Mathematics Grade 11 P2 Gauteng June 2016 Question Paper

Downloaded from hlayiso.com GAUTENG DEPARTMENT OF EDUCATION PROVINCIAL EXAMINATION JUNE 2016 GRADE 11 MATHEMATICS (Second Paper) TIME: 2 hours MARKS: 100 9 pages + 2 diagram sheets P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 2 (Second Paper) GAUTENG DEPARTMENT OF EDUCATION PROVINCIAL EXAMINATION MATHEMATICS (Second Paper) TIME: 2 hours MARKS: 100 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 7 questions. Answer ALL the questions. 2. Number your answers according to the numbering of the questions in the question paper. 3. Use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 4. Round-off the final answer correct to 2 decimal places, unless instructed otherwise. 5. Show ALL calculations, diagrams, graphs, etc. that you have used to determine your answers. 6. Answers only will not necessarily be awarded full marks. 7. Diagrams are NOT necessarily drawn to scale. 8. Reasons must accompany statements made in Questions 5, 6 and 7. 9. It is in your interest to write legibly (in blue ink) and present all answers neatly and logically. 10. Use the diagram sheets on Pages 10 and 11 for additional constructions and labelling . (SUBMIT these sheets with your ANSWER BOOK.) P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 3 (Second Paper) QUESTION 1 In the diagram below, K(-5 ; 1) , P(7 ; -2) and T(1 ; 6) are the vertices of ∆KTP. A and B are points on the y-axis and C and D are points on the x-axis. y T (1 ; 6) θ A K (-5 ; 1) C  2 1  x 1 3 2 D B P (7 ; -2) 1.1 Calculate the length of KT. (Leave the answer in simplest surd form.) (3) 1.2 Determine the equation of line KP in the form y = mx + c. (3) 1.3 Calculate the length of AB. (4) 1.4 Calculate the size of KT̂P. (6) [16] P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 4 (Second Paper) QUESTION 2 In the diagram below, the circle with centre M passes through the origin. AB is the diameter of the circle with B(8 ; 0). K A M T O B (8, 0) 2.1 Calculate the x-co-ordinate of Point M. (4) 2.2 Calculate the co-ordinates of Point A, the other y-intercept of the circle with centre M(4 ; 2). (3) 2.3 Determine the equation of the line OK which is parallel to AB. (3) 2.4 Determine the x-co-ordinate of T which lies on line OK, such that AT is the shortest distance from A to line OK. (4) [14] P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 5 (Second Paper) QUESTION 3 3.1 If 13sin α = –5 and tan α ˃ 0, use a diagram to evaluate: 3cos α. (5) 3.2 Simplify the following expressions without the use of a calculator. sin θ  180 . tan(360  θ) . sin(90  θ) 3.2.1 cos 2 (θ  180) (6) sin 210.cos400 3.2.2 sin (-50).cos120 (6) 3.3 If (4 θ – 8) sin 30° = (θ 3 – 8) and (θ 2 + 2θ + 4) = 2, determine the value of tan 240° without the use of a calculator. (5) 1 1 3.4 3.4.1 Prove that : (sin α tan α  cos α)  tan α sin α (4) 3.4.2 Determine for which value(s) of α is: (sin α tan α  cos α) undefined for   0 ; 360 . 1 tan α (2) [28] P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 6 (Second Paper) QUESTION 4 The sketch below shows the graphs of: f(x) = sin ax and g(x) = cos( x + θ) , for x   180 ; 180 A (-40o, 1) f(x) 1 B (-103.3o, y) -180 -135o -90o -45o 45o 90o 135o 180o -1 g(x) 4.1 Determine the period of f(x) = sin ax (1) 4.2 Write down the range of g(x) = cos( x + θ). (2) 4.3 Write down the values of θ and a. (2) 4.4 Calculate the value of: g(180°). (1) 4.5 Determine the values of x, if x  0;180 for which: 4.5.1 f(x) – g(x) ˃ 0 (2) 4.5.2 g(x).f(x) > 0 (2) 4.6 Determine the minimum value of: 3cos(90°–2x) (2) [12] P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 7 (Second Paper) STATEMENTS AND REASONS ARE REQUIRED WHEN ANSWERING QUESTIONS 5 , 6 AND 7. Diagram sheets are provided on Pages 10 and 11 to assist you. QUESTION 5 5.1 Complete: The line drawn through the centre of a circle to the midpoint of a chord is ... (1) 5.2 In the diagram below, AC is a chord of the circle with centre O. AC = 48 cm , BD = 16 cm and OB = x . B is the midpoint of line AC. A D B C O 5.2.1 Calculate the length of AD. (2) 5.2.2 Determine the length of OD in terms of x. (1) 5.2.3 Determine the numerical value of x. (4) [8] P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 8 (Second Paper) QUESTION 6 In the diagram below, KT is a chord of Circle O. OC is drawn perpendicular to KT and cuts KT at P. OP = x and PC = y. T C P K O 6.1 Determine PT in terms of KT. (2) 6.2 Prove that: KC2 = 2xy + 2y2 (6) [8] P.T.O.

Downloaded from hlayiso.com MATHEMATICS Grade 11 9 (Second Paper) QUESTION 7 In the accompanying figure, V and U are points on the smaller circle with centre S. T and R are the points of intersection of the two circles. PR and TQ are chords of the larger circle. VŜR  110 ; R̂ 3  20 and R̂1  30 P Q 4 2 1 3 2 K 1 V T 2 1 110 3 S 4 R U Calculate: 7.1 Û (2) 7.2 V̂2 (3) 7.3 Q̂ (3) 7.4 Given that K̂ 2  90 , calculate P̂. (4) 7.5 Prove that TR//PQ. (2) [14] TOTAL: 100 END

Downloaded from hlayiso.com MATHEMATICS Grade 11 10 (Second Paper) DIAGRAM SHEET 1 Name :__________________________________________ Grade: 11____ QUESTION 5 A D B C O QUESTION 6 T C P K O

Downloaded from hlayiso.com MATHEMATICS Grade 11 11 (Second Paper) DIAGRAM SHEET 2 Name :__________________________________________ Grade: 11____ QUESTION 7 P Q 4 2 1 3 2 K 1 V T 2 1 110 3 S 4 R U