Mathematics Grade 11 P2 Gauteng November 2022 QP (Eng)

Downloaded from hlayiso.com PROVINCIAL EXAMINATION NOVEMBER 2022 GRADE 11 MATHEMATICS (PAPER 2) TIME: 3 hours MARKS: 150 14 pages and 4 answer sheets P.T.O

Downloaded from hlayiso.com MATHEMATICS 2 (PAPER 2) GRADE 11 INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. 3. Answers only will not necessarily be awarded full marks. 4. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 5. Where necessary, round off answers to TWO decimal places, unless stated otherwise. 6. Answer sheets for answering QUESTION 1.1, QUESTION 2.2, QUESTION 9.1 and QUESTION 11.1 are provided at the end of the question paper. Write your name in the Commented [BK1]: And Question 9.1 spaces provided and submit them together with your ANSWER BOOK. 7. Diagrams are NOT necessarily drawn to scale. 8. Number the answers correctly according to the numbering system used in this question paper. 9. Write neatly and legibly. P.T.O.

Downloaded from hlayiso.com MATHEMATICS 3 (PAPER 2) GRADE 11 QUESTION 1 The data below is the annual report from the local municipalities regarding the spending of money (in thousands of rands) on electricity. 250 266 277 287 287 259 306 315 294 329 1.1 Draw a box-and-whisker diagram of the data above on the attached ANSWER SHEET A. (3) 1.2 Comment on the skewness of the data. (1) 1.3 Calculate the standard deviation. (2) 1.4 How many municipalities are within one standard deviation of the mean? (3) 1.5 Determine the semi-interquartile range of the municipalities. (3) [12] QUESTION 2 32 learners wrote a Mathematics test and the results are shown below. Marks Frequency Cumulative Frequency 20  x < 30 5 p 30  x < 40 5 10 40  x < 50 10 20 50  x < 60 7 q 60  x < 70 3 30 70  x < 80 1 31 80  x < 90 r 32 2.1 Find the values of p, q and r. (3) 2.2 Draw a cumulative frequency graph (OGIVE) of the data on the grid provided in the ANSWER SHEET B. (3) 2.3 Use the cumulative frequency graph to determine the value of the median. (2) 2.4 From the cumulative frequency graph, determine the number of learners who obtained a mark of more than 55. (2) [10] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 4 (PAPER 2) GRADE 11 QUESTION 3 In the diagram below, the points A(0 ; 7), B(10 ; 2), C(7 ; −4) and D (x ; y) form quadrilateral ABCD. AB || CD, with AC joined. The angles of inclination for AC and BC are θ and α respectively. y A(0 ; 7) B(10 ; 2) D(x ; y) O θ α x C(7 ; −4) Determine: 3.1 The length of AB (Leave your answer in simplified surd form.) (3) 3.2 The gradient of AC (2) 3.3 The size of θ, the angle of inclination of AC (3) 3.4 ˆD The magnitude (size) of BC (4) [12] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 5 (PAPER 2) GRADE 11 QUESTION 4 Consider ΔKLN drawn below. KL has the equation y = 5x + 9, while KN has the equation 5y + x – 19 = 0. y K N(4 ; 3) x O L(−2 ; −1) 4.1 Show that the coordinates of K are (–1 ; 4). (3) 4.2 Show that KL ⊥ KN. (3) 4.3 Hence, or otherwise, determine the area of ΔKNL. (4) 4.4 Determine the equation of the perpendicular bisector of LN. (5) 4.5 If L, N and P (7 ; y) are collinear, find y. (2) 4.6 Determine the coordinates of Q, if KLQN is a parallelogram. (2) 4.7 Explain, with geometric reasons, why KLQN is a square. (2) [21] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 6 (PAPER 2) GRADE 11 QUESTION 5 5.1 Determine the value of the following without the use of a calculator: tan(180 − x )cos(360 + x ) 2cos(90 + x ) (6) 5.2 Given the identity: sin − 2sin cos − sin = 2cos2 + cos − 1 cos + 1 5.2.1 Prove the above identity. (4) 5.2.2 For which values of θ will the identity be undefined? (5) 5.3 ˆ T =  with In the diagram below A is a point on the Cartesian plane such that AO OA = 4. P (k ; m) is a point such that OP = 6 and AO ⊥ OP. y • P(k ; m) A(b ; 1) • 6 4 α T• O x 5.3.1 Determine the value of b. (2) 5.3.2 Determine, without the use of a calculator, the value of: (a) tan  (2) (b) sin(– –180° + cos(–) (3) Commented [BK2]: Bracket missing 5.3.3 Without the use of a calculator, determine the values of k and m. (5) [27] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 7 (PAPER 2) GRADE 11 QUESTION 6 In the diagram below, the graphs of f(x) = a sin bx + 1 and g(x) = 2 cos (x + p) are shown for the interval −90° ≤ x ≤ 180°. y 2 1 f M(48,05° ; 0,41) x −90° −60° −30° 0° 30° 60° 90° 120° 150° 180° -1 g -2 6.1 Write down the amplitude of g. (1) 6.2 Determine the period of f. (1) 6.3 Determine the values of a, b and p. (3) 6.4 Determine the value(s) of x such that: 6.4.1 g(x) > 0 (1) 6.4.2 f(x)  g(x) (2) 6.4.3 f(x).g(x) < 0 (2) 6.5 Given h(x) = g(x + t), determine ALL values of t where h(x) = h(–x). (3) [13] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 8 (PAPER 2) GRADE 11 QUESTION 7 A camera person is stationed at the top of a vertical tower (TG) where TG = h. She films a cat starting at A, where the angle of depressionof T after A is θ. The camera follows the cat as it walks (in the same vertical plane) a distance of x metres horizontally to B, creating the angle 𝛽. T 𝜃 𝛽 h 1 2 G B x A 7.1 Determine B̂1 in terms of θ and β. (1) h sin  7.2 Show that x = sin( +  ) sin (5) 7.3 Determine the distance (x) the cat walked from A to B if the tower was 20 m high,  = 65° and  = 37° (2) [8] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 9 (PAPER 2) GRADE 11 QUESTION 8 Figure A below shows a rainwater gauge in the shape of a cone. The level of rainwater is indicated, with a diameter of 18 cm. Figure B shows the same cone after a steel ball has been dropped into the cone. The angle of the cone is P̂ = 28 , with the steel ball having a radius of r cm. Q is the point where the ball touches the side of the cone. FIGURE A FIGURE B 18cm r Q 28° P 8.1 Determine the volume of rainwater in FIGURE A. (2) 8.2 The volume of rainwater in FIGURE B is increased by 268,08 cm3 when the steel ball is placed inside the cone. Determine the length of PQ. (5) [7] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 10 (PAPER 2) GRADE 11 QUESTION 9 9.1 In the diagram, chord FD is drawn in a circle with centre O and FM = MD. O• D M F Use the diagram on the attached ANSWER SHEET C to complete the proof of the theorem which states that the line drawn from the centre of the circle to the midpoint of a chord will be perpendicular to that chord, that is, prove that OM ⊥ FD. (4) P.T.O.

Downloaded from hlayiso.com MATHEMATICS 11 (PAPER 2) GRADE 11 9.2 The points S, Q, R and P lie on the circumference of the circle with centre O. SR bisects chord PQ at M. PR and RQ are joined R̂ 1 = 62° P S O• M 2 62° 1 1 R 2 1 2 Q Calculate, with reasons, the values of the following angles: 9.2.1 M̂ 2 (2) 9.2.2 Q̂1 + Q̂2 (2) 9.2.3 S (2) [10] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 12 (PAPER 2) GRADE 11 QUESTION 10 The larger circle ADBF with centre O, intersects the smaller circle at F and B respectively. Points O and E also lie on the smaller circle, with FED and AOB joined. D̂1 = x A O 1 2 3 x 1 1 E D 1 2 3 2 F 2 2 3 1 B 10.1 Determine TWO other angles, each equal to x. (2) 10.2 Determine, with reasons, the following angles in terms of x: 10.2.1 Ô1 (2) 10.2.2 OFB (2) 10.2.3 D̂2 (2) 10.2.4 Ê 2 (4) 10.3 Prove, with reasons, that EB = ED. (4) [16] P.T.O.

Downloaded from hlayiso.com MATHEMATICS 13 (PAPER 2) GRADE 11 QUESTION 11 11.1 In the diagram, chords SR, RT and ST are drawn in a circle, with centre O. DSL is a tangent to the circle at S. L R S • O D T Use the diagram on the attached ANSWER SHEET D to prove the theorem which states that the angle between the tangent DSL and chord SR is equal to the angle in the alternate segment, that is, prove that LˆSR = ˆT . (6) P.T.O.

Downloaded from hlayiso.com MATHEMATICS 14 (PAPER 2) GRADE 11 11.2 TA is a common tangent to circles CBA and FDA at A. Chords AC and AD are drawn, with BCD a tangent to the circle at D. D 1 2 C 2 1 B A 12 4 3 F T 11.2.1 Provide the geometric reason that D̂1 = F̂ . (1) 11.2.2 Prove that Â1 = Â2 . (7) [14] TOTAL: 150 END

Downloaded from hlayiso.com MATHEMATICS 15 (PAPER 2) GRADE 11 Name and Surname: Grade: ANSWER SHEET A QUESTION 1 1.1 (3)

Downloaded from hlayiso.com MATHEMATICS 16 (PAPER 2) GRADE 11 Name and Surname: Grade: ANSWER SHEET B QUESTION 2 2.2 CUMULATIVE FREQUENCY GRAPH (OGIVE) 34 32 30 28 26 Cumulative frequency of learners 24 22 20 18 16 14 12 10 8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100 MARKS (3)

Downloaded from hlayiso.com MATHEMATICS 17 (PAPER 2) GRADE 11 Name and Surname: Grade: ANSWER SHEET C QUESTION 9.1 9.1 O• D 1 2 M F Construction: ………………………………. In OFM and ODM i. FM = MD [GIVEN] ii. OM is a common side iii. OF = OD […………………………….] [OFM  ODM] [………………….] …………………. [OFM  ODM] But M̂ 1 = M̂ 2 = 180° ['s on a straight line] Commented [BK3]: Not = but +  M̂ 1 = M̂ 2 = 90°  OM ⊥ FD (4)

Downloaded from hlayiso.com MATHEMATICS 18 (PAPER 2) GRADE 11 Name and Surname: Grade: ANSWER SHEET D QUESTION 11.1 11.1 L R S • O D T (6)