NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over

2 Mathematics/P DoE/Feb. March 009 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig the aswers. A approved scietific calculator (o-programmable ad o-graphical) may be used, uless stated otherwise. If ecessary, aswers should be rouded off to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. THREE diagram sheets for aswerig QUESTION 3.., QUESTION 8., QUESTION., QUESTION. ad QUESTION. are icluded at the ed of this questio paper. Write your eamiatio umber o these sheets i the spaces provided ad had them i together with your ANSWER BOOK. Number the aswers correctly accordig to the umberig system used i this questio paper. It is i your ow iterest to write legibly ad to preset the work eatly. Please tur over

3 Mathematics/P 3 DoE/Feb. March 009 QUESTION ABCD is a quadrilateral with vertices A( ; 6), B(3 ; 0), C(6 ; ) ad D(7 ; t ) i a Cartesia plae. AD BC. y D(7 ; t) A( ; 6) 0 B(3 ; 0) θ C(6 ; ). Calculate the gradiet of BC. (). Determie the equatio of AD i the form y = (3).3 Show that t = 8. ().4 Calculate the legths of AD, BC ad AB. (4).5 Show that AB is perpedicular to BC. (3).6 Calculate the area of the quadrilateral ABCD. (Simplify your aswer.) (4).7 Determie, the agle of icliatio of BC. (3) [] Please tur over

4 Mathematics/P 4 DoE/Feb. March 009 QUESTION A(0 ; 5) ad B( 8 ; ) are two poits o the circumferece of the circle cetre M, i a Cartesia plae. M lies o AB. DA is a taget to the circle at A. The coordiates of D are (3 ; ) ad the coordiates of C are ( ; ). Poits C ad D are joied. K is the poit (0 ; 7). CTD is a straight lie. y A(0 ; 5) M C( ; ) B( 8 ; ) 0 T D(3 ; ) K(0 ; 7 ). Show that the coordiates of M, the midpoit of AB, are ( 4 ; 3). (). Determie the equatio of the taget AD. (4).3 Determie the legth of AM. (3).4 Determie the equatio of circle cetre M i the form a + by + c + dy + e = 0 (4).5 Quadrilateral ACKD is oe of the followig: parallelogram; kite; rhombus; rectagle Which oe is it? Justify your aswer. (4) [6] Please tur over

5 Mathematics/P 5 DoE/Feb. March 009 QUESTION 3 The poit P( 3 ; ) lies i a Cartesia plae. 3. Determie the coordiates of the image of P if: 3.. P is reflected about the y-ais () 3.. P is rotated about the origi through 80º i a aticlockwise directio () 3. The vertices of a polygo PQRS are show i the grid below. The coordiates are P( 3 ; ), S( 4 ; ), R( ; 0) ad Q( ; ). Each of the poits of PQRS i the grid below is rotated about the origi i a clockwise directio through a agle 90º. 9 y P S Q R Write dow the coordiates of Q /, the image of Q. () 3.. Sketch ad label the vertices of the image P / Q / R / S / of PQRS o the grid provided o DIAGRAM SHEET. (4) / / / / 3..3 The polygo P Q R S is the elarged through the origi by a scale // // // // factor of to give the polygo P Q R S. Write dow the coordiates of // P the image of / P. () 3..4 State whether the trasformatio from PQRS to ot. Give a reaso for your aswer. // // // // P Q R S is rigid or () 3..5 Write dow the geeral trasformatio of a poit ( ; y) i PQRS to ( // ; y // ) after PQRS has udergoe the above two trasformatios, amely rotatio through 90º clockwise followed by a elargemet through the origi by a factor of. (3) 3..6 Calculate the ratio of area PQRS : Area P // Q // R // S //. () [9] Please tur over

6 Mathematics/P 6 DoE/Feb. March 009 QUESTION 4 4. Show that the coordiates of P /, the image of P( ; y) rotated about the origi through a agle of 35º, i the ati-clockwise directio, is give by y; y +. (4) y P( ; y) P / ( / ; y / ) 35º 0 4. M / is the image of M( ; 4) uder a rotatio about the origi through 35º, i the aticlockwise directio. Determie the coordiates of M /, usig the results i QUESTION 4.. () [6] QUESTION 5 Simplify each of the followig to a sigle trigoometric ratio: (Show ALL the calculatios.) ta(80 + )cos(360 ) si(80 )cos(90 + ) + cos(540 + )cos( ) cos si si cos (8) (5) [3] Please tur over

7 Mathematics/P 7 DoE/Feb. March 009 QUESTION 6 6. If si 3 = p, write dow the followig i terms of p. Do NOT use a calculator. 6.. cos 3 () 6.. cos 3 () 6..3 si 46 () 3 6. It is kow that 3 siα 5 = 0 ad ta β = where α [ 90 ; 70 ] ad 4 β [ 90 ; 70 ]. Determie, without usig a calculator, the values of the followig: 6.. cos α (3) 6.. cos( α + β ) (5) 6.3 Solve for [0 ; 360 ] if cos = 0, 435. (3) [7] Please tur over

8 Mathematics/P 8 DoE/Feb. March 009 QUESTION 7 Thadi is stadig at poit P o the horizotal groud ad observes two poles, AC ad BD, of differet heights. P, C ad D are i the same horizotal plae. From P the agles of icliatio to the top of the poles A ad B are 3 ad 8 respectively. Thadi is 8 m from the base of pole AC. The height of pole BD is 7 m. B A 7 m D C 8 m 3 4 P 8 Calculate, correct to TWO decimal places: 7. The distace from Thadi to the top of pole BD () 7. The distace from Thadi to the top of pole AC () 7.3 The distace betwee the tops of the poles, that is the legth of AB, if A P ˆB = 4 (4) [8] QUESTION 8 Cosider the fuctios defied by f() = si ad g() = ta for [ 90 ; 80 ]. 8. Sketch the graphs of f ad g o the same system of aes o DIAGRAM SHEET. (6) 8. Calculate the -coordiates of the poits of itersectio of f ad g. (0) 8.3 Determie the values of for which g() > f(). (3) [9] Please tur over

9 Mathematics/P 9 DoE/Feb. March 009 QUESTION 9 Determie the miimum ad maimum values of the followig: f ( ) = 3si + 4cos [4] QUESTION 0 The data below shows the eergy levels, i kilocalories per 00 g, of 0 differet sack foods Calculate the mea eergy level of these sack foods. () 0. Calculate the stadard deviatio. (4) 0.3 The eergy levels, i kilocalories per 00 g, of 0 differet breakfast cereals had a mea of 545,7 kilocalories ad a stadard deviatio of 8 kilocalories. Which of the two types of food show greater variatio i eergy levels? What do you coclude? () [8] Please tur over

10 Mathematics/P 0 DoE/Feb. March 009 QUESTION The heights, h, of the learers at Nkosi High School i Grades 0, ad were recorded as follows: HEIGHT (IN CM) FREQUENCY 8 h < h < h < h < h < h < 7 7 h < 8 4. Set up a cumulative frequecy table for the data o DIAGRAM SHEET. (). Draw a ogive for the data o the grid provided o DIAGRAM SHEET. (3).3 Use the ogive, or otherwise, to determie the lower quartile, media ad upper quartile. (3).4 If the miimum height was 9 cm ad the maimum height was 78 cm, draw a bo ad whisker diagram for the data. (3).5 Commet o the distributio of the heights of the learers. ().6 Approimately how may learers are betwee 38 cm ad 58 cm tall? () [3] QUESTION A motor compay did research o how the speed of a car affects the fuel cosumptio of the vehicle. The followig data was obtaied: Speed i km/h Fuel cosumptio i /00 km,5 0 8,4 9, 7,8 8,9 8,8 8,6 0,. Represet the data as a scatter plot o DIAGRAM SHEET 3. (3). Suggest whether a liear, quadratic or epoetial fuctio would best fit the data. ().3 What advice ca the compay give about the drivig speed i order to keep the cost of fuel to a miimum? () [6] TOTAL: 50

11 Mathematics/P DoE/Feb. March 009 b ± = b 4 ac a A = P( + i) A = P( i) INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE A = P( i) A = P( + i) i= i= = ar F = f i ( r ) a = r [( + i) ] i f ( + h) f ( ) '( ) = lim h 0 h i= ; r i= ( + ) i = [ ( + i) ] P = i a r ( a + ( i ) d ) = ( a + ( ) d ) i= i ar = ; < r < d = ( ) ( ) + y y M + y + y ; y = m + c y y = m ) ( a) + ( y b) = r ( y y m = m = taθ I ABC: si a A b c = = a b c = + bc. cos A area ABC = ab. si C si B si C ( α + β ) = siα.cosβ cosα. si β si( α β ) = siα.cosβ cosα. si β si + cos ( α + β ) = cosα.cosβ siα. si β cos ( α β ) = cosα.cos β + siα. si β cos α si α cosα = si α si α = siα. cosα cos α ( i ) = σ = i= f ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) ( S) y ˆ = a + b b ( ) ( y y) = ( )

12 Mathematics/P DoE/Feb. March 009 EXAMINATION NUMBER DIAGRAM SHEET QUESTION y 3 P Q S R O QUESTION 8. y

13 Mathematics/P DoE/March 009 EXAMINATION NUMBER DIAGRAM SHEET QUESTION. HEIGHT (IN CM) 8 h < 7 7 h < h < h < h < h < 7 7 h < 8 FREQUENCY CUMULATIVE FREQUENCY QUESTION Cumulative frequecy Height (i cm)

14 Mathematics/P DoE/March 009 EXAMINATION NUMBER DIAGRAM SHEET 3 QUESTION. Scatter plot of speed vs fuel cosumptio Fuel cosumptio (/00 km) Speed (km/h)

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 008 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, diagram sheets ad a formula sheet. Please tur over Mathematics/P DoE/Eemplar 008