Mathematics Grade 12 Free State November 2021 QP (Eng)

Downloaded from hlayiso.com education Department of Education FREE STATE PROVINCE PREPARATORY EXAMINATION TMU This question paper consists of 9 pages and 1 information sheet, Copyright reserved Please turn over’ as a

Downloaded from hlayiso.com Mathematics Pl 2 FS/September 2021 Grade 12 Prep. Exam. INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1, 2, 10, This question paper consists of 11 questions. Answer ALL the questions. Number the answers correctly according to the numbering system used in this question paper. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. An information shect with formulae is included at the end of this question paper. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 3 Grade 12 Prep. Exam, QUESTION 1 1.1 1.2. Solve simultaneously for x and y: y+2x=2 and y?—3yx=-2y’ 1.3 Simplify, without using a calculator: (:¥20- J, Gx) | V20 + JD.) QUESTION 2 2.1 Given the quadratic pattern: —22; -12; —6; -4;... 2.1.1 Write down the value of the next two terms of the pattern. 2.1.2 Determine an expression for the n' term of the pattern. 2.1.3. Which term of the pattern will have the maximum value? 2.2 Then" term of the arithmetic sequence is T, = 27—6(n+}) Solve for x: LLY x?-4x-21=0 1.1.2 x(5x-1)=3 (Correct to TWO decimal places) 1.1.3 2x?-9x+420 11.4 3*'-3*1-24-0 2.2.1 Write down the value of the common difference. 2.2.2. Which term of the sequence is equal to -117? Copyright reserved FS/September 2021 G3) (4) (3) (4) 65) (4) [23] @) ) @) q) (2) Please turn over

Downloaded from hlayiso.com Mathematics P1 4 FS/September 2021 Grade 12 Prep. Exam. 2.3 The series 5+9+13+... consists of ” terms. The sum of the last six terms of this series is 906. 2.3.1 Determine, in terms of », the sum to 7 terms of the series. (2) 2.3.2 -If the last six terms of the series are excluded, determine, in terms of n, the sum of the remaining terms, (2) 23. 3 Hence, determine the value of n. : (4) 119} QUESTION 3 . : . 3.3 3.1 Given the geometric series: 3; 3 re ooo -Determine the largest value of n for which T,, > 3 ; (4) 16384 2 30 1 nl 3.2 Calculate the value of p if 5°27p* = (5) 6) A= ast 19] QUESTION 4 Given the function defined by / (x) =~2x" —4x +16 4.1 Write down the y-intercept of f, Q) 42 Calculate the x-intercepts of f (2) 4.3 Determine the coordinates of the turning point of f (GB) 44 Sketch the graph of /, indicating all intercepts with the axes and the turning point. QB) 4.5 Write down the range of /. (2) (11) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 . 3 FS/September 2021 Grade 12 Prep. Exam. QUESTION 5 Sketched below are the graphs of /| (x) = (5) and g(x) = op +q. Bis the point of x+p interse¢tion of the asymptotes of g. A is the y-intercept of f The graph g passes through the origin. AB is parallel to the x axis. 5.1 Write down the equation of “, the inverse of fin the form Youn (2) 5.2 Write-down the domain of f”. (a 5.3 Sketch the graph of y= f “I (x) on your answer book. Clearly indicate the intercept with the axis and the asymptote on your graph. (2) 5.4 If h(x)=x+3 is the equation of one of the axes of symmetry of g, determine the coordinates of B. (2) a 5.5 Hence, determine the equation of g in the form & (x) Smog. x+p (4 5.6 Write down the equation of kif &(x) = g(x—3)+1 in the form K(x) = a + xtp | @) 5.7 Determine for which value(s) of x is: 571 f{x)xg(x)<0? (2) 5.7.2 gi (x)>0? (2) (17) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 6 FS/September 2021 Grade 12 Prep, Exam. QUESTION 6 61° Given: p(x) =-3x" 6.1.1 Give areason why p” is nota function (2) 6.1.2 Place the restrictions on the domain of p so that Pp’ isa function. Q) 62 Ina test, Peter was required to determine the derivative of a function f (x). However, by mistake; he determined the inverse of f(x) and his answer was raf Determine the correct answer to the question given in a test. (3) [7 QUESTION 7 7.1 An investment earns interest at 12% per annum, compounded every six months. Calculate the effective annual interest rate of this investment. Q) 7.2 Acompany bought equipment that depreciates annually at 7% on the reducing balance. How many years will it take for this equipment to be worth half the original purchase price? @) 73 Kenneth invested R250 000 in an account that paid interest at 9,5% p.a. compounded quarterly. Twelve years after making the initial investment, Kenneth bought a house that cost R2 920 000. He used all the money that had accumulated in his investment as a deposit and the bank granted him a loan for the balance. The bank charged iriterest on the loan at 10,3% p.a., compounded monthly. The loan was for a 20-years period and the first raonthly instalment was made one month after the loan was granted. 73.1 Calculate the value of Kenneth's investment after 12 years. QB) 73.2 Determine the value of the loan that Kenneth obtained. re) 73.3 Calculate the monthly repayment on the loan. (¢)) 7.3.4 Kenneth paid off the loan at the end of 20 years. How much interest did he pay on the loan? Q) [14] Copyright reserved Please tum over

Downloaded from hlayiso.com Mathematics Pi 7 FS/September 2021 Grade 12 Prep. Exam. QUESTION 8 $1 Determine f/(x) from first principles if it is given that f(x) = 3. (4) x 82 Determine: 82.1 D,(x°-2) | @) dy 4. 822 if pode -—44Ne (4) 8.3 Determine the coordinates of the point on the curve of y= 2(1 - x) where ° 4 the tangent at this point is perpendicular to Y= 2x ? 4) [15] QUESTION 9 The graph of a cubic function, fi (x) =ax +bx +ox+d, is sketched below. P( 036); Q(1;0);R and $(3;0) are points on the graph. Q and R are the turning (or stationary) points of f° ” 6 R Q s x 9 1 3 of Show that a = -2;6 =10;¢ =—14 and d =6. (5) 9.2 Determine the x-coordinate of the point of inflection of, ft (3) 9.3. For which value(s) of: 93.1 xis f(x) increasing? (5) 9.3.2 kwill (x)= & have exactly three different real roots? (3) [16] Copyright reserved Please tum over

Downloaded from hlayiso.com Mathematics P1 8 FS/September 2021 Grade 12 Prep. Exam. QUESTION 10 A farmer makes a rectangular enclosure for his sheep using 100 metres of fencing. He puts a double layer of fencing along one side to stop the sheep breaking through to his crops in the next field. A plan of his fence is shown below. A sme TIESTO Te D Double fence: the space x between is negligible B om —— a Cc The length of AB is x metres, 10.1 Show that the area of the land which can be enclosed, in this way is given by . - 3 2 the formula: A(x) =50x — 3 x Q) 10.2 Determine the length of AB if the enclosed area is to be a maximum. (3) 5] QUESTION 11 11.1. A group of 540 people with green or blue eyes were randomly selected to determine whether green or blue eyes are gender dependent. The results are tabulated below: Male Female Total Green eyes 183 147 330- Blue eyes 117 93 210 Total 300 240 540 {1.1:1 Ifa person from this group is selected at random, write down the probability that the person is a female and has green eyes. qd) 11.1.2 After analysing the results, a learner concludes that the probability of having green eyes is independent of gender. Is she correct? Support your answer with relevant calculations. (4) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 9 FS/September 2021 Grade 12 Prep. Exam. 11.2 E and F are two independent events. P(E) =x andP(F) =y. P(E and P=; and P(E or F)= aa Show that 30y’ +10=37y B) 113 Thirteen jJearners must be seated on a bus where the seats are arranged as shown below. XXXKX <— back row KX XX xX XX <— front row 11.3.1. Ifthe learners are arranged at random, how many ways can they be-seated? ° (2) 11.3.2 Five friends insist on sitting in the back row and one learner who gets sick needs to sit next to the door in the seat marked D. xX XM KX XxX XX KX %XD In how many ways can the learners now be arranged? (4A [14] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Mathematics P1 Grade 12 Prep. Exam. INFORMATION SHEET _abtvb? —4Aae a _. 2a A= P(t+ni) T, =a+(n~Nd A=P(—ni) T= ar" Ss a ” L ay te LED LO) PG) = fet, h d=y(%—mY +0, -n) mf yemxre yry, =Mx-H) (x-al +(p-bY =r? In AABC: ab _¢ sind sinB sinC a’ =b* +c? —2be.cosA area AABC =F absin€ sin(a + 8) = sina.cos 8 + cosa. sinB cos(a + 8) = cosa.cos 8 ~ sina. sin B cos’ a —sin? a cos2a = 41-2sin? a 2cos* a1 Copyright reserved _ a" =!) sre Ss, =— FS/September 2021 A=Pl-i" A=PO+i" 8, = p Ret (n-)d] j-i<r<l o r-l1 i-r Pe xf - (+ iy” | i My tX%y Yt2 ? ee) ma22 29% m=tand XQ ¥y sin(a — B) = sina. cos f — cosa@.sinB cos(@ — 8) = cos a.cos 8 + sina.sin B sin2a@ = 2sina.cosa n SG, 7 x) oo = n P(A or B) = P(A) + P(B) — P(A and B) ,_Le-Fe-7 DG@- xy