Limpopo Mathematics P1 Sept 2024 Grade 12_hlayiso.com_.pdf

_Downloaded from hlayiso.com — | oe ———— EE —= REPUBLIC OF SOUTH AFRICA | i ROBO DEPARTMENT OF EDUCATION NATIONAL SENIOR CERTIFICATE [ GRADE 12 | r MATHEMATICS P1 ’ : SEPTEMBER 2024 : eee rer rrr rrr rrr tr tt it MARKS: 150 TIME: 3 hours | MMU This question paper consists of 9 pages and 1 information sheet. Please turn over Copyright reserved

Downloaded from hlayiso.com 2 Limpopo DoE/September 2024 NSC INSTRUCTIONS AND INFORMATION Mathematics/P1 Read the following instructions carefully before answering the questions. 1, This question paper consists of 10 Questions. 2s Answer ALL the questions. 3 Number your answers correctly according to the numbering system used in this question paper. 4, Clearly show ALL calculations, diagrams and graphs that you have used in determining your answers. 3: Answers only will NOT necessarily be awarded full marks. 6. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 7. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 8. Diagrams are NOT necessarily drawn to scale. 9. Information sheet with formulae is included at the end of the question paper. 10. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P1 3 Limpopo DoE/September 2024 NSC QUESTION 1 1.1 Solve for x: Ld 2 -3x42=0 @) 1.1.2 3x? =-2-6x (Round off to TWO decimal digits) (4) 1.1.3 2x-1=Vl-x (4) 1.1.4 (x+3)G-x)<0 G3) 1.2 Solve for x and y simultaneously: 2x=y+2 y-2=x -3x (6) 1.3 An athlete calculated that if he increases his current speed of x km/h by 5 km/h, he can reduce his time (t) by 12 minutes. He will be participating in the City Marathon in Polokwane which is 72 km long. Determine the value of x. (5) [25] : Please turn over Copyright reserved ei

Downloaded from hlayiso.com Mathematics/P] 4 Limpopo DoE/September 2024 NSC QUESTION 2 2:1 The 4th term of an arithmeuc sequence is 5 and the 14th term is 15. 2.1.1 Calculate the common difference. 2.1.2 Determine the general term(Z,) of the sequence. 2413 Calculate the sum of the first 22 terms. 2.2 A quadratic pattern has the following properties: hax Q=7 LT, =7x 7, -T, =6 Determine the value of x. QUESTION 3 - a f 22 Given the geometric series; 2+ = + 9 HP esteetesevnieivers 3.1 Determine the sum to infinity. 32 Show that the sum of the first _» terms of the series is given by 3 -3(3) i 33 Calculate the smallest value of » for which the sum of the first 1 terms is greater than 2,99. Copyright reserved Please turn over (4) (2) (2) (4) [12] (3) (3) (5) (11)

Downloaded from hlayiso.com Mathematics/P 1 5 Limpopo DoE/September 2024 NSC QUESTION 4 4.1 The graphs of f(x)=-x? 4412 and g(x)=mx+k are sketched below. B and Care x-intercepts and A is the y-intercept. E isa point on f and F 1S a point on g. EG is parallel to the y-axis. B I [ 4.1.1 Determine the coordinates of B. (2) 4.1.2 Calculate the values of m and k. (3) 4.1.3 If OG =2 units, calculate : (a) The length of EF. (2) (b) The area of AOGF. (3) 4.1.4 Determine the coordinates of the point of intersection of g and the tangent of f at C. (6) 4.1.5 Determine which value(s) of x will LO)’. 02 (3) f') 42 Given: f(x)= 2 49 f iven: 7) 4.2.1 Draw a sketch graph of f clearly showing the intercepts and the asymptotes. (4) 4.2.2 Determine for which values of x will: = 2-2 (2) = x 4.2.3 Determine the equation of the axes of symmetry for which the gradient is negative. (2) [27] Copyright reserved Please turn over

~ Downloaded from hlayiso.com Mathematics/P] 6 Limpopo DoE/September 2024 NSC QUESTION 5 The figure below shows the graph of h(x) =log, x. Point A(16;4) lies on the curve and B is the x-intercept of h. 4 A(16;4 x 0 5.1 Calculate the value of a. 5:2 Write down the coordinates of B. 5.3 Determine the equation of h7', in the form Av'(x) =.......... 5.4 Write down the range of h™. Copyright reserved Please turn over @) (1) (2) (l) (71

Downloaded from hlayiso.com Mathematics/P | 7 Limpopo DoE/September 2024 NSC QUESTION 6 Thapelo and Mahlatse invest their inheritance of R200 000 at 11, 5% p.a compounded quarterly. After 10 years, they use their return on their investment to build a house at a cost of R1 850 000. They borrow the balance needed to build the house from a bank. The bank grants them a loan at an interest rate of 12% p.a compounded monthly. They must repay * the loan back over a period of 25 years. 6.1 Calculate the effective interest rate. 3) 6.2 Calculate the value of the investment after 10 years. (2) 63 Determine the loan amount needed to finish building the house. (2) 6.4 Calculate their monthly payment on the loan. (4) 6.5 Calculate the balance on the loan after 15 years. (3) 6.6 Determine how much interest they will pay on the loan in the 25 years’ time. (2) [16] QUESTION 7 TA Determine f’(x) from first principles if it given that f(x) = 3x. (5) 72 p Determine: (721 f(x) if fF) =(@-DOE +x tx4 4 42x? +241) (3) Sua? +x 72.2 D, brcan (4) x+l by ey 123 ety ae 4) [16] Copyright reserved Please turn over

YVownloaded rom niayiso.com Mathematics/P] aig * Limpopo DoE/September ‘QUESTION 8 "Given: f(x) =2x(x? -9x +24) 8.1 Show that PG ; 36) isa point on the graph of /. 8.244 Calculate the coordinates of the turning points of the graph of /. Sa Draw a neat sketch graph off. Indicate the coordinates of any intercepts iy with the axes and of the turning points ; Cea Determine the value(s) of & forwhich 2x(x?-9x+24)=k has three unequal roots. 5 185) Determine the maximum value of f(x) if x¢[0 ; 5]. | QUESTION 9 Ht two circles and has the shape shown in the figure below. The larger circle has radius R _ the smaller circle has radius r.The values of R and r must vary, and R+ r=200 m m To minimise costs, Mashudu Business Enterprise has also stated that the area of the shay must la a minimum. 9.1) Show that the area(A), of the figure is given by: A=2z7(R’ —200R + 20000) Pa 9.2 Determine the values of R and r if the area, of the figure is a minimum. 9.3 Hence, at ti why the shape suggested by the company is not possible if you ¢ want to ae, a minimum area. 4; Please turn over

Downloaded from hlayiso.com Mathematics/P 1 9 Limpopo DoE/September 2024 NSC QUESTION 10 10.1 Given P(A)=0,45 and P(B) =0,25. 10.2 10.3 Determine P(AorB) if A and B are mutually exclusive events. A survey was conducted in one of the schools in Limpopo province with a population of 50 educators. 30 of the educators indicated that they each own a car. Two educators were randomly selected one after the other without repetition. 10.2.1 Represent the given information on a tree diagram. Clearly indicate the possible outcomes of the event. 10.2.2 Find the probability that only one of the educators selected owns a car. 10.2.3 Find the probability that the two educators selected, each owns a car. You require a password for an online account. The password must have 3 numerical values, followed by 2 vowels. 10.3.1 How many passwords are possible if repetitions are allowed? 10.3.2 Determine the number of possible passwords with the following conditions: e If repetitions are not allowed, e The password does not start with a zero and e It ends with a vowel a. Copyright reserved Please turn over (2) (3) (3) (2) (2) (3) [15]

Downloaded from hlayiso.com Mathematics/P1 1 Limpopo DoE/September 2024 NSC INFORMATION SHEET xe —b+Vb? —4ac 2a A= P(1+ni) A=P(1-ni) A=P(i-i)” A=P(1+i)” T, =a+(n-l)d S, =5[24+@-Dd] T, =ar"! 5 _ alr” -1) : r#l Ss, =-—— ;-I<r<l " r-1 l-r yn _ xfl-(+i-1 Nx) = li S(xth)- fF) _ x{ (1+4)"-1] pA f'(3)= jim i d= (x -2¥ +02- 9) M| = 24d} 2 2 y=mx+ce y-y, =m(x-x,) m= m=tan@ X2—% (x-a/y t+(y-byY =r? In AABC: a bie sinA sinB sinC a =)? +c’-2bc.cos A areaAABC = 5 absinc sin(a+ 8) =sina.cos 8 +cosa.sin B sin (a - 8) =sina.cos B—cosa.sin B cos (a +B) =cosa.cos # -sina.sin f cos (a— 8) =cosa@.cos #8 +sina.sin B cos’ @—sin’? a cos2a =41-2sin’ a sin2 a =2sina.cosa 2cos’ a-1 . 2 een o? = +1 n Pia) = P (Aor B) = P(A) + P (B) -— P (A and B) ni x-X)\(y-y proatbx pees )O-9) Yi(e-x) Copyright reserved