Department of Mathematics

Transcription

1 WYNBERG BOYS HIGH SCHOOL Department of Mathematics GRADE 11 MATHEMATICS P2 21 NOVEMBER AM 12.00PM MARKS: 150 TIME: 3 hours EXAMINER: Hu MODERATOR: Lr This question paper consists of printed 16 pages, including this cover page.

2 11 Mathematics/P2 2 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 11 questions. 2. Answer ALL the questions in BLUE or BLACK ink only. Graphs must be drawn in pencil when required. 3. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. 4. In Questions 9 11, you are required to provide reasons for the statements you make. 5. Answers only will not necessarily be awarded full marks. 6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 7. If necessary, round off answers to TWO decimal places, unless stated otherwise. 8. Number the answers correctly according to the numbering system used in this question paper. 9. Write neatly and legibly.

3 11 Mathematics/P2 3

4 11 Mathematics/P2 4 QUESTION The following stem and leaf display represents the marks obtained by 50 students in a recent surprise test Determine the mean of the given data. (2) Calculate the standard deviation of the data. (2) Determine the number of students whose test scores are greater than ONE standard deviation from the mean. (1) The 5-number summary for the data is as follows: Min = 1, Max.= 49, Q1 = 16, Q2 = 30, Q3 = 39 Draw a box and whisker diagram to represent the data. (2) Comment on whether the data is positively or negatively skewed and provide a reason for your answer. (2)

5 11 Mathematics/P The table below shows the final Mathematics and Accounting results of a small group of Grade 11 students. Mathematics % Accounting % A scatter plot is given below. 100 Scatter plot of final Mathematics vs Accounting results for a group of Grade 11 students Accounting % Mathematics % Describe the correlation. (1) The line of best fit has the equation ŷ = a + bx. Determine the values of a and b. (2) A twelfth student wrote the Mathematics exam but not the Accounting exam. Calculate what his Accounting mark is predicted to be if he achieved 68% for Mathematics. (2) [14]

6 11 Mathematics/P2 6 QUESTION 2 A group of Grade 11 students was interviewed about their Instagram use. The number of Instagram posts, p, posted by each student per month was summarised in the histogram below Complete the cumulative frequency table provided in your ANSWER BOOK. (2) 2.2. Use the grid provided in your ANSWER BOOK to draw an ogive (cumulative frequency curve) to represent the data. (3) 2.3. Use the ogive to identify the median for the data. (1) 2.4. Use the ogive to estimate the percentage of the learners who post more than 11 Instagram posts per month. (1) [7]

7 11 Mathematics/P2 7 QUESTION 3 A, B, C and D are the vertices of a trapezium with AD BC. E is the midpoint of BC and is the angle of inclination of BC as shown. y D 45 A (1; 6) G C (12; 3) B (3; 0) E x F 3.1. Calculate the coordinates of E. (2) 3.2. Calculate the gradient of BC. (2) 3.3. Calculate the size of. (2) 3.4. Prove that AD AB. (3) 3.5. The line passing through A and F, with y-intercept G, makes an angle of 45 to AD. Determine the coordinates of G. (5) [14]

8 11 Mathematics/P2 8 QUESTION 4 In the diagram below, P( 3; 9), Q, O and S are the vertices of a parallelogram. OQ and OS are segments on the lines y = -x and y = 1 2 x respectively. P ( 3; 9) y Q S O 26,57 x 4.1. Determine the equation of the line passing through P and Q in the form y = mx +c. (3) 4.2. Hence show that Q has coordinates ( 6; 6). (4) 4.3. Calculate the length of OQ and leave your answer in simplified surd form. (2) 4.4. Given that OS = 20 units, calculate the length of QS. (5) [14]

9 11 Mathematics/P2 9 QUESTION Given that 13sinq = -5 and tanq > Represent this information using a neat sketch diagram, clearly indicating the correct positioning of. (2) Use your diagram to calculate the value of 3cosq. (2) 5.2. If cos46 = t, determine the following in terms of t: cos226 (2) sin44 (2) 5.3. Simplify: sin(q -180 ) tan(360 -q) sin(90 -q). (5) cos 2 (180 +q) Use Q to determine the value of sin(-135 )tan(315 )sin(45 ). (1) cos 2 (225 ) 5.4. Given the identity: sin 4 x +sin 2 x cos 2 x cos 2 x = tan 2 x Prove the identity. (4) For which value(s) of x is the identity undefined, where 0 x 180? (2) 5.5. Find the general solution to the equation 2sin xcosx -cos 2 x = 0 (5) [25]

10 11 Mathematics/P2 10 QUESTION 6 The diagram shows the graphs of f (x) = asinbx and g(x) = cos(x + p). The graphs intersect at turning points A ( 30 ; 1) and B, of g(x), as shown. y f (x) A g(x) C (90 ; t) x B 6.1. Write down the values of a, b and p. (3) 6.2. Write down the coordinates of B. (2) 6.3. Write down the range of f (x). (2) 6.4. C (90 ; t) is a point on g(x). Calculate the value of t. (1) 6.5. Describe the transformation(s) applied to f (x) to obtain g(x). (2) [10]

11 11 Mathematics/P2 11 QUESTION Use the triangle alongside to prove that R sin ˆP p sin ˆQ = q P Q (5) 7.2. Mr Hull recently purchased a plot of land (called an erf) with measurements as shown below. He has neighbouring plots on three sides with an edge along a road. neighbour C neighbour B 29,7 m 58 neighbour 34,6 m 42 D 28,3 m A 87 road Calculate the length of the wall to be built along the boundary with the adjoining plots. (3) The city council charges a property tax based on the erf size. In this case, they claim that the area is 759 m 2 (nearest square metre). Are they correct? (3)

12 11 Mathematics/P A is the highest point of a vertical tower AT. At point N on the tower, n metres below the top of the tower, a bird has made its nest. The angle of inclination from G to point A is α and the angle of inclination from G to point N is β Show that the height (H) of the nest above the ground is given by the formula H = ncosa sinb sin(a - b) (5) Calculate H if n = 10m, α = 68 and β = 40. (2) [18]

13 11 Mathematics/P2 13 QUESTION 8 A solid metallic hemisphere has a radius of 3 cm. It is made of metal A. To reduce its weight a conical hole is drilled into the hemisphere (as shown in the diagram) and it is completely filled with a lighter metal B. The conical hole has a radius of 1,5 cm and a depth of 25 mm Calculate the total surface area of the object. (3) 8.2. Calculate the ratio of the volume of metal A to that of metal B. (6) [9]

14 11 Mathematics/P2 14 In Questions 9 11, you are required to provide reasons for all of the statements you make. QUESTION Complete the following statement: A line drawn through the centre of a circle, perpendicular to a chord, will... (1) 9.2. In the diagram below, diameter AB is perpendicular to chord CD at E. O is the centre of the circle, AB = 20cm and EB = 3cm. C B E O D A Calculate the length of ED (Leave your answer in surd form). (3) Write down the length of CE. (2) Show that AC AE = (4) [9]

15 11 Mathematics/P2 15 QUESTION In the diagram below, O is the centre of the circle. Prove that AÔB = 2 AĈB. C O B A (5) In the diagram, M is the centre of the circle. A, B, C, K and T lie on the circle. AT produced and CK produced meet in N. Also NA = NC and A ˆBK= 38. N 2 4 K C M 38 B T 1 Calculate the size of: A A ˆMK (2) ˆT2 (2) Ĉ (2) ˆK4 (2) Show that NK = NT. (2) Prove that AMKN is a cyclic quadrilateral. (3) [18]

16 11 Mathematics/P2 16 QUESTION 11 In the diagram, HS is a tangent to the circle with centre O at S. NOUH is a straight line. NU MS and U ˆNS = x. H U O E S x N M Find 3 other angles equal to x. (6) Write down the size of NĤS in terms of x. (3) Hence prove that MU is a tangent to the circle passing through S, H and U. (3) [12]