Mathematics Grade 12 P2 Limpopo Pre June 2024 Question Paper

Downloaded from hlayiso.com ee LIMPOPO PROVINCIAL GOVERNMENT REPUSLIC OF SOUTH AFRICA DEPARTMENT OF EDUCATION NATIONAL SENIOR CERTIFICATE GRADE 12 PUREE EEE MATHEMATICS P2 PRE —- JUNE EXAMINATION 2024 SERB eunmone? MARKS: 150 TIME: 3 Hours This question paper consists of 8 pages and 1 information sheet. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 Limpopo DoE / Pre June 2024 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. 10. This question paper consists of 10 Questions. Answer ALL the questions. Number your answers correctly according to the numbering system used in this question paper. Clearly show ALL calculations, diagrams and graphs that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. Information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Limpopo DoE / Pre June 2024 Mathematics/P2 QUESTION 1 3 Fifty motorists were asked to record the number of kilometres travelled in one week. The following table shows the results: Number of kilometres | Number of motorists | Cumulative frequency 10<x<20 2 20<x<30 7 30<x<40 4 40<x<50 13 50<x<60 16 60<x<70 8 1.1 Complete the Cumulative frequency column. 1.2 Draw the cumulative frequency curve (ogive curve) 1.3 Use your graph to estimate the median number of kilometres travelled per week. 1.4 What percentage of motorists travelled more than 50km in one week? QUESTION 2 2.1 The maximum daily temperatures in degrees Celsius for Polokwane for the first 10 days in July were recorded in the following table: Day 1 | Day2 | Day3 | Day4 | Day5 | Day6 | Day7 | Day8 | Day9 | Day 10 24 25 22 28 27 21 18 17 24 25 2.1.1. Calculate the mean temperature for this data. 2.1.2 Calculate the standard deviation from the mean for this data. 2.1.3. How many days did the temperature lie outside one standard deviation of the mean? 2.2 The box and whisker plots given below represent the Mathematics marks of the same 25 students at the end of their Grade 10 and 11 years at school. 2.2.1 2.2.2 Copyright reserved 35 45 60 70 85 _ Ge; |}_{_\_ __ —— — _— % = 46% Sc 10: | [J a Gro ! ¥ = 56% 30 38 48 65 16 85 Calculate the inter-quartile range of each set of data. Comment on the way in which the distribution of the marks changes from the end of the Grade 10 year to the end of the Grade 11 year. Please turn over (2) (4) (2) (2) [10] (2) (2) (4) (4) (3) [15]

Downloaded from hlayiso.com Mathematics/P2 4 Limpopo DoE / Pre June 2024 QUESTION 3 In the diagram below ABCD is a parallelogram. A (14) D(-7;3) C (-3;-4) 3.1 Give the co-ordinates of point B. 3.2 Find E, the midpoint of AC. 3.3 Show that ACLDB. 3.4 Hence give a reason why ABCD is a rhombus. 3.5 Find the size of angle ADC. QUESTION 4 4.1 Theequationofacircleis x?7+ y? — 2x + 4y-4= 0. ¥ mn A (x;y) Vv 4.1.1. Determine the coordinates of A, the centre of the circle and the length of the radius, r. 4.1.2 Calculate the value of p if N (1; p) with p > 0 is a point on the circle. 4.1.3 Determine the equation of the tangent to the circle at N. 4.2 Asecond circle, centre B, with equation (x — 4)? + y? = k? cuts the circle given in (4.1) twice. Determine the values of k for which point A will be inside the circle B. Copyright reserved Please turn over (2) (3) (3) (1) (6) [15] (5) (1) (2) (6) [14]

Downloaded from hlayiso.com Mathematics/P2 5 Limpopo DoE / Pre June 2024 QUESTION 5 5.1 Ifsina = = and cosa > 0, calculate the values of the following without the use of a calculator: S.1.1 tana (3) 5.1.2 2sina.cosa (3) 5.1.3 sin?a+cos*a (4) 5.2 Given that cos(a— 8) =cosa.cos #+sina.sin £ , derive the identity for sin(a — f) (4) 5.3 Determine, without the use of a calculator, the value of : cos 35°. sin 25° — cos(—205°).cos 55° 6) 5.4 Consider the following identity: 2sin’ x+sin 2xcosx _ 2tanx cos x 5.4.1 Prove the identity. (4) 5.4.2 For which values of x, x € (-180°; 180°), is this identity not valid? (2) 5.5 Determine the general solution for cos 2x — cosx = 2 (7) [32] QUESTION 6 The diagram shows the graphs of f(x) = sin(x + 30°) and h(x) = tans x. h “270 —B0° “80 0 3 Se ae f 6.1 Write down the domain of (x). (2) 6.2 Write down the period of h(x). (2) 63 Determine the equation of the function, (x), obtained when h(x) is translated 45° to the right and then 2 units downwards. (2) 6.4 Determine a value of x for which f(x) — h(x) = ; (2) 6.5 Determine the distance between (x) and A(x) if x = —135°. (3) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 Limpopo DoE / Pre June 2024 QUESTION 7 The diagram below shows a vertical tower PR, with points Q, R and S all on horizontal ground. The angle of elevation of P from Q is «, the length QS = 14m. RQS = x and RSQ = y. / f \ fe / st \ We Sie a my [Hs IN Q 14m $ TA Express PR in terms of QR and « (2) 7.2 Show that QR = iésiny (3) cosy 73 Ifx = y, show that PR = 7 tones (4) cosy [9] QUESTION 8 8.1 In the diagram below, P, Q, R and S are points that lie on the circumference of the circle with centre O. Given below is the partially completed proof of the theorem that states that P +R = 180°. o (7) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 Limpopo DoE / Pre June 2024 8.2 In the figure below, UVWX is a cyclic quadrilateral, with UW||YZ and tangent YXZ touching the circle at X. UWX = 58°. Y x z Determine the values of the following angles, showing all steps and reasons: 8.2.1 (2) 8.2.2 ¥ (2) 8.2.3 &, (2) 8.2.4 (2) [15] QUESTION 9 In the diagram below, AC is a chord of circle ABCDE. AFD 1 EFB, ZD1 = 40° and ZE = 2B1. In the following questions, give a reason for each statement: 9.1 Name THREE angles each equal to 50°. (3) 9.2 Calculate the size of DCB. (3) 93 Prove that EB||DC. (3) 9.4 Prove that AC is a diameter of the circle. (4) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 Limpopo DoE / Pre June 2024 QUESTION 10 In the diagram, WPTU is a cyclic quadrilateral with UW = UT. Chords WT and PU intersect at Q. PW extends to S such that SU||WT. Ss W, (/ P U T Prove that: 10.1 US is a tangent to circle PWUT at U. 10.2 ASPU ||| ASUW 10.3 SU = SP.SW 10.4 sUu?.QU = PU.SW? Copyright reserved Please turn over 6) (4) Q) 4) [16]

Downloaded from hlayiso.com Mathematics/P2 Limpopo DoE / Pre June 2024 INFORMATION SHEET —b+vb2 —4ac 2a A= P(1+ni) A= P(1-ni) A=P(1-i)" A=P(\+i)" T, =a+(n-l)d 8, =5(2a+(n-1)d) T, =ar"™" Ss _ alr" -1) ; r#¥l1 S, = a j;-l<r<l " r-1 l-r pasa =! pola] fe) = lim LE+H- SO) i i ho0 h d= (x, -x,)? +02 -y,)" y=mxt+e (ra) +00) =P a b c In AABC: - =— =— snd snB snC a” =b* +c” —2be.cos A area AABC = . ab.sin C sin(a + B) =sina.cos 8 +cosa.sin B cos(a + 8) = cos a.cos 8 sin a.sin 8 cos’ a—sin? a cos2a@ =41-2sin? a 2cos* a-1 x= n _ mA) ar) y=a+bx Copyright reserved v-Y =m(x-X;) x, +x. + m| SE %2.. YF 2 > ma22—2L m=tand aoa 2 sin(a - B) =sina.cos B —cosa.sin B cos(a — B)=cosa.cos B+ sin a.sin B sin 2a = 2sina.cosa@ P(A or B) = P(A) + P(B) — P(A and B) X-¥Ky-¥) Yiw-xy