Mathematics Grade 12 Paper 2 KwaZulu Natal Mock 2016 Memo

Mathematics P2 Downloaded from hlayiso.com NSC-KZN Basic Education KwaZulu-Natal Department of Basic Education REPUBLIC OF SOUTH AFRICA MATHEMATICS P2 PREPARATORY EXAMINATION SEPTEMBER 2016 NATIONAL SENIOR CERTIFICATE GRADE 12 MARKS: 150 TIME: 3 hours This question paper consists of 14 pages, 1 information sheet and an answer book of 21 pages.

Mathematics P2 Downloaded from 2 hlayiso.com Preparatory Examination September 2016 NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 11 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK. 3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. 4. Answers only will not necessarily be awarded full marks. 5. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 6. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. Copyright reserved Please turn over

Mathematics P2 Downloaded from 3 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 1 Two schools, M-cee-nai High and Bee Vee high are in competition to see which school performed better in mathematics in the June Examination. The marks of the leaners at M-cee-nai High school are recorded below. The box whisker diagram below illustrates the results of Bee Vee High School. Both schools have 25 learners. (Marks are given in %). Marks of M-cee-nai High School The box and whisker diagram learners for the Bee Vee High School Learners 1.1 Write down the five-number summary for M-cee-nai High School. (4) 1.2 Draw the box and whisker diagram that represents M-cee-nai High School marks. Clearly indicate ALL relevant values. (3) 1.3 Comment on the skewness of the data of M-cee-nai High School. (1) 1.4 Calculate the mean mark of M-cee-nai High School. (2) 1.5 Determine which school performed better in the June Examination and give reasons for your conclusion. (3) [13] Copyright reserved Please turn over

Mathematics P2 Downloaded from 4 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 2 A survey was conducted indicating the number of bees that visited flowers over a period of 12 days. The information is represented in the table and in the scatter plot below. No. of Flowers 4 10 7 12 1 6 2 5 11 9 8 3 No. of bees 30 22 20 38 65 160 48 62 61 74 88 86 SCATTER PLOT Number of bees Number of flowers 2.1 Write down the coordinates of the outlier. (2) 2.2 Determine the equation for the least squares regression line. (4) 2.3 Calculate the correlation co-efficient if the outlier is excluded. (2) [8] Copyright reserved Please turn over

Mathematics P2 Downloaded from 5 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 3 In the diagram below, A (–5; 1) , B(1; 6) and C(7; –2) are vertices of ∆ ABC with AB produced to D. BD forms an angle, β , with the negative x – axis and BC forms an angle, α , with the positive x – axis. AB̂C = θ y B(1; 6) θ A(–5; 1) β O α D x C(7; –2) Determine: 3.1 the length AC (2) 3.2 the equation of line BC (3) 3.3 AB̂C (5) 3.4 the midpoint P of AB (2) 3.5 the equation of the line parallel to AC and passing through the point (–1; 3) (3) 3.6 Show that AB is perpendicular to 6x + 5y = 18 (3) [18] Copyright reserved Please turn over

Mathematics P2 Downloaded from 6 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 4 In the diagram below, centre W of the circle lies on the straight line 3x + 4y + 7 = 0 The straight line cuts the circle at V and Z(–1; –1). The circle touches the y-axis at G(0; 2) y V M(a ;; b) M b) W G(0; 2) x Z (–1; –1) 4.1.1 Determine the equation of the circle in the form (x – a)2 + (y – b)2 = r2. (5) 4.1.2 Determine the length of diameter VZ (1) 4.1.3 Calculate the gradient of GZ. (2) 4.1.4 Write down the coordinates of the midpoint of GZ. (1) 4.1.5 Determine the equation of the line that is the perpendicular bisector of GZ. (3) 4.1.6 Show that the line in QUESTION 4.1.5 and straight line VZ intersect at W. (2) 4.2 The circle defined by (x + 2)2 + (y – 1)2 = 25 has centre M, and the circle defined by (x –1)2 + (y – 3)2 = 9 has centre N. 4.2.1 Show that the circles intersect each other at two distinct points. (6) 4.2.2 Determine the equation of the common chord. (3) [23] Copyright reserved Please turn over

Mathematics P2 Downloaded from 7 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 5 5.1 ( ) In the diagram below, P 2 3 ; − 2 is a point in the cartesian plane, with reflex angle QÔP = α . Q is the point on the x – axis so that OP̂Q = 90 o y α O x β Q ( P 2 3;− 2 ) Calculate without measuring: 5.1.1 β. (3) 5.1.2 the length of OP. (2) 5.1.3 the co-ordinates of Q. (3) 5.2 If cos α + 3 sin α = k sin (α + β ) . Calculate the values of k and β . (5) [13] Copyright reserved Please turn over

Mathematics P2 Downloaded from 8 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 6 6.1 On the same system of axes, sketch the graphs of f (x) = 3 cos x and 1 g (x) = tan x for –180° ≤ x ≤ 360°. Clearly show the intercepts with the axes 2 and all turning points. (5) Use the graphs in 6.1 to answer the following questions. 6.2 Determine the period of g. (1) 6.3 Determine the co-ordinates of the turning points of f on the given interval. (2) 6.4 For which values of x will both functions increase as x increases for –180° ≤ x ≤ 360°? (2) 6.5 If the y–axis is moved 45o to the left, then write down the new equation of f in the form y = ….. (1) [11] Copyright reserved Please turn over

Mathematics P2 Downloaded from 9 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 7 7.1 Determine the general solution of: cos 54o . cos x + sin 54o . sin x = sin 2x (5) 7.2 ABCD is a trapezium with AD || BC, BÂD = 90o and BĈD = 150 o . CD is produced to E. F is point on AD such that BFE is a straight line, and CB̂E = α . The angle of elevation of E from A is θ , BC = x and CE = 18 – 3x. E 18 – 3x A θ F D α 150o B x C AB cos θ 7.2.1 Show that: BE = (5) sin (α − θ ) 9 3x 2 7.2.2 Show that the area of Δ BCE = x− (3) 2 4 7.2.3 Determine, without the use of a calculator, the value of x for which the area of ΔBCE will be maximum. (3) 7.2.4 Calculate the length of BE if x = 3. (3) [19] Copyright reserved Please turn over

Mathematics P2 Downloaded from 10 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 8 In the diagram below, PQ is the chord of circle O. OR is perpendicular to PQ and OR intersects PQ at T. If the radius of the circle is 13 cm and PT = 12 cm. Q O R T P Calculate the length of: 8.1 PQ (2) 8.2 PR (4) [6] Copyright reserved Please turn over

Mathematics P2 Downloaded from 11 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 9 9.1 Complete the statement so that it is true: The exterior angle of a cyclic quadrilateral is equal to … (1) 9.2 In the diagram below the circle with centre O passes through L, N and P. KLM is a tangent to the circle at L. NP, NL and LP are joined. N O P
x 1 K L M Using the above diagram, prove the theorem that states that PL̂M = N̂ . (5) Copyright reserved Please turn over

Mathematics P2 Downloaded from 12 hlayiso.com Preparatory Examination September 2016 NSC 9.3 In the diagram below, BAED is a cyclic quadrilateral with BA || DE. BE = DE and AÊD = 70 o . The tangent to the circle at D meets AB produced at C. D E 1 1 2 70o 2 2 1 3 C B A Calculate, with reasons the sizes of the following angles. 9.3.1 Â (2) 9.3.2 B̂1 (2) 9.3.3 D̂ 2 (2) 9.3.4 B̂ 2 (2) 9.3.5 D̂ 1 (3) [17] Copyright reserved Please turn over

Mathematics P2 Downloaded from 13 hlayiso.com Preparatory Examination September 2016 NSC QUESTION 10 In the diagram below, SP is a tangent to the circle at P and PQ is a chord. Chord QF produced meets SP at S and chord RP bisects QP̂S . PR produced meets QS at B. BC || SP and cuts the chord QR at D. QR produced meets SP at A. Let B̂ 2 = x . Q 2 1 F B 3 D 3 2 C 1 2 1 3 R 2 1 2 1 S A P 10.1 Name, with reasons, 3 angles equal to x. (4) 10.2 Prove that PC = BC (2) 10.3 Prove that RCQB is a cyclic quadrilateral. (2) 10.4 Prove that Δ PBS ||| Δ QCR. (5) 10.5 Show that PB.CR = QB.CP (4) [17] Copyright reserved Please turn over

Mathematics P2 Downloaded from 14 hlayiso.com Preparatory Examination September 2016 NSC K QUESTION 11 In the diagram alongside, Δ KLM, 6 9 DE || LM, LD̂M = MD̂E = x. KD = 9, EM = 8 and EK = 6 D x E x Calculate, with reasons LM. 8 L M (5) TOTAL: [150] Copyright reserved Please turn over

Mathematics P2 Downloaded from hlayiso.com September 2016 Preparatory Examination NSC-KZN INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE − b ± b 2 − 4ac x= 2a A = P (1 + ni ) A = P (1 − ni ) A = P(1 − i) n A = P(1 + i) n n Tn = a + (n − 1)d Sn = (2a + (n − 1)d ) 2 T n = ar n −1 Sn = ( a r n −1 ) ; r ≠1 S∞ = a ; −1 < r < 1 r −1 1− r F= [ x (1 + i ) − 1 n ] P= x[1 − (1 + i)−n ] i i f ( x + h) − f ( x ) f ' ( x) = lim h→ 0 h  x + x2 y1 + y 2  d = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 M  1 ;   2 2  y 2 − y1 y = mx + c y − y1 = m( x − x1 ) m= m = tan θ x 2 − x1 ( x − a )2 + ( y − b )2 = r 2 a b c 1 In ∆ABC: = = a 2 = b 2 + c 2 − 2bc. cos A area ∆ABC = ab. sin C sin A sin B sin C 2 sin(α + β ) = sinα. cosβ + cosα.sin β sin(α − β ) = sinα.cos β − cosα.sin β cos(α + β ) = cosα. cos β − sinα.sin β cos(α − β ) = cosα. cos β + sinα.sin β cos 2 α − sin 2 α  cos 2α = 1 − 2 sin 2 α sin 2α = 2 sin α . cosα  2 cos α − 1 2 n 2 ∑x ∑ (xi − x ) x= σ 2 = i =1 n n n(A) P ( A) = P(A or B) = P(A) + P(B) - P(A and B) n (S ) yˆ = a + bx b= ∑ (x − x )( y − y ) ∑ (x − x) 2