Mathematics P2 Feb March 2015 Eng_hlayiso.com_.pdf

Downloaded from hlayiso.com kL ey basic education WH & y Department: Gy) Basic Education WWE REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P2 i" FEBRUARY/MARCH 2015 “ EER SEES EEOEEE EEE EERE Eee ee eee eee eee eer ee) MARKS: 150 TIME: 3 hours This question paper consists of 14 pages, 5 diagram sheets and 1 information sheet. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 DBE/Feb—Mar. 2015 NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 10 questions. 2. Answer ALL the questions. Clearly show ALL calculations, diagrams, graphs, et cetera which you have used in determining your answers. ioe 4. Answers only will NOT necessarily be awarded full marks. 5. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 6. If necessary, round off answers to TWO decimal places, unless stated otherwise. 7. FIVE diagram sheets for QUESTIONS 1.3, 7, 8, 9.2, 9.3 and 10 are attached at the end of this question paper. Write your centre number and examination number on these sheets in the spaces provided and insert them inside the back cover of your ANSWER BOOK. 8. Diagrams are NOT necessarily drawn to scale. 9. Number the answers correctly according to the numbering system used in this question paper. 10. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 NSC QUESTION 1 DBE/Feb.-Mar. 2015 The table below shows the distances (in kilometres) travelled daily by a sales representative for 21 working days in a certain month. 131 132 140 140 141 144 146 147 149 150 151 159 167 169 169 172 174 175 178 187 189 1.1 Calculate the mean distance travelled by the sales representative. 1.2 Write down the five-number summary for this set of data. 1.3 Use the scaled line on DIAGRAM SHEET 1 to draw a box-and-whisker diagram for this set of data. 1.4 Comment on the skewness of the data. 1.5 Calculate the standard deviation of the distance travelled. 1.6 The sales representative discovered that his odometer was faulty. The actual reading on each of the 21 days was p km more than that which was indicated. Write down, in terms of p (if applicable), the: 1.6.1 Actual mean 1.6.2 Actual standard deviation Copyright reserved Please turn over (2) (4) 2) 0) (2) (1) (1) [13]

Downloaded from hlayiso.com Mathematics/P2 4 DBE/Feb.—Mar. 2015 NSC QUESTION 2 An ice-cream shop recorded the sales of ice cream, in rand, and the maximum temperature, in °C, for 12 days in a certain month. The data that they collected is represented in the table and scatter plot below. Temperature in °C 24,2 | 26,4 | 21,9 | 25,2 | 28,5 | 32,1 | 29,4 | 35,1 | 33,4 | 28,1 | 32,6 | 27,2 Sales of ice cream in rand | 215 | 325 | 185 | 332 | 406 | 522 | 412 | 614 | 544 | 421 | 445 | 408 Scatter plot 650 ; 1 600 ° | 550 ° e 500 3 t I : | | = 400 + : o (e @ : 350 - i; T 3 i 2 | : bad = 300 ml . | & 250 - S I n | e 200 + 150 + - = 24 26 28 30 32 34 36 38 Temperature in °C 2.1 Describe the influence of temperature on the sales of ice cream in the scatter plot. qd) 22 Give a reason why this trend cannot continue indefinitely. qd) 2.3 Calculate an equation for the least squares regression line (line of best fit). (4) 2.4 Calculate the correlation coefficient. () 2.5 Comment on the strength of the relationship between the variables. qd) [8] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 DBE/Feb—Mar. 2015 NSC QUESTION 3 In the diagram below points P(5 ; 13), Q(-1;5) and S(7,5; 8) are given. SR||PQ where R is the y-intercept of SR. The x-intercept of SR is B. QR is joined. as P(S; 13) 8(7,5 ; 8) Q(-1 ; 5) 0 a B ey R a. Calculate the length of PQ. (3) 3.2 Calculate the gradient of PQ. (2) 3.3 Determine the equation of line RS in the form ax + by+e¢=0. (4) 3.4 Determine the x-coordinate of B. (2) 3.5 Calculate the size of ORB. (3) 3.6 Prove that QBSP isa parallelogram. (4) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 DBE/Feb.—Mar. 2015 NSC QUESTION 4 4.1 In the diagram below, the circle centred at M(2 ; 4) passes through C(-1 ; 2) and cuts the y-axis at E. The diameter CMD is drawn and ACB is a tangent to the circle. y D M(2 ; 4) x (6) B 4.1.1 Determine the equation of the circle in the form (x—a)> +(y—b)* =r’. (3) 4.1.2 Write down the coordinates of D. (2) 4.1.3 Determine the equation of AB in the form y= mx +c. (5) 4.1.4 Calculate the coordinates of E. (4) 4.1.5 Show that EM is parallel to AB. (2) 42 Determine whether or not the circles having equations (x +2)? +(y—4)? =25 and (x-5)? +(y +1)? =9 will intersect. Show ALL calculations. (6) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 DBE/Feb.-Mar. 2015 NSC QUESTION 5 5.1 If x=3 sin 6 and y=3 cos @, determine the value of x? + y’. G3) 5.2 Simplify to a single term: sin(40° — x).sin(—x) — cos(1 80° — x).sin(90° + x) (6) 5.3 In the diagram below, T(x ; p) is a point in the third quadrant and it is given that sina = —2 y+ p? ; on . Tx; p) $.3.1 Show that x =-l. (3) 5.3.2 Write cos(180°+a@) in terms of p in its simplest form. Q) . l-p’ 5.33 Show that cos 2@ can be written as = (3) l+p : . 2tanx —sin2x . 5.4 5.4.1 For which value(s) of x will Sein be undefined in the sin’ x interval 0° <x < 180°? (3) 2t —sin2 5.4.2 Prove the identity: ke ee = tanx 2sin? x (6) [26] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 DBE/Feb.—Mar. 2015 NSC QUESTION 6 6.1 In the figure, points K, A and F lie in the same horizontal plane and TA represents a vertical tower. ATK =x, KAF=90°+x and KFA=2x where 0° <x <30°. TK =2 units. 6.1.1 Express AK in terms of sin x. (2) 6.1.2 Calculate the numerical value of KF. (5) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 DBE/Feb.—Mar. 2015 NSC 6.2 In the diagram below, a circle with centre O passes through A, B and C. BC = AC= 15 units. BO and OC are joined. OB = 10 units and BOC=x. 10 jI B 15 i Cc Calculate: 6.2.1 The size of x (4) 6.2.2 The size of ACB (3) 6.2.3 The area of AABC (2) [16] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 DBE/Feb.—Mar. 2015 NSC GIVE REASONS FOR YOUR ANSWERS IN QUESTIONS 7, 8, 9 AND 10. QUESTION 7 In the diagram, AB is a chord of the circle with centre O. M is the midpoint of AB. MO is produced to P, where P is a point on the circle. OM = x units, AB = 20 units and —— = — OM 2 A B P 7.1 Write down the length of MB. d2 Give a reason why OM AB. 3x . ves! Show that OP = > units. 74 Calculate the value of x. Copyright reserved Please turn over (1) (1) (3) [7]

Downloaded from hlayiso.com Mathematics/P2 I DBE/Feb.—Mar. 2015 NSC QUESTION 8 In the diagram below, the circle with centre O passes through A, B, C and D. AB ||DC and BOC =110°. The chords AC and BD intersect at E. EO, BO, CO and BC are joined. 8.1 Calculate the size of the following angles, giving reasons for your answers: 8.1.1 D (2) 8.1.2 A (2) 8.1.3 E, (4) 8.2 Prove that BEOC is a cyclic quadrilateral. (2) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 DBE/Feb.—Mar. 2015 NSC QUESTION 9 9.1 Complete the statement of the following theorem: The exterior angle of a cyclic quadrilateral is equal to ... qd) 9.2 In the diagram below the circle with centre O passes through points S, T and V. PR isa tangent to the circle at T. VS, ST and VT are joined. Given below is the partially completed proof of the theorem that states that VIR =S. Using the above diagram, complete the proof of the theorem on DIAGRAM SHEET 3. Construction: Draw diameter TC and join CV. Statement | Reason Let: VIR= 7, =x a Te S00 ae sccccurecsrccacussrasunzncussnsasssannoeseenoounecennsensesnsnesccnessnsouaessusassenarse C= sess Sum of the angles of a triangle SuBimae | asssssessensnecnensvszzensnnnssnnnesns stan vsuvavanrnsceasanesseessnogactaneeonei .VIR=S 6) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 13 DBE/Feb.—Mar. 2015 NSC 93 In the figure, TRSW is a cyclic quadrilateral with TW = WS. RT and RS are produced to meet tangent VWZ at V and Z respectively. PRQ is a tangent to the circle at R. RW is joined. R, = 30° and R, =50°. 9.3.1 Give a reason why R, =30°, (1) 9.3.2 State, with reasons, TWO other angles equal to 30°. (3) 9.3.3 Determine, with reasons, the size of: @ 8, (3) (b) v (4) 9.3.4 Prove that WR?=RV x RS. (5) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 14 DBE/Feb.—Mar. 2015 NSC QUESTION 10 In ATRM, M= 90°. NP is drawn parallel to TR with N on TM and P on RM. It is further given that RT = 3PN. 10.1 Give reasons for the statements below. Use DIAGRAM SHEET 5. Statement Reason In APNM and ARTM : 10.1.1] N,=T Mis common 10.1.2] «. APNM ||| ARTM (2) 10.2 Prove that a = i (2) RM 3 10.3 Show that RN? — PN? = 2RP?. (4) [8] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/Feb.—Mar. 2015 NSC CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 1 QUESTION 1.3 f T T T T T mT T nana! 120 130 140 150 160 170 180 190 200 QUESTION 7 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/Feb.—Mar. 2015 NSC CENTRE NUMBER: EXAMINATION NUMBER: | DIAGRAM SHEET 2 QUESTION 8 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 QUESTION 9.2 DBE/Feb.—Mar. 2015 NSC CENTRE NUMBER: : EXAMINATION NUMBER: [ | DIAGRAM SHEET 3 Cc S Vv ] P T R Construction: Draw diameter CT and join CV. Statement | Reason Sum of the angles of a triangle Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/Feb.—Mar. 2015 NSC CENTRE NUMBER: EXAMINATION NUMBER: [| |] | | DIAGRAM SHEET 4 QUESTION 9.3 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/Feb.—Mar. 2015 NSC CENTRE NUMBER: | | EXAMINATION NUMBER: | [ | | DIAGRAM SHEET 5 QUESTION 10 10.1 Statement Reason In APNM and ARTM : VOLT | N= P| ececceesssesssssesssssssssssessssssnssesesstuuussessssesssseseseee Miscommon 10.1.2 J APNM [I] ARTM | i issessseessecsseesseecsnesssessussssessssesssstessecsseceseces Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/Feb.—Mar. 2015 NSC INFORMATION SHEET -b4Vb7 —4ac 2a A=P(l+ni) A= P(1—ni) A= P(-iy" A=P(+i)" T, =a+(n-l)d 8, = jba+(n—na] T, =ar"' Ss aaa! sre Sn = es -1eret ” re I Li r paisa = p-G+i" i i . +h)- fa PQ) = tim LEtW-LO) h>0 h ADA: + ty, d= (4 -%)' +O, -y) M2 222) y=mxt+e y-y, =m(x-x,) m=22 1 m=tan@ %2~*| (x-af +(y-bY =r? b In AABC: 2— =? = © sind sinB sinC a@ =b? +c? —2be.cosA area MABC => absinc sin(a + £)=sina.cosB +cosa.sinB sin(a —B)=sina.cosB —cosa.sinB cofa+ B)= cosa.cosf#—sina.sin B cota -£) =cosa@.cos#+sina.sinB cos’ a -sin’ a cos2@ =41-2sin’? a sin2o = 2sina.cosa 2cos* a-1 vA DG, -¥° x= o n n P(A) = “iS P(A ot B) = P(A) + P(B) — P(A and B) AMS p<athy p= Lk V-D) Sa-xy Copyright reserved