves given in parametric form. 3U Find the equation of a tangent/normal to a curve given in parametric form. (iii) 3U Write down the parametric co-ordinates 2ap, ap 2 of a point on x 2 4ay . (iv) derive the equation of a chord: 1 ( p q) x y apq . 2 3U (v) 3U (vi) 3U (vii) 3U Implications, considerations and implementations Given x x (t ) and y y (t ) , dy dy dt dx dt dx derive the equation of the tangent to the parabola: px y ap 2 . derive the equation of the normal to the parabola: x py 2ap ap 3 . derive the equation of the chord of contact drawn from the external point x0 , y 0 : xx0 2a y y0 Student should also recognise that this equation also represents the equation of the tangent at the point x0 , y 0 (viii) prove simple geometrical properties such as: 3U (a) the tangent to a parabola at a given point is equally inclined to the axis and the focal chord through the point. Practical applications. (b) the tangents at the extremities of a focal chord intersect at right angles on the directrix. (ix) 3U solve exercises on simple geometrical properties. JRAHS Ext I (16.12.05) - Prelim 2006 46 (x) Find the locus of a point where: 3U (a) only one parameter is involved Students should be encouraged to check that all points of the equation derived from the elimination of the parameter in the equations are on the locus. (b) two parameters are involved and either one parameter becomes a constant or there is a relationship between the two parameters and they can be eliminated through the use of the identity: p q 2 p 2 q 2 2 pq JRAHS Ext I (16.12.05) - Prelim 2006 47 ----------------------------------------------------------------------------------------------------------------------------- END OF PRELIMINARY COURSE ---------------------------------------------------------------------------------------------------------------------------- JRAHS Ext I (16.12.05) - Prelim 2006 48