MATRIC MATHEMATICS. NSC CAPS EXAM PAPERS 2014 Booklet

Transcription

1 MATRIC MATHEMATICS NSC CAPS EXAM PAPERS 2014 Booklet

2 These are past Grade 12 Mathematics Papers, with a difference... there are video solutions for each question embedded in the Papers! To access the videos... Scan the QR-codes next to the question by using the Paper Video App OR Enter the 4 digit alphanumerics, underneath the QR-codes, at papervideo.co.za OR Click on the QR-codes if you are using a Pdf

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4 Contents Papers 2014 Exemplar Paper Exemplar Paper November CAPS Aligned Paper November CAPS Aligned Paper February CAPS Aligned Paper February CAPS Aligned Paper Solutions 2014 Exemplar Paper 1 - Solutions Exemplar Paper 2 - Solutions November CAPS Aligned Paper 1 - Solutions November CAPS Aligned Paper 2 - Solutions February CAPS Aligned Paper 1 - Solutions February CAPS Aligned Paper 2 - Solutions...81 Formula Sheet Formula Sheet...84

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6 2014 Exemplar PAPER 1 Question 1: 1.1) Solve for x: 1.1.1) 3x 2 4x = 0 (2) 1.1.2) x x = 0 ; x 0. (leave your answer correct to TWO decimal places.) (4) 1.1.3) x 2 3 = 4 (2) 1.1.4) 3 x (x 5) < 0 (2) 1.2) Solve for x and y simultaneously: y = x 2 x 6 and 2x y = 2 (6) 1.3) Simplify, without the use of a calculator: x+1 2 2x (3) 1.4) Given: f(x) = 3(x 1) and g(x) = 3: 1.4.1) Is it possible for f(x) = g(x)? Give a reason for your answer. (2) Algebraic Approach Graphical Approach 1

7 2014 Exemplar PAPER ) Determine the value(s) of k for which f(x) = g(x) + k has TWO unequal real roots. (2) Algebraic Approach Graphical Approach [23] Question 2: 2.1) Given the arithmetic series: : 2.1.1) Determine the number of terms in this series. (3) 2.1.2) Calculate the sum of the series. (2) 2.1.3) Calculate the sum of all the whole numbers (N), up to and including 300, that are NOT divisible by 6. (4) 2.2) The first three terms of an infinite geometric sequence are 16, 8 and 4 respectively ) Determine the n th term of the sequence. (2) 2.2.2) Determine all possible values of n for which the sum of the first n terms of this sequence is greater than 31. (3) 2.2.3) Calculate the sum to infinity of this sequence. (2) [16] 2

8 2014 Exemplar PAPER 1 Question 3: 3.1) A quadratic number pattern T n = an 2 + bn + c has a first term equal to 1. The general term of the first differences is given by 4n ) Determine the value of a. (3) 3.1.2) Determine the formula for T n. (4) 3.2) Given the series: (1 2) + (5 6) + (9 10) + (13 14) (81 82) Write the series in sigma notation. (It is not nessesary to calculate the value of the series.) (4) [10] Question 4: 4.1) Given f(x) = 2 x ) Calculate the coordinates of the y-intercept of f. (2) 4.1.2) Calculate the coordinates of the x-intercept of f. (2) 4.1.3) Sketch the graph of f, showing clearly the asymptotes and the intercepts with the axes. (3) 3

9 2014 Exemplar PAPER ) One of the axes of symmetry of f is a decreasing function. Write down the equation of this axis of symmetry. (2) 4.2) The graph of an increasing exponential function with equation f(x) = a.b x + q has the following properties: Range y > 3 The points (0; 2) and (1; 1) lie on the graph of f ) Determine the equation that defines f. (4) 4.2.2) Determine the translation from f(x) to h(x) = 2.2 x + 1 (2) [15] Question 5: The sketch below shows the graph of f(x) = 2x 2 5x + 3 and g(x) = ax + q. The angle of inclination of graph g is 135 in the direction of the positive x-axis. P is the point of intersection of f and g such that g is a tangent to the graph of f at P. 5.1) Calculate the coordinates of the turning point of the graph of f. (3) 4

10 2014 Exemplar PAPER 1 5.2) Calculate the coordinates of P, the point of contact between f and g. (4) 5.3) Hence, or otherwise, determine the equation of g. (2) 5.4) Determine the values of d for which the line k(x) = x + d will not intersect the graph of f. (1) [10] Question 6: The graph of g is defined by the equation g(x) = ax. The point (8; 4) lies on g. 6.1) Calculate the value of a. (2) 6.2) If g(x) > 0, for what values of x will g be defined? (1) 6.3) Determine the range of g. (1) 6.4) Write down the equation of g 1, the inverse of g, in the form y =... (2) 5

11 2014 Exemplar PAPER 1 6.5) If h(x) = x 4 is drawn, determine ALGEBRAICALLY the point(s) of intersection of h and g. (4) 6.6) Hence, or otherwise, determine the values of x for which g(x) > h(x) (2) [12] Question 7: Siphokazi bought a house. She paid a deposit of R , which is equivalent to 12% of the selling price of the house. She obtained a loan from the bank to pay the balance of the selling price. The bank charges her interest of 9% per annum, compounded monthly. 7.1) Determine the selling price of the house. (1) 7.2) The period of the loan is 20 years and she starts repaying the loan one month after it was granted. Calculate her monthly instalment. (4) 7.3) How much interest will she pay over the period of 20 years? Round your answer correct to the nearest rand. (2) 7.4) Calculate the balance of her loan immediately after her 85 th instalment. (3) Present Value Approach Future Value Approach 7.5) She experienced financial difficulties after the 85 th instalment and did not pay any instalments for 4 months (that is months 86 to 89). Calculate how much Siphokazi owes on her bond at the end of the 89 th month. (2) 6

12 2014 Exemplar PAPER 1 7.6) She decides to increase her payments to R8 500 per month from the end of the 90 th month. How many months will it take to repay her bond after the new payment of R8 500 per month? (4) [16] Question 8: 8.1) Determine f (x) from first principles if f(x) = 3x 2 2. (5) 8.2) Determine dy dx if y = 2x 4 x 5 (2) [7] Question 9: Given: f(x) = x 3 4x 2 11x ) Use the fact that f(2) = 0 to write down a factor of f(x). (1) 9.2) Calculate the coordinates of the x-intercepts of f. (4) 9.3) Calculate the coordinates of the stationary points of f. (5) 9.4) Sketch the curve of f. Show all intercepts with the axes and turning points clearly. (3) 9.5) For which value(s) of x will f (x) < 0? (2) [15] 7

13 2014 Exemplar PAPER 1 Question 10: Two cyclists start to cycle at the same time. One starts at point B and is heading due north to point A, whilst the other starts at point D and is heading due west to point B. The cyclist starting from B cycles at 30 km/h while the cyclist starting from D cycles at 40 km/h. The distance between B and D is 100km. After time t (measured in hours), they reach points F and C respectively. 10.1) Determine the distance between F and C in terms of t. (4) 10.2) After how long will the two cyclists be closest to each other? (4) 10.3) What will the distance between the cyclists be at the time determined in QUESTION 10.2? (2) [10] Question 11: 11.1) Events A and B are mutually exclusive. It is given that: P(B) = 2P(A) P(A or B) = 0,57 Calculate P(B). (3) 8

14 2014 Exemplar PAPER ) Two identical bags are filled with balls. Bag A contains 3 pink and 2 yellow balls. Bag B contains 5 pink and 4 yellow balls. It is equally likely that Bag A or Bag B is chosen. Each ball has an equal chance of being chosen from the bag. A bag is chosen at random and a ball is then chosen at random from the bag ) Represent the information by means of a tree diagram. Clearly indicate the probability associated with each branch of the tree diagram and write down all the outcomes. (4) ) What is the probability that a yellow ball will be chosen from bag A? (1) ) What is the probability that a pink ball will be chosen? (3) [11] Question 12: Consider the word M A T H S. 12.1) How many different 5-letter arrangements can be made using all the above letters? (2) 12.2) Determine the probability that the letters S and T will always be the first two letters of the arrangements in QUESTION (3) [5] 9

15 2014 Exemplar PAPER 2 Question 1: Twelve athletes trained to run the 100m sprint event at the local athletics club trials. Some of them took their training more seriously than others. The following table and scatter plot shows the number of days that an athlete trained and the time taken to run the event. The time taken, in seconds, is rounded to one decimal place. 1.1) Discuss the trend of the data collected. (1) 1.2) Identify any outliers(s) in the data. (1) 1.3) Calculate the equation of the least squares regression line. (4) 10

16 2014 Exemplar PAPER 2 1.4) Predict the time taken to run the 100m sprint for an athlete training for 45 days. (2) 1.5) Calculate the correlation coefficient. (2) 1.6) Comment on the strength of the relationship between the variables. (1) [11] Question 2: The following table shows the amount of time (in hours) that learners aged between 14 and 18 spent watching television during 3 weeks of the holiday. 2.1) Draw an ogive (cumulative frequency curve) to represent the above data, on the diagram provided. (3) 2.2) Write down the modal class of the data. (1) 11

17 2014 Exemplar PAPER 2 12

18 2014 Exemplar PAPER 2 2.3) Use the ogive (cumulative frequency curve) to estimate the number of learners who watched television more than 80% of the time. (2) 2.4) Estimate the mean time (in hours) that learners spent watching television during 3 weeks of the holiday. (4) [10] Question 3: In the diagram below, M, T ( 1; 5), N(x; y) and P (7; 3) are vertices of trapezium M T N P having T N MP. Q(1; 1) is the midpoint of MP. P K is a vertical line and S ˆP K = θ. The equation of NP is y = 2x ) Write down the coordinates of K. (1) 3.2) Determine the coordinates of M. (2) 13

19 2014 Exemplar PAPER 2 3.3) Determine the gradient of P M. (2) 3.4) Calculate the size of θ. (3) 3.5) Hence, or otherwise, determine the length of P S. (3) Trigonometric Approach Distance Formula Approach 3.6) Determine the coordinates of N. (5) Gradient Approach Points of Intersection Approach 3.7) If A(a; 5) lies in the Cartesian plane: 3.7.1) Write down the equation of the straight line representing the possible positions of A. (1) 3.7.2) Hence, or otherwise, calculate the value(s) of a for which T ÂQ = 45. (5) [22] 14

20 2014 Exemplar PAPER 2 Question 4: In the following diagram, the equation of the circle having centre M is (x + 1) 2 + (y + 1) 2 = 9. R is a point on chord AB such that MR bisects AB. ABT is a tangent to the circle having centre N(3; 2) at point T (4; 1). 4.1) Write down the coordinates of M. (1) 4.2) Determine the equation of AT in the form y = mx + c. (5) ) If it is further given that MR = units, calculate the length of AB. Leave your answer in 2 surd form. (4) 15

21 2014 Exemplar PAPER 2 4.4) Calculate the length of MN. (2) 4.5) Another circle having centre N touches the circle having centre M at point K. Determine the equation of the new circle. Write your answer in the form x 2 + y 2 + Cx + Dy + E = 0. (3) [15] Question 5: 5.1) Given that sin a = 4 5 and 90 < a < 270. WITHOUT using a calculator, determine the value of each of the following in its simplest form: 5.1.1) sin( a) (2) 5.1.2) cos(a) (2) 5.1.3) sin(a 45 ) (3) 5.2) Consider the identity: 8 sin(180 x) cos(x 360 ) sin 2 x sin 2 (90 + x) = 4 tan 2x 5.2.1) Prove the identity. (6) 5.2.2) For which value(s) of x in the interval 0 < x < 180 will the identity be undefined? (2) 16

22 2014 Exemplar PAPER 2 5.3) Determine the general solution of cos 2θ + 4 sin 2 θ 5 sin θ 4 = 0 (7) [22] Question 6: In the following diagram, the graphs of f(x) = tan bx and g(x) = cos(x 30 ) are drawn on the same set of axes for 180 x 180. The point P (90 ; 1) lies on f. Use the diagram to answer the following questions. 6.1) Determine the value of b. (1) 6.2) Write down the coordinates of A, a turning point of g. (2) 6.3) Write down the equation of the asymptote(s) of y = tan b(x + 20 ) for x [ 180 ; 180 ]. (1) 17

23 2014 Exemplar PAPER 2 6.4) Determine the range of h if h(x) = 2g(x) + 1. (2) [6] Question 7: 7.1) Prove that in any acute-angled ABC, sin A a = sin B. (5) b 7.2) The framework for a construction consists of a cyclic quadrilateral P QRS in the horizontal plane and a vertical post T P as shown in the following figure. From Q, the angle of elevation of T is y. P Q = P S = k units, T P = 3 units and S ˆRQ = 2x ) Show, giving reasons, that P ŜQ = x. (2) 7.2.2) Prove that SQ = 2k cos x. (4) Cosine Rule Approach Sine Rule Approach 18

24 2014 Exemplar PAPER ) Hence, prove that SQ = 6 cos x tan y. (2) [13] Give reasons for your statements in QUESTIONS 8,9 and 10. Question 8: 8.1) Complete the following statement: The angle between the tangent and the chord at the point of contact is equal to... (1) 8.2) In the diagram, A, B, C, D and E are points on the circumference of the circle such that AE BC. BE and CD produced meet at F. GBH is a tangent to the circle at B. ˆB1 = 68 and ˆF = 20. Determine the size of each of the following: 8.2.1) Ê 1 (2) 8.2.2) ˆB3 (1) 19

25 2014 Exemplar PAPER ) ˆD1 (2) 8.2.4) Ê 2 (1) 8.2.5) Ĉ (2) [9] Question 9: In the diagram, M is the centre of the circle and diameter AB is produced to C. M E is drawn perpendicular to AC such that CDE is a tangent to the circle at D. ME and the chord AD intersect at F. MB = 2BC. 9.1) If ˆD 4 = x, write down, with reasons, TWO other angles each equal to x. (3) 9.2) Prove that CM is a tangent at M to the circle passing through M, E and D. (4) 20

26 2014 Exemplar PAPER 2 9.3) Prove that F M BD is a cyclic quadrilateral. (3) 9.4) Prove that DC 2 = 5BC 2. (3) 9.5) Prove that DBC DF M. (4) 9.6) Hence, determine the value of DM F M. (2) [19] Question 10: 10.1) In the diagram, points D and E lie on sides AB and AC respectively of ABC such that DE BC. Use Euclidean Geometry methods to prove the theorem which states that AD DB = AE EC. (6) 21

27 2014 Exemplar PAPER ) In the diagram, ADE is a triangle having BC ED and AE GF. It is also given that AB : BE = 1 : 3, AC = 3 units, EF = 6 units, F D = 3 units and CG = x units. Calculate, giving reasons: ) the length of CD (3) ) The value of x (4) ) The length of BC (5) ) The value of Area ABC Area GF D (5) Geometric Approach Trigonometric Approach [23] 22

28 2013 November CAPS Aligned PAPER 1 Question 1: 1.1) Solve for x in each of the following: 1.1.1) x 2 x 12 = 0 (3) 1.1.2) i. 2x 2 5x 11 = 0 (4) ii. 2x 3 5x 2 11x = 0 (2) 1.1.3) 3(x + 7)(x 5) < 0 (4) 1.2) Given: y + 2 = x and y = x 2 x 10 Solve for x and y simultaneously. (6) 1.3) Simplify: (3) 1.4) For which value(s) of k will the roots of 6x = 4kx be real and equal? (4) [26] Question 2: 2.1) Given the geometric sequence: 7; x; 63;... Determine the possible values of x. (3) 2.2) The first term of a geometric sequence is 15. If the second term is 10, calculate: 2.2.1) T 10 (3) 23

29 2013 November CAPS Aligned PAPER ) S 9 (3) 2.3) Given: 0; 1 2 ; 0; 1 2 ; 0; 3 2 ; 0; 5 2 ; 0; 7 ; 0;... 2 Assume that this number pattern continues consistently ) Write down the value of the 191 st term of this sequence. (1) 2.3.2) Determine the sum of the first 500 terms of this sequence. (4) 2.4) Given: 20 k=2 (4x 1) k ) Calculate the first term of the series (4x 1) k if x = 1. (2) k= ) For which values of x will (4x 1) k exist? (3) k=2 [18] Question 3: 3.1) Given the arithmetic sequence: 3; 1; 5;...; ) Determine a formula for the n th term of the sequence. (2) 3.1.2) Write down the 4 th, 5 th, 6 th and 7 th terms of the sequence. (2) 24

30 2013 November CAPS Aligned PAPER ) Write down the remainders when each of the first seven terms of the sequence is divided by 3. (2) 3.1.4) Calculate the sum of the terms in the arithmetic sequence that are divisible by 3. (5) [10] Question 4: Given: f(x) = 2x 2 5x ) Write down the coordinates of the y-intercept of f. (1) 4.2) Determine the coordinates of the x-intercepts of f. (3) 4.3) Determine the coordinates of the turning point of f. (3) 4.4) Sketch the graph of y = f(x), clearly showing the coordinates of the turning points and the three intercepts with the axes. (3) [10] 25

31 2013 November CAPS Aligned PAPER 1 Question 5: 5.1) Sketched below are the graphs of g(x) = k x, where k > 0 and y = g 1 (x). (2; 36) is a point on g ) Determine the value of k. (2) 5.1.2) Give the equation of g 1 in the form y =... (2) 5.1.3) For which value(s) of x is g 1 0? (2) 5.1.4) Write down the domain of h if h(x) = g 1 (x 3). (1) 5.2) 5.2.1) Sketch the graph of the inverse of y = 1. (2) 26

32 2013 November CAPS Aligned PAPER ) Is the inverse of y = 1 a function? Motivate your answer. (2) [11] Question 6: A sketch of the hyperbola f(x) = x d, where d and p are constants, is given below. The dotted lines x p are the asymptotes. The asymptotes intersect at P and B(2; 0) is a point on f. 6.1) 6.1.1) Determine the values of d and p. (2) 6.1.2) Show that the equation of f can be written as y = (2) x ) Write down the coordinates of P. (2) 27

33 2013 November CAPS Aligned PAPER ) Write down the coordinates of the image of B(2; 0) if B is reflected about the axis of symmetry y = x + 2. (2) 6.2) The exponential function, g(x) = p.2 x +q has a horizontal asymptote at y = 1 and passes through (0; 2). Determine the values of p and q. (3) [11] Question 7: 7.1) Mpho invests R for exactly k years. She earns interest at a rate of 9% per annum, compounded quarterly. At the end of k years, her investment is worth R ) Calculate the effective annual interest rate of Mpho s investment. (2) 7.1.2) Determine the value of k. (5) 7.2) Darrel is planning to buy his first home. The bank will allow him to use a maximum of 30% of his monthly salary to repay the bond ) Calculate the maximum amount that the bank will allow Darrel to spend each month on his bond repayments, if Darrel earns R per month. (1) 7.2.2) Suppose, at the end of each month, Darrel repays the maximum amount allowed by the bank. How much money does Darrel borrow if he takes 25 years to repay the loan at a rate of 8% p.a., compounded monthly? (The first repayment is made one month after the loan is granted.) (4) [12] Question 8: 8.1) Given: f(x) = 3x ) Determine f (x) from first principles. (5) 28

34 2013 November CAPS Aligned PAPER ) A(x; 23), where x > 0, and B( 2; y) are points on the graph of f. calculate the numerical value of the average gradient of f between A and B. (4) 8.2) Differentiate y = x + 5 x with respect to x. (3) 8.3) Determine the gradient of the tangent of the graph of f(x) = 3x 3 4x + 5 at x = 1. (4) Question 9: The function defined by f(x) = x 3 + ax 2 + bx 2 is sketched below. P ( 1; 1) and R are the turning points of f. [16] 29

35 2013 November CAPS Aligned PAPER 1 9.1) Show that a = 1 and b = 1. (5) 9.2) Hence, or otherwise, determine the x-coordinate of R. (3) 9.3) Write down the coordinates of a turning point of h if h is defined by h(x) = 2f(x) 4. (2) [10] Question 10: An industrial process requires water to flow through its system as part of the cooling cycle. Water flows continuously through the system for a certain period of time. The relationship between the time (t) from when the water starts flowing and the rate (r) at which the water is flowing through the system is given by the equation: where t is measured in seconds. r = 0, 2t t 10.1) After how long will the water be flowing at the maximum rate? (3) 10.2) After how many seconds does the water stop flowing? (3) [6] Question 11: 11.1) Given that P (A) = 0, 6; P (B ) = 0, 15; P (A B) = 0.1 and P (A and B) = x: ) Determine the value of x. (3) ) Determine P (A B). (2) 30

36 2013 November CAPS Aligned PAPER ) Determine P (A B). (1) ) Determine whether A and B are complementary events. (1) 11.2) Researchers conducted a study to test how effective a certain inoculation is at preventing malaria. part of their data is shown below: Malaria No Malaria Total Male a b 216 Female c d 648 Total ) Calculate the probability that a randomly selected study participant will be female. (1) ) Calculate the probability that a randomly selected study participant will have malaria. (1) ) If being female and having malaria are independent events, show that the value of c is 81. (2) 31

37 2013 November CAPS Aligned PAPER ) Using the value of c, determine the values of a, b and d. (2) ) What is the probability of someone being male if we know they have malaria? (2) [15] Question 12: As a Paper Video developer, you need to determine how many different 4-letter codes you can make if you had a selection of 28 letters and numbers. 12.1) How many codes can you make if repetitions of letters and numbers is allowed? (1) 12.2) How many codes can you make if repetitions are NOT allowed? (2) 12.3) How many codes can you make if each code must start with a P and end with a V, and repetitions are allowed? (2) [5] 32

38 2013 November CAPS Aligned PAPER 2 Question 1: 1.1) The relationship between blood alcohol levels and the risk of having a car accident has been studied for years. research has shown the following results: 1.1.1) Draw a scatter plot to represent the data, on the diagram sheet provided. (3) 33

39 2013 November CAPS Aligned PAPER ) Draw a line (or curve) of best fit on your scatter plot from question (1) 1.1.3) Describe the trend in the data. (1) 1.1.4) Estimate the probability of having a car accident when one s blood alcohol level is 0,18%. (The legal limit of the blood alcohol level is 0,05%.) (2) 1.2) The following table shows the number of calories and the total fat content (in grams) for some sandwiches sold in restaurants: 1.2.1) Calculate the equation of the least squares regression line. (4) 1.2.2) Calculate the correlation coefficient. (2) 1.2.3) Explain the correlation between the number of calories and the total fat content (in grams) of the sandwiches. (2) [15] 34

40 2013 November CAPS Aligned PAPER 2 Question 2: The cumulative frequency curve (ogive) drawn below shows the time taken (in minutes) for 140 patrons to leave an auditorium after watching a show. 2.1) Estimate the number of people who took more than 15 minutes to leave the auditorium. (2) 2.2) Estimate the number of people who took between 8 and 12 minutes to leave the auditorium. (2) 2.3) Write down the modal class of the data. (1) 2.4) Use the ogive to approximate the value of the median of the data set. (1) [6] 35

41 2013 November CAPS Aligned PAPER 2 Question 3: In the following diagram, P is a point (-5; 0). The inclination of line P T is 63, 43. S is the midpoint and the y-intercept of P T. R is a point on the x-axis such that P O : OR = 2 : ) Determine: 3.1.1) The gradient of P T, correct to the nearest integer value (2) 3.1.2) The equation of P T in the form y = mx + c (2) 3.1.3) The distance P S (3) Geometric Approach Trigonometric Approach 3.1.4) The coordinates of T (2) 36

42 2013 November CAPS Aligned PAPER 2 3.2) Determine the coordinates of R. (2) 3.3) Calculate the area of P T R. (4) [15] Question 4: In the following diagram, M is the centre of the circle having the equation x 2 + y 2 6x + 2y 8 = 0. The circle passes through R(0; 4) and N(p; q). R ˆMN = 90. The tangents drawn to the circle at R and N meet at P. 4.1) Show that M is the point (3; 1). (4) 4.2) Determine the equation of MR in the form y = mx + c. (3) 4.3) Show that q = 2 p. (4) 37

43 2013 November CAPS Aligned PAPER 2 4.4) Determine the values of p and q. (5) 4.5) Determine the equation of the circle having centre O and passing through N. (2) 4.6) Calculate the area of the circle centred at M. (2) 4.7) Calculate the ratio in its simplest form: NP MP (4) [24] Question 5: 5.1) In the following diagram, reflex T ÔP = α and P has coordinates ( 5; 12). Determine the value of each of the following trigonometric ratios WITHOUT using a calculator: 5.1.1) cos(α) (3) 38

44 2013 November CAPS Aligned PAPER ) tan(180 α) (2) 5.1.3) sin(30 α) (3) 5.2) Prove the following identity: cos 2 (90 + θ) cos( θ) + sin(90 θ) cos θ = 1 cos θ 1 (6) 5.3) Determine the general solution of tan x sin x + cos x tan x = 0. (7) [21] Question 6: 6.1) Draw the graphs of f(x) = tan x + 1 amd g(x) = cos 2x for x [ 180 ; 180 ] on the same system of axes, provided on the diagram sheet. Clearly show all intercepts with the axes, turning points and asymptotes. (6) 39

45 2013 November CAPS Aligned PAPER 2 6.2) Write down the period of g. (1) 6.3) If h(x) = cos 2(x + 10 ), describe fully, in words, the transformation from g to h. (2) 6.4) For which values of x, where x > 0, will f (x)g(x) > 0? (4) [13] Question 7: the Great Pyramid at Giza in Egypt was built around BC. The pyramid has a square base (ABCD) with sides 232,6 metres long. The distance from each corner of the base to the apex (E) was originally 221,2 meters. 7.1) Calculate the size of the angle at the apex of a face of the pyramid (for example CÊB). (3) 7.2) Calculate the angle each face makes with the base (for example E ˆF G, where EF AB in AEB). (6) [9] 40

46 2013 November CAPS Aligned PAPER 2 Give reasons for your statements in QUESTIONS 8,9 and 10. Question 8: In the following diagram, O is the centre of the circle. BD is a diameter of the circle. GEH is a tangent to the circle at E. F and C are two points on the circle and F B, F E, BC, CE and BE are drawn. Ê 1 = 32 and Ê3 = ) Ê 2 (2) 8.2) E ˆBC (3) 8.3) ˆF (4) [9] 41

47 2013 November CAPS Aligned PAPER 2 Question 9: 9.1) In the following diagram, O is the centre of the circle and OB is perpendicular to the chord AC. Prove, using Euclidean geometry methods, the theorem that states AB = BC. (5) 9.2) In the diagram below, two circles intersect at K and Y. The larger circle passes through O, the centre of the smaller circle. T is a point on the smaller circle such that KT is a tangent to the larger circle. T Y produced meets the larger circle at W. W O produced meets KT at E. Let Ŵ1 = x ) Determine FOUR other angles, each equal to x. (8) 42

48 2013 November CAPS Aligned PAPER ) Prove that ˆT = 90 x. (3) 9.2.3) Prove that KE = ET. (3) 9.2.4) Prove that KE 2 = OE.W E (6) [25] Question 10: 10.1) In the following diagram, V RK has P on V R and T on V K such that P T RK. V T = 4 units, P R = 9 units, T K = 6 units and V P = 2x 10 units ) Calculate the value of x. (4) 43

49 2013 November CAPS Aligned PAPER ) Write down the value of Area of V P T Area of V RK (2) 10.2) In ADC, E is a point on AD and B is a point on AC such that EB DC. F is a point on AD such that F B EC. It is also given that AB = 2BC ) Determine the value of AF : F E (2) ) Calculate the length of ED if AF = 8cm (4) [12] 44

50 2013 February CAPS Aligned PAPER 1 Note: Extra Question refers to a question that has been added to the paper but exceeds the number of marks that you can expect in that section in the final exam. Question 1: 1.1) Solve for x: 1.1.1) (x 2 9)(2x + 1) = 0 (3) 1.1.2) x 2 + x 13 = 0 (Leave your answer correct to TWO decimal places.) (4) 1.1.3) 2.3 x = 81 3 x (4) 1.1.4) (x + 1)(4 x) > 0 (3) 1.2) 2 x + 2 x+2 = 5y ) Express 2 x in terms of y. (2) 1.2.2) Solve for x if y = 4? (Extra Question) (2) 1.2.3) Solve for x if y is the largest possible integer value for which 2 x + 2 x+2 = 5y + 20 will have solutions. (Extra Question) (3) 45

51 2013 February CAPS Aligned PAPER 1 1.3) The equation, x x = 3kx 2 + 2, has real roots ) Find the greatest value of k such that k Z. (4) 1.3.2) Find one rational value of k for which the above equation has rational roots. (2) Question 2: 2.1) Given the geometric series: p [27] 2.1.1) Determine the value of p. (3) 2.1.2) Calculate the sum of the first 8 terms of the series. (3) 2.1.3) Why does the sum to infinity for this series exist? (1) 2.1.4) Calculate S (3) 2.2) Consider the arithmetic sequence: -8;-2;4;10; ) Write down the next term of the sequence. (1) 2.2.2) If the n th term of the sequence is 148, determine the value of n. (3) 46

52 2013 February CAPS Aligned PAPER ) Calculate the smallest value of n for which the sum of the first n terms of the sequence will be greater than (Extra Question) (5) ) Calculate (3k + 5) (3) k=1 [22] Question 3: Consider the sequence: 3; 9; 27;... Jacob says that the fourth term of the sequence is 81. Vusi disagrees and says that the fourth term of the sequence is ) Explain why Jacob and Vusi could both be correct. (2) 3.2) Jacob and Vusi continue with their number patterns. Determine a formula for the n th term of: 3.2.1) Jacob s sequence (1) 3.2.2) Vusi s sequence (4) Question 4: The graph of f(x) = ( ) x 1 is sketched below. 3 [7] 47

53 2013 February CAPS Aligned PAPER 1 4.1) Write down the domain of f. (1) 4.2) Write down the equation of the asymptote of f. (1) 4.3) Write down the equation of f 1 in the form y =... (2) 4.4) Sketch the graph of f 1. Indicate the x-intercept and ONE other point. (3) 4.5) Write down the equation of the asymptote of f 1 (x + 2). (2) 4.6) Prove that: [f(x)] 2 [f( x)] 2 = f(2x) f( 2x) for all values of x. (3) [12] 48

54 2013 February CAPS Aligned PAPER 1 Question 5: Sketched below is the graph of g(x) = a x p + q. C(2; ( 6) is ) the point of intersection of the asymptotes of g. 5 B 2 ; 0 is the x-intercept of g. 5.1) Determine the equation for g in the form g(x) = a x p + q (4) 5.2) F is the reflection of B across C. Determine the coordinates of F. (2) [6] 49

55 2013 February CAPS Aligned PAPER 1 Question 6: S(1; 18) is the turning point of the graph of f(x) = ax 2 + bx + c. P and T are x-intercepts of f. The graph of g(x) = 2x + 8 has an x-intercept at T. R is a turning point of the intersection of f and g. 6.1) Calculate the coordinates of T. (2) 6.2) Determine the equation for f in the form f(x) = ax 2 + bx + c. Show ALL your working. (4) x-intercept Approach Turning Point Approach 6.3) If f(x) = 2x 2 + 4x + 16, calculate the coordinates of R. (4) 6.4) Use your graphs to solve x where: 6.4.1) f(x) g(x) (2) 50

56 2013 February CAPS Aligned PAPER ) 2x 2 + 4x 2 < 0 (4) [16] Question 7: 7.1) Raeesa invests R4 million into an account earning interest of 6% per annum, compounded annually. How much will her investment be worth at the end of 3 years? (3) 7.2) Joanne invests R4 million into an account earning interest of 6% per annum, compounded monthly ) She withdraws an allowance of R per month. The first withdrawal is exactly one month after she has deposited the R4 million. How many such withdrawals will Joanne be able to make? (6) 7.2.2) If Joanne withdraws R per month, how many withdrawals will she be able to make? (3) [12] Question 8: Jeffrey invests R700 per month into an account earning interest at a rate of 8% per annum, compounded monthly. His friend also invests R700 per month and earns interest compounded semi-annually (that is every six months) at r% per annum. Jeffrey and his friend s investments are worth the same at the end of 12 months. Calculate r. [3] Question 9: 9.1) use the definition of the derivative (first principles) to determine f (x) if f(x) = 2x 3 (5) 9.2) Determine dy dx if y = 2 x + 1 x 2 (4) 9.3) Calculate the values of a and b if f(x) = ax 2 + bx + 5 has a tangent at x = 1 which is defined by the equation y = 7x + 3. (6) [15] 51

57 2013 February CAPS Aligned PAPER 1 Question 10: Given: f(x) = x 3 x 2 + x ) Write down the coordinates of the y-intercept of f. (1) 10.2) Show that (2; 0) is the only x-intercept of f. (4) 10.3) Calculate the coordinates of the turning points of f. (6) 10.4) Sketch the graph of f. Show all intercepts with the axes and all turning points. (3) Question 11: A rectangular box is constructed in such a way that the length (l) of the base is three times as long as its width. The material used to construct the top and the bottom of the box costs R100 per square metre. The material used to construct the sides of the box costs R50 per square metre. The box must have a volume of 9m 3. Let the width of the box be x metres. [14] 11.1) Determine an expression for the hight (h) of the box in terms of x. (3) 52

58 2013 February CAPS Aligned PAPER ) Show that the cost to construct the box can be expressed as C = 1200 x + 600x2. (3) 11.3) Calculate the width of the box (that is the value of x) if the cost is to be a minimum. (4) [10] Question 12: 12.1) Given that P (A) = 0, 75 and P (B) = 0, 15, determine: ) P (A B), given that A and B are independent. (2) ) P (A B), given that A and B are mutually exclusive. (1) ) P (A B), given that A and B are independent. (Extra Question) (2) ) Whether A and B are complementary events. (Extra Question) (2) 12.2) There are 25 boys and 15 girls in the English class. Each lesson, two learners are randomly chosen to do an oral ) Represent this situation in a tree diagram showing all possible outcomes and the probabilities of each branch. It is not necessary to calculate the probability of each outcome. (4) 53

59 2013 February CAPS Aligned PAPER ) Calculate the probability that a boy and girl are chosen to do an oral in any particular lesson. (3) ) Calculate the probability that at least one of the learners chosen to do an oral in any particular lesson, is male. (Extra Question) (2) ) Are the events, picking a boy first and picking a girl second, independent or dependent? Justify you answer with a calculation. (Extra Question) (2) [18] Question 13: Julius, Jacob, Helen, Mamphela, Mangosuthu, Terror and Desmond sit down in a row of seven chairs to watch a movie together. Helen and Mamphela are both girls, the rest are boys. 13.1) How many ways can they sit next to each other if there are no restrictions on who can sit where? (2) 13.2) Julius and Jacob want to sit next to each other. Helen and Mamphela also want to sit next to each other. If they take up the first four seats, how many ways can everyone now sit in the row? (4) 13.3) Julius and Jacob have a fight. Helen and Mamphela also have a fight. Julius demands to sit in the first seat and Mamphela chooses to sit in the last seat. If Jacob can t sit next to Julius and Helen can t sit next to Mamphela, how many ways can every one now sit in the row? (Extra Question, quite tricky, don t worry if you can t do it) (4) [10] 54

60 2013 February CAPS Aligned PAPER 2 Question 1: The following table gives the average rand/dollar exchange rate and the average monthly oil price for the year ) Draw a scatter plot to represent the exchange rate (in R/$) versus the oil price (in $), on the diagram sheet provided. (3) 55

61 2013 February CAPS Aligned PAPER 2 1.2) Describe the relationship between the exchange rate (in R/$) and the oil price (in $). (2) 1.3) Determine the mean oil price. (2) 1.4) Determine the standard deviation of the oil price. (2) 1.5) Calculate the equation of the least squares regression line. (4) 1.6) Calculate the correlation coefficient. (2) 1.7) Comment on the strength of the relationship between the variables. (1) 1.8) Generally there is concern from the public when the oil price is higher than two standard deviations from the mean. In which month(s) would there have been concerns from the public? (2) [18] 56

62 2013 February CAPS Aligned PAPER 2 Question 2: The average percentage of 150 learners for all their subjects is summarised in the cumulative frequency table below. 2.1) Draw the ogive (cumulative frequency graph) representing the data, on the diagram sheet provided. (2) 57

63 2013 February CAPS Aligned PAPER 2 2.2) Use the ogive to approximate the following: 2.2.1) The number of learners who scored less than 85% (2) 2.2.2) The interquartile range (Show ALL calculations.) (3) [7] Question 3: In the following diagram, trapezium ABCD with AD BC is drawn. The coordinates of the vertices are A(1; 7); B(p; q); C( 2; 8) and D( 4; 3). BC intersects the x-axis at F. DĈB = α. 3.1) Calculate the gradient of AD. (2) 3.2) Determine the equation of BC in the form y = mx + c. (3) 58

64 2013 February CAPS Aligned PAPER 2 3.3) Determine the coordinates of point F. (2) 3.4) Show that α = 48, 37. (4) 3.5) Calculate the area of DCF. (6) [17] Question 4: Circles C 1 and C 2 in the figure below have the same centre M. P is a point on C 2. P M intersects C 1 at D. The tangent DB ot C 1 intersects C 2 at B. The equation of circle C 1 is given by x 2 +2x+y 2 +6y+2 = 0 and the equation of line P M is y = x 2. 59

65 2013 February CAPS Aligned PAPER 2 4.1) Determine the following: 4.1.1) The coordinates of centre M (3) 4.1.2) The radius of circle C 1 (1) 4.2) Determine the coordinates of D, the point where line P M and circle C 1 intersect. (5) 4.3) If it is given that DB = 4 2, determine MB, the radius of circle C 2. (3) 4.4) Write down the equation of C 2 in the form (x a) 2 + (y b) 2 = r 2 (2) 4.5) Is the point F (2 5; 0) inside circle C 2? Support your answer with calculations. (4) [18] Question 5: 5.1) If sin 61 = p, determine the following in terms of p: 5.1.1) sin 241 (2) 60

66 2013 February CAPS Aligned PAPER ) cos 61 (2) 5.1.3) cos 73 cos 15 + sin 73 sin 15 (3) 5.2) Show that cos(x + 45 ) cos(x 45 ) = 1 cos 2x (4) 2 5.3) 5.3.1) Prove the following identity: cos x + sin x cos x sin x = 2 tan 2x (6) cos x sin x cos x + sin x 5.3.2) Determine a value of x in the interval [0 ; 180 ] for which the identity is not valid. (2) 5.4) 5.4.1) Given: sin x = cos 2x 1. Show that 2 sin 2 x + sin x = 0. (1) 5.4.2) Determine the general solution of the equation: sin x = cos 2x 1. (6) [26] 61

67 2013 February CAPS Aligned PAPER 2 Question 6: The graph of f(x) = sin 2x for 180 x 90 is shown in the following sketch. 6.1) Write down the range of f. (2) 6.2) Determine the period of f ( ) 3 2 x. (2) 6.3) Draw the graph of g(x) = cos(x 30 ) for 180 x 90, on the same system of axes. Clearly label ALL x-intercepts and turning points. (4) 6.4) Describe the transformation that graph f has to undergo to form y = sin(2x + 60 ). (2) [10] 62

68 2013 February CAPS Aligned PAPER 2 Question 7: In the following diagram, ABC is a right-angled triangle. KC is the bisector of AĈB. AC = r units and BĈK = x. 7.1) Write down AB in terms of x and r. (2) 7.2) Give the size of A ˆKC in terms of x. (1) 7.3) If it is given that AK AB = 2, calculate the value of x. (8) 3 [11] 63

69 2013 February CAPS Aligned PAPER 2 Give reasons for your statements in QUESTIONS 8,9 and 10. Question 8: In the following diagram, O is the centre of the circle KT UV. P KR is a tangent to the circle at K. OÛV = 48 and K ˆT U = 120. Calculate, with reasons, the sizes of the following angles: 8.1) ˆV (2) 8.2) KÔU (2) 8.3) Û 2 (2) 8.4) ˆK1 (2) 64

70 2013 February CAPS Aligned PAPER 2 8.5) ˆK2 (2) [10] Question 9: In the figure, AGDE is a semicircle. AC is the tangent to the semicircle at A and EG produced intersects AC at B. AD intersects BE in F. AG = GD. Ê 1 = x. 9.1) Write down, with reasons, FOUR other angles each equal to x. (8) 9.2) Prove that BE.DE = AE.F E (7) 9.3) Prove that ˆB 1 = ˆD 1 (4) [19] 65

71 2013 February CAPS Aligned PAPER 2 Question 10: 10.1) If in LMN and F GH it is given that ˆL = ˆF and ˆM = Ĝ, prove the theorem that states LM F G = LN F H. (7) 10.2) In P QR, B lies on PR such that 2P B = BR. A lies on P Q such that P A : P Q = 3 : ) Write down the value of Area of P RA Area of QRA (2) ) Calculate the value of the ratio BD. Show all working to support BQ your answer. (5) [14] 66

72 2014 Exemplar PAPER 1 - Solutions 2014 Exemplar Paper 1 - Solutions 67

73 2014 Exemplar PAPER 1 - Solutions 68

74 2014 Exemplar PAPER 1 - Solutions 69

75 2014 Exemplar PAPER 2 - Solutions 2014 Exemplar Paper 2 - Solutions 70

76 2014 Exemplar PAPER 2 - Solutions 71

77 2014 Exemplar PAPER 2 - Solutions 72

78 2013 November CAPS Aligned PAPER 1 - Solutions 2013 November CAPS Aligned Paper 1 - Solutions 73

79 2013 November CAPS Aligned PAPER 1 - Solutions 74

80 2013 November CAPS Aligned PAPER 1 - Solutions 75

81 2013 November CAPS Aligned PAPER 2 - Solutions 2013 November CAPS Aligned Paper 2 - Solutions 76

82 2013 November CAPS Aligned PAPER 2 - Solutions 77

83 2013 February CAPS Aligned PAPER 1 - Solutions 2013 February CAPS Aligned Paper 1 - Solutions 78

84 2013 February CAPS Aligned PAPER 1 - Solutions 79

85 2013 February CAPS Aligned PAPER 1 - Solutions 80

86 2013 February CAPS Aligned PAPER 2 - Solutions 2013 February CAPS Aligned Paper 2 - Solutions 81

87 2013 February CAPS Aligned PAPER 2 - Solutions 82

88 2013 February CAPS Aligned PAPER 2 - Solutions 83

89 Mathematics/P1 NSC Grade 12 Exemplar Formula Sheet INFORMATION SHEET b ± b 2 4 ac x = 2a n A = P( 1+ ni) A = P( 1 ni) A = P( 1 i) A = P( 1+ i) n DBE/2014 T n a + ( n 1) d = S = [ 2a + ( n 1 d] n 2 n ) n n 1 a( r 1) T n = ar S n = x F = f n [( 1+ i) 1] i f ( x + h) f ( x) '( x) = lim h 0 h r 1 ; r 1 n x[1 (1 + i) ] P = i ( ) 2 ( ) 2 x1 + x2 y1 + y2 d = x2 x1 + y 2 y1 M ; 2 2 y = mx + c y y = m x ) ( x a) + ( y b) = r In ABC: sin a A area ABC 1 ( x1 b c = = a 2 = b 2 + c 2 2bc. cos A sin B sin C 1 = 2 ab. sin C 2 S a = ; 1 < r < 1 1 r y2 y1 m = m = tanθ x x ( α + β ) = sinα.cos β cosα. sin β sin( α β ) = sinα.cos β cosα. sin β sin + cos ( α + β ) = cosα.cos β sinα. sin β cos ( α β ) = cosα.cos β + sinα. sin β 2 2 cos α sin α 2 cos 2α = 1 2sin α sin 2α = 2sinα. cosα 2 2cos α 1 ( xi x) = 2 σ = i= 1 fx x n n( A) P( A) = P(A or B) = P(A) + P(B) P(A and B) n y ˆ = a + bx ( S ) b n n ( x x) 2 ( x x) 1 ( y y) = 2 Copyright reserved 84