Mathematics Grade 11 P2 Gauteng November 2023 QP (Eng)

Downloaded from hlayiso.com PROVINCIAL EXAMINATION NOVEMBER 2023 GRADE 11 MATHEMATICS PAPER 2 TIME: 3 hours MARKS: 150 12 pages + 1 information sheet and 3 answer sheets P.T.O.

Downloaded from hlayiso.com MATHEMATICS 2 (PAPER 2) GRADE 11 INSTRUCTIONS AND INFORMATION 1. This question paper consists of 11 questions. 2. Answer ALL the questions. 3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. 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. Answer sheets for answering QUESTIONS 1.1, 1.2 and 9.1 are provided at the end of the question paper. Write you name in the spaces provided on the ANSWER SHEETS and submit them together with 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. P.T.O.

Downloaded from hlayiso.com MATHEMATICS 3 (PAPER 2) GRADE 11 QUESTION 1 A small tuck shop displayed a record of daily sales in rands, for the past two months (60 days) using the following histogram. 25 20 Freqency(in days) 15 10 5 0 300 400 300 500 400 600 500 700 600 800 700 900 800 1000 900 Daily sales (in rands) 1.1 Complete the following table. Use the table provided on ANSWER SHEET A. Class interval Frequency Cumulative frequency 300 < x  400 4 4 400 < x  500 500 < x  600 600 < x  700 700 < x  800 800 < x  900 (2) 1.2 Draw a cumulative frequency curve for the sales over the past two months. Use the graph sheet provided on ANSWER SHEET B. (3) 1.3 Use the graph in QUESTION 1.2 and determine the estimated median value for the daily sales. (2) 1.4 The tuck shop must make R475 in sales per day to break-even. On how many days did the tuck shop make a profit? (2) 1.5 On the first day of the following month, the tuck shop made R725 in sales. Does this day lie within the top 25% of sales from the previous two months? (2) [11] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 4 (PAPER 2) GRADE 11 QUESTION 2 Five data values are represented as follows: 2k ; k + 1 ; k + 2 ; k – 3 ; 2k – 2 2.1 If the mean of the data set is 15, show that k = 11. (3) 2.2 Calculate the standard deviation ()for this data, rounded-off to one decimal place. (2) 2.3 If t units are subtracted from each data value in the set, without further calculation, explain how the mean and standard deviation would be affected in terms of t. (2) [7] QUESTION 3 In the diagram below, RPQ is drawn, with P(0 ; 6), Q(4 ; –2). M(–2 ; 0) is the midpoint of RQ. y P(0 ; 6) R x O M(–2 ; 0) Q(4 ; –2) 3.1 Determine the gradient of the line MQ. (2) 3.2 Determine the equation of the line MP, in the form y = mx + c. (3) 3.3 Determine the coordinates of R. (3) 3.4 Calculate the length of PQ, in simplified surd form. (2) 3.5 Given that RPQT is a parallelogram, determine the coodinates of T if point T is in the third quadrant. (2) 3.6 Explain why RPQT is a rhombus. (2) [14] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 5 (PAPER 2) GRADE 11 QUESTION 4 In the diagram below, lines AC and BD intersect at B, where AC ⊥ BD. C and D lie on the y-axis, while A lies on the x-axis. The equation of AC is py – x – 5p = 0, while  is the angle of inclination for AC, with CD̂B = . y C B O x A 4.1 Determine the coordinates of C. (3) 4.2 1 If the gradient of AC is , show that p = 2. 2 (2) 4.3 Calculate the coordinates of B. (5) 4.4 Determine the size of . (2) 4.5 Hence, or otherwise, prove that ABOD is a cyclic quadrilateral. (3) 4.6 Determine the coordinates of the centre of the circle which passes through D, B and C. (2) [17] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 6 (PAPER 2) GRADE 11 QUESTION 5 5.1 If 3sin  = 2. and cos  < 0, determine with the aid of a diagram and without the use of a calculator, the value of: 5.1.1 3cos2  – 1 (4) 5.1.2 tan (– – 180) (3) 5.2 Given: t cos 15 = 4 Determine the following in terms of t, without the use of a calculator: 5.2.1 sin 15 (3) 5.2.2 sin 75 (2) 5.2.3 1 – tan2 15 (Give the answer as a single fraction.) (3) 5.3 Simplify the following to a single trigonometric function, without the use of a calculator. cos (90 − α) sin ( −α − 540 ) tan 225 + sin α.sin (180 + α) (5) 5.4 Given: 1 – cos = 2 sin2 5.4.1 Show that the equation can be written as: (2cos + 1)(cos – 1) = 0. (2) 5.4.2 Hence, determine the general solution of (2cos + 1)(cos – 1) = 0. (5) [27] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 7 (PAPER 2) GRADE 11 QUESTION 6 1 In the diagram below, the graphs of f (x) = tan x – 1 and g(x) = – cos 2x are drawn, 2 where x [–90 ; 180]. y f 2 1 g g -45º O 45º 90º 135º 180º -90º x -1 -2 f 6.1 Write down the period of g. (1) 6.2 Determine the range of g(x). (2) 6.3 Use the graphs to determine graphically the values of x where: 6.3.1 f (x)  0 (2) 6.3.2 f (x).g(x) > 0 (2) 6.3.3 2tan x + cos 2x = 2 (3) 6.4 sin x + cos x If h(x) = , describe the vertical translation of h from f. cos x (2) 6.5 Determine the maximum value of p(x) = 4g(x). (2) [14] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 8 (PAPER 2) GRADE 11 QUESTION 7 In the figure below, ABC is drawn where AB⊥ BC and AĈB = p, with CB = 3AB. DCB is drawn such that DĈB = 135, and BD = 121 m A C p B 135 121 m D 7.1 Determine the value of p, correct to 3 decimal places. (2) 7.2 If p = 18,4, determine the length of CD. (3) [5] QUESTION 8 The diagram below represents an open tank with a square base (side dimensions of x cm) and a height of h cm. The tank has a volume of 490 cm3. h cm V = Area of base x Height x cm x cm 8.1 Determine the height (h) of the tank in terms of x. (2) P.T.O.

Downloaded from hlayiso.com MATHEMATICS 9 (PAPER 2) GRADE 11 8.2 Show that A, the external surface area of the tank, is given by the formula: A = x2 + 1 960 cm2 x (2) 8.3 Given that the tank is 10 cm high, calculate the external surface area of the tank. (4) [8] QUESTION 9 9.1 In the diagram below, O is the centre of the circle passing through P, Q and R. Chords PQ and PR are drawn, with OQ and OR joined. Q O P R Use the diagram provided in ANSWER SHEET C to prove the theorem that states that the angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle, that is RÔQ = 2 P̂ (6) P.T.O.

Downloaded from hlayiso.com MATHEMATICS 10 (PAPER 2) GRADE 11 9.2 In the diagram below, DF is a diameter of the circle with centre O. Chord EG intersects DF at H such that DF ⊥ EG. Chords EF and GF are drawn. EĜF = 55. E 1 2 3 D 32 H 1 1 2 O 1 2 G F 9.2.1 Determine, giving reasons, the size of: (a) D̂ (2) (b) Ô 2 (2) (c) Ê 2 (2) (d) Ê 3 (3) 9.2.2 Determine the length of OH, if the diameter of the circle is 10 units and GE = 9,4 units. (4) [19] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 11 (PAPER 2) GRADE 11 QUESTION 10 In the diagram below, AT is a tangent to the circle at T. Chords BT, BV and VC are drawn. CB is extended to A, such that AC || TV. BC = CV and BV̂T = x. A T 2 1 B 3 2 1 C x 1 2 V 10.1 Determine, with reasons, 3 angles equal to x. (6) 10.2 If ATVC is a parallelogram, prove that AT = BT. (5) 10.3 Determine the size of x. (2) [13] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 12 (PAPER 2) GRADE 11 QUESTION 11 DE is a tangent to the larger circle at E. DH is a tangent to the smaller circle at H. Chord HK is extended to meet the larger circle at E. F and K are the points of intersections between the circles, with FK produced to D. GK is a chord of the smaller circle with FE a chord of the larger circle. HF, GH and GF are joined. D 3 1 2 1 E K 4 2 H 1 1 3 2 2 4 3 2 1 G 2 1 3 F 11.1 Complete the following: D̂ 3 = ... + ... (ext  ) (1) 11.2 Prove that DEFH is a cyclic quadrilateral. (4) 11.3 Prove that DF bisects HF̂E . (3) 11.4 If K̂1 = Ê1 , prove that GK is a tangent to the larger circle at K. (7) [15] TOTAL: 150 END

Downloaded from hlayiso.com INFORMATION SHEET − b  b 2 − 4ac x= 2a A = P(1 + ni) A = P(1 − ni) A = P(1 − i ) n A = P(1+ i ) n Tn = a + (n − 1)d n Sn = 2a + (n − 1)d  2 Tn = ar n −1 Sn = ( ) a rn − 1 ;r 1 S = a ; −1  r  1 r −1 1− r F=  x (1 + i )n − 1 P=  x 1 − (1 + i ) −n  i i f ( x + h) − f ( x) f ' ( x ) = lim h→ 0 h  x + x2 y1 + y 2  d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 M 1 ;   2 2  y − y1 y = mx + c y − y1 = m( x − x1 ) m= 2 m = tan x2 − x1 (x − a )2 + ( y − b)2 = r 2 a b c In ABC: = = a 2 = b2 + c 2 − 2bc.cosA sinA sinB sinC 1 area ΔABC = ab.sinC 2 sin (α + β ) = sin α.cos β + cosα.sin β sin (α − β ) = sin α.cos β − cosα.sin β cos(α + β ) = cosα.cos β − sin α.sin β cos(α − β ) = cosα.cos β + sin α.sin β cos 2 α − sin 2 α  cos2α = 1 − 2sin 2 α sin2 α = 2sin α.cos α  2cos α − 1 2 n  fx  ( xi − x ) 2 x= σ 2 = i =1 n n n (A) P(A) = P(A or B) = P(A) + P(B) – P(A and B) n (S) yˆ = a + bx b=  (x − x )( y − y ) ( x − x )2

Downloaded from hlayiso.com Name and Surname: Grade: ANSWER SHEET A QUESTION 1 1.1 Class interval Frequency Cumulative frequency 300 < x  400 4 4 400 < x  500 500 < x  600 600 < x  700 700 < x  800 800 < x  900 (2)

Downloaded from hlayiso.com Name and Surname: Grade: ANSWER SHEET B 1.2 70 60 50 Cumulative Frequency 40 30 20 10 0 0 100 200 300 400 500 600 700 800 900 1000 Daily Sales (3)

Downloaded from hlayiso.com Name and Surname: Grade: ANSWER SHEET C QUESTION 9 9.1 Q O P R (6)