Gr11 Mathematics Revision Document 2019

GRADE 11 REVISION MATERIAL KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES 10 – 12 DIRECTORATE 2019 Compiled by KZN FET Mathematics Advisors

Mathematics/Grade 11 2 2019 KZN FET MATHEMATICS TABLE OF CONTENTS ASPECT PAGE A). GRADE 11 EXAMINATION GUIDELINES 3 B). MATHEMATICS PAPER 1 (DBE NATIONAL PAPERS 2016 – 2018) 13 Algebra, Equations and Inequalities 13 Patterns and Sequences 16 Functions and Graphs 18 Finance, Growth and Decay 24 Probability 26 C). MATHEMATICS PAPER 2 (DBE NATIONAL PAPERS 2016 – 2018) 30 Statistics 30 Analytical Geometry 35 Trigonometry 41 Euclidean Geometry and Measurement 50 D). ANSWERS TO QUESTION PAPERS 66 2016 DBE National P1 & P2 Answers 66 2017 DBE National P1 & P2 Answers 70 2018 DBE National P1 & P2 Answers 75 Copyright reserved Please turn over

Mathematics/Grade 11 3 2019 KZN FET MATHEMATICS A. GRADE 11 EXAMINATION GUIDELINES (2015) 1. INTRODUCTION The Curriculum and Assessment Policy Statement (CAPS) for Mathematics outlines the nature and purpose of the subject Mathematics. This guides the philosophy underlying the teaching and assessment of the subject in Grade 11. The purpose of these Examination Guidelines is to: • Provide clarity on the depth and scope of the content to be assessed in the Grade 11 common national examination in Mathematics. • Assist teachers to adequately prepare learners for the examinations. This document deals with the final Grade 11 final examinations. It does not deal in any depth with the School-Based Assessment (SBA). These Examination Guidelines should be read in conjunction with: • The National Curriculum Statement (NCS) Curriculum and Assessment Policy Statement (CAPS): Mathematics • The National Protocol of Assessment: An addendum to the policy document, the National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework (NQF), regarding the National Protocol for Assessment (Grades R–12) • The national policy pertaining to the programme and promotion requirements of the National Curriculum Statement, Grades R–12 Included in this document is a list of Euclidean Geometry reasons which should be used as a guideline when teaching learners Euclidean Geometry. Copyright reserved Please turn over

Mathematics/Grade 11 4 2019 KZN FET MATHEMATICS 2. ASSESSMENT IN GRADE 11 All candidates will write two external papers as prescribed. 2.1 Format of question papers for Grade 11 Paper Topics Duration Total Date Marking Patterns and Sequences Finance, Growth and Decay 1 Functions and Graphs 3 hours 150 October/November Internally Algebra, Equations and Inequalities Probability Euclidean Geometry and Measurement Analytical Geometry 2 3 hours 150 October/November Internally Statistics Trigonometry Questions in both Papers 1 and 2 will assess performance at different cognitive levels with an emphasis on process skills, critical thinking, scientific reasoning and strategies to investigate and solve problems in a variety of contexts. Copyright reserved Please turn over

Mathematics/Grade 11 5 2019 KZN FET MATHEMATICS 2.2 Weighting of cognitive levels Papers 1 and 2 will include questions across four cognitive levels. The distribution of cognitive levels in the papers is given below. Approximate number of Cognitive level Description of skills to be demonstrated Weighting marks in a 150-mark paper • Recall • Use of the correct formula (no changing of the subject) • Use of mathematical facts Knowledge 20% 30 marks • Appropriate use of mathematical vocabulary • Algorithms • Estimation and appropriate rounding of numbers • Proofs of prescribed theorems and derivation of formulae • Perform well-known procedures • Simple applications and calculations which Routine might involve few steps 35% 52–53 marks Procedures • Derivation from given information may be involved • Identification and use (after changing the subject) of correct formula • Generally similar to those encountered in class • Problems involve complex calculations and/or higher order reasoning • There is often not an obvious route to the solution • Problems need not be based on a real world Complex context 30% 45 marks Procedures • Could involve making significant connections between different representations • Require conceptual understanding • Learners are expected to solve problems by integrating different topics. • Non-routine problems (which are not necessarily difficult) • Problems are mainly unfamiliar • Higher order reasoning and processes are involved Problem Solving 15% 22–23 marks • Might require the ability to break the problem down into its constituent parts • Interpreting and extrapolating from solutions obtained by solving problems based in unfamiliar contexts. Copyright reserved Please turn over

Mathematics/Grade 11 6 2019 KZN FET MATHEMATICS 3. ELABORATION OF CONTENT/TOPICS The purpose of the clarification of the topics is to give guidance to the teacher in terms of depth of content necessary for examination purposes. Integration of topics is encouraged as learners should understand Mathematics as a holistic discipline. Thus questions integrating various topics can be asked. FUNCTIONS 1. Candidates must be able to use and interpret functional notation. In the teaching process learners must be able to understand how f (x) has been transformed to generate f (− x) , − f (x) , f ( x + a) , f ( x) + a and af (x) where a  R . 2. Trigonometric functions will ONLY be examined in Paper 2. 3. Not more than two transformations will be applied to a function simultaneously. NUMBER PATTERNS 1. The sequence of first differences of a quadratic number pattern is linear. Therefore, knowledge of linear patterns can be tested in the context of quadratic number patterns. 2. Recursive patterns will not be examined explicitly. 3. Links must be clearly established between patterns done in earlier grades. 4. Questions need not be limited to only quadratic patterns. Questions can be formed by using combinations of quadratic patterns and patterns done in earlier grades. FINANCE, GROWTH AND DECAY 1. Understand the difference between nominal and effective interest rates and convert fluently between them for the following compounding periods: monthly, quarterly and half-yearly or semi-annually. 2. With the exception of calculating n in the formulae: A = P(1 + i) n and A = P(1 − i) n , candidates are expected to calculate the value of any of the other variables. ALGEBRA 1. Solving quadratic equations using the substitution method (k-method) is examinable. 2. Equations involving surds that lead to a quadratic equation are examinable. 3. Solution of non-quadratic inequalities should be seen in the context of functions. 4. Nature of the roots will be tested intuitively with the solution of quadratic equations and in all the prescribed functions. PROBABILITY 1. Dependent events are examinable but conditional probabilities are not part of the syllabus. 2. Dependent events in which an object is not replaced is examinable. Copyright reserved Please turn over

Mathematics/Grade 11 7 2019 KZN FET MATHEMATICS EUCLIDEAN GEOMETRY & MEASUREMENT 1. Measurement can be tested in the context of Trigonometry and Euclidean Geometry. 2. Composite shapes could be formed by combining a maximum of TWO of the stated shapes. 3. Candidates must know the formulae for the surface area and volume of the right prisms. 4. If the question is based on the surface area and/or volume of the cone, sphere and/or pyramid, a list of the relevant formulae will be provided in that question. Candidates will be expected to select the correct formula from this list. 5. The following proofs of theorems are examinable: • The line drawn from the centre of a circle perpendicular to a chord bisects the chord • The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre) • The opposite angles of a cyclic quadrilateral are supplementary • The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment 6. Corollaries derived from the theorems and axioms are necessary in solving riders: • Angles in a semi-circles • Equal chords subtend equal angles at the circumference of a circle • Equal chords subtend equal angles at the centre of a circle • In equal circles equal chords subtend equal angles at the circumference • In equal circles equal chords subtend equal angles at the centre. • The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle of the quadrilateral. • If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic • Tangents drawn from a common point outside the circle are equal in length 7. The theory of quadrilaterals will be integrated into questions in the examination. 8. Concurrency theory is excluded. TRIGONOMETRY 1. The reciprocal ratios: cosec , sec  and cot  can be used by candidates in the answering of problems but will not be explicitly tested. 2. The focus of trigonometric graphs is on the relationships, simplification and determining points of intersection by solving equations, although characteristics of the graphs should not be excluded. ANALYTICAL GEOMETRY 1. Prove the properties of polygons by using analytical methods. 2. The concept of collinearity must be understood. 3. Candidates are expected to be able to integrate Euclidean Geometry axioms and theorems into Analytical Geometry problems. 4. Concepts involved with concurrency will not be examined. Copyright reserved Please turn over

Mathematics/Grade 11 8 2019 KZN FET MATHEMATICS STATISTICS 1. Candidates should be encouraged to use the calculator to calculate standard deviation and variance. 2. The interpretation of standard deviation in terms of normal distribution is not examinable. 3. Candidates are expected to identify outliers intuitively in the box and whisker diagram. In the case of the box and whisker diagram, observations that lie outside the interval (lower quartile – 1,5 IQR ; upper quartile + 1,5 IQR) are considered to be outliers. However, candidates will not be penalised if they did not make use of this formula in identifying outliers. Copyright reserved Please turn over

Mathematics/Grade 11 9 2019 KZN FET MATHEMATICS 4. ACCEPTABLE REASONS: EUCLIDEAN GEOMETRY In order to have some kind of uniformity, the use of the following shortened versions of the theorem statements is encouraged. THEOREM STATEMENT ACCEPTABLE REASON(S) LINES The adjacent angles on a straight line are supplementary. s on a str line If the adjacent angles are supplementary, the outer arms of these adj s supp angles form a straight line. The adjacent angles in a revolution add up to 360. s around a pt OR s in a rev Vertically opposite angles are equal. vert opp s = If AB || CD, then the alternate angles are equal. alt s; AB || CD If AB || CD, then the corresponding angles are equal. corresp s; AB || CD If AB || CD, then the co-interior angles are supplementary. co-int s; AB || CD If the alternate angles between two lines are equal, then the lines are alt s = parallel. If the corresponding angles between two lines are equal, then the corresp s = lines are parallel. If the co-interior angles between two lines are supplementary, then co-int s supp the lines are parallel. TRIANGLES The interior angles of a triangle are supplementary.  sum in  OR sum of s in ∆ OR int s  The exterior angle of a triangle is equal to the sum of the interior ext  of  opposite angles. The angles opposite the equal sides in an isosceles triangle are equal. s opp equal sides The sides opposite the equal angles in an isosceles triangle are equal. sides opp equal s In a right-angled triangle, the square of the hypotenuse is equal to the Pythagoras OR sum of the squares of the other two sides. Theorem of Pythagoras If the square of the longest side in a triangle is equal to the sum of converse Pythagoras the squares of the other two sides, then the triangle is right-angled. OR converse Theorem of Pythagoras If three sides of one triangle are respectively equal to three sides of SSS another triangle, the triangles are congruent. If two sides and an included angle of one triangle are respectively SAS OR SS equal to two sides and an included angle of another triangle, the triangles are congruent. If two angles and one side of one triangle are respectively equal to AAS OR S two angles and the corresponding side in another triangle, the triangles are congruent. If in two right angled triangles, the hypotenuse and one side of one RHS OR 90°HS triangle are respectively equal to the hypotenuse and one side of the other, the triangles are congruent The line segment joining the midpoints of two sides of a triangle is midpt Theorem parallel to the third side and equal to half the length of the third side Copyright reserved Please turn over

Mathematics/Grade 11 10 2019 KZN FET MATHEMATICS THEOREM STATEMENT ACCEPTABLE REASON(S) The line drawn from the midpoint of one side of a triangle, parallel line through midpt || to 2nd side to another side, bisects the third side. If two triangles are equiangular, then the corresponding sides are in ||| s OR equiangular ∆s proportion (and consequently the triangles are similar). If the corresponding sides of two triangles are proportional, then the sides of ∆ in prop triangles are equiangular (and consequently the triangles are similar). If triangles (or parallelograms) are on the same base (or on bases of same base; same height OR equal length) and between the same two parallel lines, then the equal bases; equal height triangles (or parallelograms) have equal areas. CIRCLES The tangent to a circle is perpendicular to the radius/diameter of the tan ⊥ radius circle at the point of contact. tan ⊥ diameter If a line is drawn perpendicular to a radius/diameter at the point line ⊥ radius OR where the radius/diameter meets the circle, then the line is a tangent converse tan ⊥ radius OR to the circle. converse tan ⊥ diameter The line drawn from the centre of a circle to the midpoint of a chord line from centre to midpt of chord is perpendicular to the chord. The line drawn from the centre of a circle perpendicular to a chord line from centre ⊥ to chord bisects the chord. The perpendicular bisector of a chord passes through the centre of perp bisector of chord the circle; The angle subtended by an arc at the centre of a circle is double the  at centre = 2 × at circumference size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre) The angle subtended by the diameter at the circumference of the s in semi-circle OR circle is 90. diameter subtends right angle OR 1  in  2 If the angle subtended by a chord at the circumference of the circle chord subtends 90 OR is 90, then the chord is a diameter. converse s in semi-circle Angles subtended by a chord of the circle, on the same side of the s in the same seg chord, are equal If a line segment joining two points subtends equal angles at two line subtends equal s OR points on the same side of the line segment, then the four points are converse s in the same seg concyclic. Equal chords subtend equal angles at the circumference of the circle. equal chords; equal s Equal chords subtend equal angles at the centre of the circle. equal chords; equal s Equal chords in equal circles subtend equal angles at the equal circles; equal chords; equal s circumference of the circles. Equal chords in equal circles subtend equal angles at the centre of equal circles; equal chords; equal s the circles. The opposite angles of a cyclic quadrilateral are supplementary opp s of cyclic quad Copyright reserved Please turn over

Mathematics/Grade 11 11 2019 KZN FET MATHEMATICS THEOREM STATEMENT ACCEPTABLE REASON(S) If the opposite angles of a quadrilateral are supplementary then the opp s quad supp OR quadrilateral is cyclic. converse opp s of cyclic quad The exterior angle of a cyclic quadrilateral is equal to the interior ext  of cyclic quad opposite angle. If the exterior angle of a quadrilateral is equal to the interior opposite ext  = int opp  OR angle of the quadrilateral, then the quadrilateral is cyclic. converse ext  of cyclic quad Two tangents drawn to a circle from the same point outside the circle tans from common pt OR are equal in length tans from same pt The angle between the tangent to a circle and the chord drawn from tan chord theorem the point of contact is equal to the angle in the alternate segment. If a line is drawn through the end-point of a chord, making with the converse tan chord theorem OR chord an angle equal to an angle in the alternate segment, then the  between line and chord line is a tangent to the circle. QUADRILATERALS The interior angles of a quadrilateral add up to 360. sum of s in quad The opposite sides of a parallelogram are parallel. opp sides of ||m If the opposite sides of a quadrilateral are parallel, then the opp sides of quad are || quadrilateral is a parallelogram. The opposite sides of a parallelogram are equal in length. opp sides of ||m If the opposite sides of a quadrilateral are equal, then the opp sides of quad are = quadrilateral is a parallelogram. OR converse opp sides of a parm The opposite angles of a parallelogram are equal. opp s of ||m If the opposite angles of a quadrilateral are equal then the opp s of quad are = OR quadrilateral is a parallelogram. converse opp angles of a parm The diagonals of a parallelogram bisect each other. diag of ||m If the diagonals of a quadrilateral bisect each other, then the diags of quad bisect each other quadrilateral is a parallelogram. OR converse diags of a parm If one pair of opposite sides of a quadrilateral are equal and parallel, pair of opp sides = and || then the quadrilateral is a parallelogram. The diagonals of a parallelogram bisect its area. diag bisect area of ||m The diagonals of a rhombus bisect at right angles. diags of rhombus The diagonals of a rhombus bisect the interior angles. diags of rhombus All four sides of a rhombus are equal in length. sides of rhombus All four sides of a square are equal in length. sides of square The diagonals of a rectangle are equal in length. diags of rect The diagonals of a kite intersect at right-angles. diags of kite A diagonal of a kite bisects the other diagonal. diag of kite A diagonal of a kite bisects the opposite angles diag of kite Copyright reserved Please turn over

Mathematics/Grade 11 12 2019 KZN FET MATHEMATICS 5. GENERAL GUIDELINES FOR MARKING • If a learner makes more than one attempt at answering a question and does not cancel any of them out, only the first attempt will be marked irrespective of which of the attempt(s) may be the correct answer. • Consistent Accuracy marking regarding calculations will be followed in the following cases: - Subquestion to subquestion: When a certain variable is incorrectly calculated in one subquestion and needs to be substituted into another subquestion full marks can be awarded for the subsequent subquestions provided the methods used are correct and the calculations are correct. - Assuming values/answers in order to solve a problem is unacceptable. 6. CONCLUSION This Examination Guidelines document is meant to articulate the assessment aspirations espoused in the CAPS document. It is therefore not a substitute for the CAPS document which educators should teach to. Qualitative curriculum coverage as enunciated in the CAPS cannot be over-emphasised. Copyright reserved

Mathematics/Grade 11 13 2019 KZN FET MATHEMATICS B). MATHEMATICS PAPER 1 (DBE NATIONAL PAPERS 2016 – 2018) 1. ALGEBRA, EQUATIONS AND INEQUALITIES DBE 2016 Copyright reserved

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Mathematics/Grade 11 66 2019 KZN FET MATHEMATICS D). ANSWERS TO EXAM QUESTIONS MATHEMATICS PAPER 1 (DBE NATIONAL PAPER (2016) Algebra, Equations and Inequalities 1.1.1 x = 1.85 or x = - 0,18 1.1.2 x = 4 or x = 2 1.1.3 x < 0 or x > 2 1.1.4 2 𝑥=3 1.1.5 𝑥 ≠ 5 𝑜𝑟 𝑥 = 10 1.2 y = 8 or y = - 7; x = - 3 or x = 2 1.3 Proof 2.1.1 1 5 2.1.2 1 −2 2.2 Proof Patterns and Sequences 3.1 x = 27 3.2 Tn = 2n 2 - 3n - 8 3.3 Tn = 2n 2 - 3n - 5 3.4 −27 𝑛 = 15 𝑜𝑟 𝑛 ≠ 2 4.1.1 6 4.1.2 Tn = -4n + 22 4.1.3 n = 23 4.1.4 - 2014 4.2.1 minimum value 4.2.2 Tn = n 2 - 22n + 114 Functions and Graphs 5.1 49 𝑦= 8 5.2 (0; 6) 5.3 3 x = 2 or x = − 2 5.4 Copyright reserved

Mathematics/Grade 11 67 2019 KZN FET MATHEMATICS 5.5 49 𝑘= 8 5.6 9 2 57 y = −2 (𝑥 − 4) + 8 6.1 x = - 3 or y = - 1 6.2 𝑥 ≠ −3 ; 𝑥 ∈ 𝑅 6.3.1 2 units 3 6.3.2 2 units 6.4 1 𝐶 (−1; − 2) 𝐷 (−4 ; −2) 6.5 𝑥 ≤ 4 𝑜𝑟 − 3 < 𝑥 ≤ −1 7.1 𝑓(𝑥) = 2. 3𝑥+1 + 2 7.2 y=4 7.3 m=8 7.4 ℎ(𝑥) = −2. 3𝑥+1 + 2 Finance, Growth and Decay 8.1 17143,74 8.2 13,07 % 8.3 7,3% 8.4 R17448,46 Probability 9.1.1 0,1 9.1.2 Independent 9.2.1 a = 15, b = 1, c = 38, d =3, e =37 9.2.2 1 4 9.3.1 9.3.2 0,38 Problem Solving 10 y = 7m Copyright reserved

Mathematics/Grade 11 68 2019 KZN FET MATHEMATICS MATHEMATICS PAPER 2 (DBE NATIONAL PAPER (2016) Statistics 1.1 Range = 70 1.2 SD = 19,56 1.3 Median = 25 1.4 Q1 = 11,5, Q3 = 33,5, IQR = 22 1.5 1.6 Skewed to the right 1.7 75 2.1 2.2 2.3 118 2.4 50 Analytical Geometry 3.1 13 M (4 ; 2 ) 3.2 m=2 Copyright reserved

Mathematics/Grade 11 69 2019 KZN FET MATHEMATICS 3.3 3 𝑦 = 2𝑥 − 2 3.4 1 N (1 ; 2) 3.5 D (0; -14) 4.1 k = -3 4.2 𝜃 = 119,74° 4.3 11 units 4.4 22 square units Trigonometry 5.1.1 t = - 15 5.1.2 (a) 8 − 17 5.1.2 32 17 (b) 5.2.1 5.2.2 √1 − 𝑎2 5.2.3 2a 2 5.4 Proof 5.5 𝑥 = 90° + 𝑘. 360° 𝑜𝑟 𝑥 = 270° + 𝑘. 360° 𝑜𝑟 𝑥 = 30° + 𝑘. 360° 𝑜𝑟 𝑥 = 150° + 𝑘. 360° 𝑘 ∈ 𝑍 6.1 b = 30° 6.2 360° 6.3 30° 6.4 −150° < 𝑥 < 30° 6.5 𝑦 ∈ [2; 4] 7.1.1 Proof 7.1.2 Proof 7.2.1 164,92m 7.2.2 177,34m Euclidean Geometry & Measurement. 8.1 0,7 m 8.2 1,5 hrs 9.1 Equal to twice the angle subtended by the arc at the circumference 9.2.1 𝑅̂ = 30° 9.2.2 𝑁𝑆̂𝑇 = 30° 10.1 170° − 2𝑥 10.2 110° 11.1 42,83° 11.2 8 units 12.1 Proof 12.2.1 Proof 12.2.2 Proof 12.2.3 Proof Copyright reserved

Mathematics/Grade 11 70 2019 KZN FET MATHEMATICS MATHEMATICS PAPER 1 (DBE NATIONAL PAPER (2017) Algebra, Equations and Inequalities 1.1.1 x = 1.5 or x = - 7 1.1.2 x = - 0. 79 or x = 0, 36 1.1.3 x = 5 only 1.1.4 x < - 3 or x > 6 1.2 y = 1 or y = - 1 ; x = 3 or x = - 1 1.3 k = +/- 0,5 2.1 1 2.2.1 1 16 2.2.2 x = 1 2.3 24,5 2.4 120 Patterns and Sequences 3.1.1 Tn = 5n + 7 3.1.2 67 3.1.3 n = 33 3.2 x = 12 and y = 21 4.1 76 4.2 Tn = 3n 2 - 60n + 301 4.3 n = 23 only 4.4 Between T85 & T86 4.5 Proof Functions and Graphs 5.1 4 5.2 x=-3 5.3 x = - 2 and y = 1 5.4 y>-4 5.5 1 𝑦 = −2 x = 1 5.6 y=-x-1 Copyright reserved

Mathematics/Grade 11 71 2019 KZN FET MATHEMATICS 5.7 5.8 6.1 x = - 3 or x = 2 6.2 1 𝑥 = −2 6.3 1 𝑦 ≤ 64 6.4 y=x+6 6.5 1 6.6 6.7 6.8 3 Finance, Growth and Decay 7.1 24,21 % 7.2 5,09 % 7.3 R58 017, 51 7.4 R52 636, 74 Probability 8.1.1 Copyright reserved

Mathematics/Grade 11 72 2019 KZN FET MATHEMATICS 8.1.2 3 10 8.2.1 Not Mutually Exclusive. 8.2.2 A & B are Independent Events 9.1 9.2 x=8 9.3 18 9.4 49 80 Copyright reserved

Mathematics/Grade 11 73 2019 KZN FET MATHEMATICS MATHEMATICS PAPER 2 (DBE NATIONAL PAPER (2017) Statistics 1.1.1 Min = 12, Q1 = 17, Median =30, Q3 = 38, Max = 65 1.1.2 21 1.1.3 Skewed Right 1.2.1 y = 94 1.2.2 30, 94 1.2.3 33,74 2.1 2.2 n = 80 2.3 40 < 𝑥 ≤ 50 2.4 2.5 27,5% Analytical Geometry 3.1 m=2 3.2 y =2x - 1 Copyright reserved

Mathematics/Grade 11 74 2019 KZN FET MATHEMATICS 3.3 T (4; 7) 3.4.1 C (10; 19) 3.4.2 3√5 3.4.3 7,5 units each 4.1 5 3 4.2 16 E (− 5 ; 0) 4.3 𝛼 = 59° 4.4 y=-x-8 Trigonometry 5.1 -1 5.2 − √2 3 5.3.1 Tan ∝ = 3 4 5.3.2 − 3 5 5.3.3 P (- 8; - 6) 5.4 Proof 6.1 𝑥 = 40° + 𝑘. 120° 𝑜𝑟 𝑥 = −120° + 𝑘. 360° 6.2.1 180° 6.2.2 −1 ≤ 𝑦 ≤ 1 6.2.3 6.2.4 𝑥 = −80° ; 𝑥 = 40° ; 𝑥 = 160° 7.1 𝑥 2 +3 𝑐𝑜𝑠𝜃 = 4𝑥 7.2.1 𝜃 = 24,15 7.2.2 2,95 units 7.3 1<𝑥<3 Euclidean Geometry & Measurement. 8.1 h = 7,5cm 8.2 606,91 cm2 9.1 Proof 9.2 3 units 10.1 𝑅̂2 = 55° 10.2 𝑄̂1 = 35° 10.3 𝑂̂2 = 110° 11.1 Proof 11.2 𝐶̂ = 90° 12.1 𝑃̂1 ; 𝐶̂2 ; 𝐸 & 𝑃̂5 12.2 Proof 12.3 𝑥 + 𝑦 = 60° Copyright reserved

Mathematics/Grade 11 75 2019 KZN FET MATHEMATICS MATHEMATICS PAPER 1 (DBE NATIONAL PAPER (2018) Algebra, Equations and Inequalities 1.1.1 x = 0 or x = - 1 2 1.1.2 x = 0,91 or x = - 1,31 1.1.3 𝑥 ≤ − 1 𝑜𝑟 𝑥 ≥ 2 2 1.1.4 x = - 2 1.2 y = - 2 or y = 3; x = 4 or x = - 1 1.3 1<𝑚<4 1.4 max = 4 2.1 −47 2 2.2.1 x = 2 2.2.2 x = 0 or x = 3 2.2.3 x = 2 5 2.3 (p + q )3 Patterns and Sequences 3.1.1 Tn = -5n + 12 3.1.2 - 88 3.1.3 T30 3.2 x=1 4.1 Proof 4.2 1 3 Tn = 2 n 2 - 2 n + 2 4.3 n = 20 only 4.4 35 Functions and Graphs 5.1 x = 3 and y = 2 5.2 x=1 5.3 2 (0; ) 3 5.4 5.5 𝑥=4 x=-1 5.6 −1 < 𝑥 < 3 5.7 Q(5; 4) 6.1 y=-4 6.2 D (3 ; 25) Copyright reserved

Mathematics/Grade 11 76 2019 KZN FET MATHEMATICS 6.3 𝑦 ≤ 25 6.4 19 units 6.5 x>3 6.6 3 𝑚̅ = −2 6.7 y<7 6.8 Turning Point (3 ; 27) 6.9 k = 20 Finance, Growth and Decay 7.1 12,13 % 7.2 R2124,97 7.3.1 R23 068, 12 7.3.2 R30 871, 48 Probability 8.1 n(S) = 32 8.2 8.3 49 124 9.1 Events are Independent. 9.2.1 Events are not Mutually exclusive 9.2.2 P (A) = 0,19 P (B) = 0,76 9.2.3 0, 81 Copyright reserved

Mathematics/Grade 11 77 2019 KZN FET MATHEMATICS MATHEMATICS PAPER 2 (DBE NATIONAL PAPER (2018) Statistics 1.1 4 minutes 1.2 Mean = 16,9 1.3 SD = 5,79 1.4 1 team member 1.5 x = 186 2.1.1 50 players 2.1.2 30 players 2.1.3 2.1.4 Copyright reserved

Mathematics/Grade 11 78 2019 KZN FET MATHEMATICS 2.2 The claim is not valid Marks of Class A is more spread than the marks of class B Class A performed worse at the bottom end. Analytical Geometry 3.1 4 m=3 3.2 4 14 𝑦 = 3𝑥 + 3 3.3 7 −2 3.4 5 2 3.5 D (- 8; - 6) 3.6 Proof 3.7 1 M ( -1; - 2 ) 4.1 4 3 4.2 y=-5 4.3 𝜃 = 50,91° 4.4 6,44 units Trigonometry 5.1.1 x = - 8 5.1.2 𝑇𝑎𝑛𝜃 = 15 −8 5.1.3 8 17 5.1.4 225 289 5.1.5 15 a= 4 5.2 3 −4 5.3 Proof 5.4 𝑥 = 0° + 𝑘. 360° 𝑜𝑟 𝑥 = 180° + 𝑘. 360° 𝑜𝑟 𝑥 = 48,19° + 𝑘. 360° 𝑜𝑟 𝑥 = 311,81° + 𝑘. 360° 𝑘 ∈ 𝑍 6.1 −3 ≤ 𝑦 ≤ 3 6.2 c=1 6.3 a = 3 and b = 1 6.4 Q (122°; - 1,6) 6.5 3,12 6.6 𝑓(𝑥) = 3cos(𝜃 − 45°) 7.1 8,49cm 7.2.1 𝜃 − 𝛼 7.2.2 Proof 7.2.3 10,59 units 7.2.4 47,70 units 2 Euclidean Geometry & Measurement. 8.1 Proof 8.2 282,74 cm2 9.1.1 𝑂̂1 = 78° 9.1.2 𝑂̂2 = 78° 9.2 x = 84° 9.3 Proof Copyright reserved

Mathematics/Grade 11 79 2019 KZN FET MATHEMATICS 10.1 Proof 10.2.1 𝑄̂1 = 24° (a) 10.2.1 𝑅̂ = 24° (b) 10.2.1 ̂1 = 48° 𝑀 (c) 10.2.2 Proof 11.1 Proof 11.2 Proof ALL DBE November National Papers from 2016 – 2018 were utilized when compiling this document. Copyright reserved