GRADE 11 - Crystal Math

NATIONALSENIOR CERTIFICATEGRADE 11MATHEMATICS P2NOVEMBER 2016MARKS: 150TIME: 3 hoursThis question paper consists of 13 pages and a 22-page answer book.Copyright reservedPlease turn over

Mathematics/P22NSC – Grade 11DBE/November 2016INSTRUCTIONS AND INFORMATIONRead the following instructions carefully before answering the questions.1.This question paper consists of 12 questions.2.Answer ALL the questions in the SPECIAL ANSWER BOOK provided.3.Clearly show ALL calculations, diagrams, graphs et cetera that you used to determinethe answers.4.Answers only will NOT necessarily be awarded full marks.5.If necessary, round off answers to TWO decimal places, unless stated otherwise.6.Diagrams are NOT necessarily drawn to scale.7.You may use an approved scientificnon-graphical), unless stated otherwise.8.Write neatly and legibly.Copyright reservedcalculator(non-programmableandPlease turn over

Mathematics/P23NSC – Grade 11DBE/November 2016QUESTION 1The table below shows the number of cans of food collected by 9 classes during a charitydrive.58152025273136751.1Calculate the range of the data.(1)1.2Calculate the standard deviation of the data.(2)1.3Determine the median of the data.(1)1.4Determine the interquartile range of the data.(3)1.5Use the number line provided in the ANSWER BOOK to draw a box and whiskerdiagram for the data above.(3)1.6Describe the skewness of the data.(1)1.7Identify outliers, if any exist, for the above data.(1)[12]QUESTION 2The table below shows the time (in minutes) that 200 learners spent on their cellphones duringa school day.2.1TIME SPENT(IN MINUTES)FREQUENCY95 x 10515105 x 11527115 x 12543125 x 13552135 x 14528145 x 15521155 x 16510165 x 1754Complete the cumulative frequency column in the table provided in the ANSWERBOOK.(2)2.2Draw a cumulative frequency graph (ogive) of the data on the grid provided.(3)2.3Use the cumulative frequency graph to determine the value of the lower quartile.(2)2.4Determine, from the cumulative frequency graph, the number of learners who usedtheir cellphones for more than 140 minutes.Copyright reservedPlease turn over(2)[9]

Mathematics/P24NSC – Grade 11DBE/November 2016QUESTION 3In the diagram, A(6 ; – 2), B(2 ; 15) and C(– 4 ; 3) are the vertices of ABC.M is the midpoint of AB. N is a point on CA such that MN BC.yB(2 ; 15)// M C(–4 ; 3)//NxOA(6 ; –2)3.1Determine the coordinates of M, the midpoint of AB.(2)3.2Determine the gradient of line MN.(3)3.3Hence, or otherwise, determine the equation of line MN, in the form y mx c.(2)3.4Calculate, with reasons, the coordinates of point N.(4)3.5If ABCD (in that order) is a parallelogram, determine the coordinates of point D.(4)[15]Copyright reservedPlease turn over

Mathematics/P25NSC – Grade 11DBE/November 2016QUESTION 4In the diagram, R and A are the x- and y-intercepts respectively of the straight line AR.1The equation of AR is y x 4 . Another straight line cuts the y-axis at P(0 ; 2) and2passes through the points M(k ; 0) and N(3 ; 4). and are the angles of inclination of the lines MN and AR respectively.yAN(3 ; 4) P(0 ; 2) M(k ; 0) O xR4.1Given that M, P and N are collinear points, calculate the value of k.(3)4.2Determine the size of 𝜃, the obtuse angle between the two lines.(4)4.3Calculate the length of MR.(3)4.4Calculate the area of MNR.(3)[13]Copyright reservedPlease turn over

Mathematics/P26NSC – Grade 11DBE/November 2016QUESTION 55.1In the diagram below,P(–8 ; t)OP 17 units and reflex XÔP .is a point in the Cartesian plane such thaty Ox17P(– 8 ; t )5.25.35.1.1Calculate the value of t.(2)5.1.2Determine the value of each of the following WITHOUT using acalculator:(a)cos( )(2)(b)1 sin (2)If sin 17 a , WITHOUT using a calculator, express the following in termsof a :5.2.1tan 17 (3)5.2.2sin 107 (2)5.2.3cos 2 253 sin 2 557 (4)Simplify fully, WITHOUT the use of a calculator:cos( 225 ). sin 135 sin 330 tan 225 1 1 2(cos x 1)(cos x 1) tan x.cos 2 x5.4Prove the identity:5.5Determine the general solution for 2sin x. cos x cos x.Copyright reserved(6)(4)(6)[31]Please turn over

Mathematics/P27NSC – Grade 11DBE/November 2016QUESTION 6In the diagram the graphs ofthe interval 180 x 90 .f ( x) cos x andg ( x) sin( x b) are drawn .56.1Write down the value of b.(1)6.2Write down the period of g.(1)6.3Write down the value(s) off ( x) g ( x) 0 .xin the interval 180 x 90 for which(2)6.4For which values of x in the interval 180 x 90 is sin(90 x) g ( x) ?(3)6.5The graph of h is obtained by shifting f 3 units upwards. Determine the range of h.(2)[9]Copyright reservedPlease turn over

Mathematics/P28NSC – Grade 11DBE/November 2016QUESTION 77.1In the figure below, acute-angled ABC is drawn having C at the origin.BAC7.27.1.1Prove that c 2 a 2 b 2 2ab cos C.7.1.2Hence, deduce that1 cos C (6)(a b c)(a b c)2ab(4)BC 235 m and AB 90,52 m. It is alsoˆ D 48,88 .given that AD̂B 31,23 ; DÂB 109,16 and CBQuadrilateral ABCD is drawn withA109,16 90,52 mD31,23 48,88 B235 mCDetermine the length of:7.2.1BD(3)7.2.2CD(3)[16]Copyright reservedPlease turn over

Mathematics/P29NSC – Grade 11DBE/November 2016QUESTION 8The diagram below shows a water tank which is made up of a cylinder and cone having equalradii. The height of the tank is 1,8 m and the radius is 0,5 m. The angle between theperpendicular height, AB, and the slant height, AC, of the conical section is 35,5 .Volume of cone ATotal surface area of cone Volume of cylinder 35,5 B0,5 mTotal surface area of cylinder C1,8 m8.1Calculate the perpendicular height, AB, of the cone.8.2When the tank is full, an electric pump switches on and pumps the water from thetank into an irrigation system at a rate of 0,52 m2/h. The pump automatically1switches off when the tank isfull.4Calculate how long, in hours, the pump feeds water into the irrigation system.Copyright reserved(2)Please turn over(4)[6]

Mathematics/P210NSC – Grade 11DBE/November 2016Give reasons for your statements and calculations in QUESTIONS 9, 10, 11 and 12.QUESTION 99.1Complete the statement so that it is TRUE:The angle subtended by an arc at the centre of a circle is .9.2(2)O is the centre of circle TNSPR. PÔS 60 and PS NT.TNOR60 PSCalculate the size of:9.2.1PR̂S(2)9.2.2NŜT(2)[6]Copyright reservedPlease turn over

Mathematics/P211NSC – Grade 11DBE/November 2016QUESTION 10D, E, F, G and H are points on the circumference of the circle.Ĝ1 x 20 and Ĥ 2 x 10 . DE FG.EFx 20 1212G2x 10 DH10.1Determine the size of DÊG in terms of x .10.2ˆ G.Calculate the size of DHCopyright reserved(2)(4)[6]Please turn over

Mathematics/P212NSC – Grade 11DBE/November 2016QUESTION 11O is the centre of the circle PTR. N is a point on chord RP such that ON PR.RS and PS are tangents to the circle at R and P respectively.ˆ S 42,83 .RS 15 units; TS 9 units; RPP42,83 ONT9S15R11.1Calculate the size of NÔR.(5)11.2Calculate the length of the radius of the circle.(4)[9]Copyright reservedPlease turn over

Mathematics/P213NSC – Grade 11DBE/November 2016QUESTION 1212.1Use the diagram below to prove the theorem which states that EF̂G ED̂F.DEF12.2(5)GIn the diagram below, BOC is a diameter of the circle. AP is a tangent to the circleat A and AE EC.DPA32 13214E1 322123C13OBProve that:12.2.1BA OD(4)12.2.2AOCD is a cyclic quadrilateral(5)12.2.3DC is a tangent to the circle at C(4)[18]TOTAL:Copyright reserved150

QUESTION 11 O is the centre of the circle PTR. N is a point on chord RP such that ON PR. RS and PS are tangents to the circle at R and P respectively. RS 15 units; TS 9 units; RÖPS 3 . 11.1 Calculate the size of NOˆR. (5) 11.2 Calculate the length of the radius of the circle. (4) [9] S 9 O P 42,83 N R 15 T