NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P FEBRUARY/MARCH 014 MARKS: 150 TIME: 3 hours This questio paper cosists of 1 pages, 3 diagram sheets ad 1 iformatio sheet. Please tur over
Mathematics/P DBE/Feb. Mar. 014 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios. 1.. 3. 4. 5. 6. 7. 8. 9. 10. This questio paper cosists of 11 questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig the aswers. Aswers oly will ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad o-graphical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. Diagram sheets for QUESTION 1.4, QUESTION.1, QUESTION. ad QUESTION 6..(b) are attached at the ed of this questio paper. Write your cetre umber ad examiatio umber o these diagram sheets i the spaces provided ad isert the diagram sheets iside the back cover of your ANSWER BOOK. A iformatio sheet with formulae is icluded at the ed of this questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly. Please tur over
Mathematics/P 3 DBE/Feb. Mar. 014 QUESTION 1 The tuck shop at Great Future High School sells cas of soft driks. The Evirometal Club at the school decided to have a ca-collectio project for three weeks to make learers aware of the effects of litter o the eviromet. The data below shows the umber of cas collected o each school day of the three-week project. 58 83 85 89 94 97 98 100 105 109 11 113 114 10 145 1.1 Calculate the mea umber of cas collected over the three-week period. () 1. Calculate the stadard deviatio. () 1.3 Determie the lower ad upper quartiles of the data. () 1.4 Use the scaled lie o DIAGRAM SHEET 1 to draw a box ad whisker diagram to represet the data. (3) 1.5 O how may days did the umber of cas collected lie outside ONE stadard deviatio of the mea? (3) [1] Please tur over
Mathematics/P 4 DBE/Feb. Mar. 014 QUESTION The histogram below shows the time, i miutes, spet by customers while shoppig at Excellet Supermarket. 100 90 80 Histogram showig time spet shoppig 79 93 70 Number of customers 60 50 40 30 48 9 0 10 1 9 0 10 0 30 40 50 60 70 Time (i miutes).1 Complete the frequecy colum ad cumulative frequecy colum i the table o DIAGRAM SHEET 1. (3). Use the grid o DIAGRAM SHEET to draw the ogive of the above data. (4).3 Use the ogive to estimate the media time that customers spet at this supermarket. ().4 Commet o the skewess of the data. (1) [10] Please tur over
Mathematics/P 5 DBE/Feb. Mar. 014 QUESTION 3 The scatter plot below shows the age ad the time take for each of the first te swimmers of a swimmig club to complete a ope water swimmig evet. The time take is rouded to the earest half-miute. 30 Scatter plot showig age ad time take for each of the first te swimmers Time take (i miutes) 0 10 0 0 5 10 15 0 5 30 35 40 45 Age of swimmer 3.1 Write dow the coordiates of a outlier i the scatter plot. (1) 3. Which of the followig fuctios will best fit the data: liear, quadratic or expoetial? (1) 3.3 Give a explaatio for the tred observed i this set of data. () 3.4 If the two worst (logest) times are disregarded from the set of data, how will this affect the followig: 3.4.1 The stadard deviatio of the origial set of data (1) 3.4. The mea of the origial set of data (1) [6] Please tur over
Mathematics/P 6 DBE/Feb. Mar. 014 QUESTION 4 I the diagram below, A( 1 ; 3), B ad C are the vertices of a triagle. P(,5 ; 1) is the midpoit of AB. CA exteded cuts the y-axis at D. The equatio of CD is y = 3x + k. C ÂB = θ. α ad β are the agles that AB ad AC respectively make with the x-axis. y C B β O α P(,5 ; 1) x θ A( 1 ; 3) D 4.1 Determie the value of k. () 4. Determie the coordiates of B. () 4.3 Determie the gradiet of AB. () 4.4 Calculate the size of θ. (5) 4.5 Calculate the legth of AD. Leave your aswer i surd form. () 4.6 If AC = AD ad AB = 113, calculate the legth of CB. (5) [18] Please tur over
Mathematics/P 7 DBE/Feb. Mar. 014 QUESTION 5 I the diagram below, the equatio of the circle with cetre M is (x 8) + (y + 4) = 45. PT is a taget to this circle at T ad PT is parallel to OM. Aother circle, havig cetre O, touches the circle havig cetre M at N. y T O N x M P 5.1 Write dow the coordiates of M. (1) 5. Calculate the legth of OM. Leave your aswer i simplest surd form. () 5.3 Calculate the legth of ON. Leave your aswer i simplest surd form. (3) 5.4 Calculate the size of O Mˆ T. () 5.5 Determie the equatio of MT i the form y = mx + c. (5) 5.6 Calculate the coordiates of T. (6) [19] Please tur over
Mathematics/P 8 DBE/Feb. Mar. 014 QUESTION 6 6.1 The diagram below shows polygos A, B ad C. P(3 ; 5) is a vertex of polygo A. 5 y P(3 ; 5) B 4 3 A 1 0-7 -6-5 -4-3 - -1 1 3 4 5 6 7-1 x - -3-4 C -5 6.1.1 Fully describe the trasformatio from polygo A to polygo B. () 6.1. Write dow the rule that trasforms polygo B to polygo C. () 6.1.3 P / is the image of P whe polygo A is rotated about the origi through 180. Write dow the coordiates of P /. () 6. P(3 ; ), Q( 1 ; ), R( ; 1) ad S( 1 ; 0) are the vertices of quadrilateral PQRS. 6..1 PQRS is elarged through the origi by a scale factor of to obtai P / Q / R / S /. Write dow the coordiates of Q /. (1) 6.. P // Q // R // S // is the image whe P / Q / R / S / is reflected about the y-axis ad the traslated 3 uits to the right ad 1 uit upwards. (a) Write dow the sigle rule that trasforms PQRS to P // Q // R // S //. (3) (b) Use the grid o DIAGRAM SHEET 3 to draw P // Q // R // S //. Label the vertices. (5) 6..3 Solve for t i the equatio: Perimeter of PQRS = t perimeter of P // Q // R // S // () [17] Please tur over
Mathematics/P 9 DBE/Feb. Mar. 014 QUESTION 7 y A(8,4 ; 0,33) x B The STOP sig is a regular octago. Cosider A(8,4 ; 0,33) as a vertex o a STOP sig havig its cetre at the origi. If the STOP sig is rotated about the origi i a aticlockwise directio such that A coicides with poit B, determie the coordiates of B. [6] QUESTION 8 8.1 If 3 si A = ad cos A < 0, determie, WITHOUT usig a calculator, the value of: 5 8.1.1 si( A) () 8.1. ta A (3) 8. If cos 34 = p, WITHOUT usig a calculator, write dow the followig i terms of p: 8..1 cos 14 () 8.. cos 68 () 8..3 ta 56 (4) 8.3 WITHOUT usig a calculator, determie the value of the followig expressio: cos 350 si 40 cos 440 cos 40 (5) [18] Please tur over
Mathematics/P 10 DBE/Feb. Mar. 014 QUESTION 9 The graphs of f ( x) = cos( x 45 ) ad g( x) = si x are draw below for x [ 180 ;180 ]. The poit T is a x-itercept of f as idicated o the diagram. y f 180 O T 180 x g 9.1 Show that cos( x 45 ) = si x ca be writte as ta x = 0, 61. (4) 9. Solve the equatio: cos( x 45 ) = si x for x [ 180 ;180 ]. (3) 9.3 Write dow the coordiates of poit T. () 9.4 Write dow the iterval for which f ( x) g( x). () 9.5 Write dow the iterval for which both f ad g are strictly icreasig. (3) 9.6 The graph h is obtaied whe the graph f is shifted 45 to the right. Write dow the equatio of h i its simplest form. () [16] Please tur over
Mathematics/P 11 DBE/Feb. Mar. 014 QUESTION 10 I the diagram below, RS is the height of a vertical tower. T ad Q are two poits i the same horizotal plae as the foot S of the tower. From poit T the agle of elevatio to the top of the tower is 60. R Tˆ Q = θ, RQˆ T = 60 ad TQ = k metres. R T 60 θ S k 60 Q 10.1 Express TR i terms of θ ad k. (3) 10. Show that RS = ( 3k 3 cosθ + siθ ). (7) [10] Please tur over
Mathematics/P 1 DBE/Feb. Mar. 014 QUESTION 11 11.1 Cosider the fuctio: f ( x) = 3 si x 11.1.1 Determie the rage of f. (4) 11.1. For which value(s) x, x [ 180 ; 180 ], will f have a miimum value? (3) 11. 11..1 Show that 1 cosq = si Q. (1) 11.. Give: Pˆ + Qˆ + Rˆ = 180 (a) Show that si R = si(p + Q). (3) (b) Hece, show that si P + si Q + si R = 4 si P si Q si R. (7) [18] TOTAL: 150
Mathematics/P DBE/Feb. Mar. 014 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 1 QUESTION 1.4 50 60 70 80 90 100 110 10 130 140 150 QUESTION.1 Time (i miutes) 0 < x 10 10 < x 0 0 < x 30 30 < x 40 40 < x 50 50 < x 60 Frequecy Cumulative frequecy
Mathematics/P DBE/Feb. Mar. 014 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION. 300 Cumulative frequecy curve of time spet shoppig 70 40 10 Cumulative frequecy 180 150 10 90 60 30 0 0 10 0 30 40 50 60 70 Time (i miutes)
Mathematics/P DBE/Feb. Mar. 014 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 3 QUESTION 6..(b) 6 y 5 4 3 1-10 -9-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 9 10 - -3-4 -5-6 x
Mathematics/P DBE/Feb. Mar. 014 INFORMATION SHEET b ± b 4 ac x = a A = P( 1+ i) A = P( 1 i) A = P( 1 i) A = P( 1+ i) i= 1 1 = i= 1 ( + 1) i = 1 T = ar a( r 1) S = T a + ( 1) d = S = ( a + ( 1 d ) ; r 1 r 1 x[ ( i) ] F 1 1 + x[ 1 (1 + i) ] = P = i i f f ( x + h) f ( x) '( x) = lim h 0 h ( ) ( ) x1 + x y1 + y d = x x1 + y y1 M ; y = mx + c y y = m x ) ( x a) + ( y b) = r I ABC: si a A 1 ( x1 S ) a = ; 1 < r < 1 1 r y y1 m = m = taθ x x b c = = a b c 1 = + bc. cos A area ABC = ab. si C si B si C ( α + β ) = siα.cos β cosα. si β si( α β ) = siα.cos β cosα. si β si + cos ( α + β ) = cosα.cos β siα. si β cos ( α β ) = cosα.cos β + siα. si β cos α si α cos α = 1 si α si α = siα. cosα cos α 1 ( x ; y) ( x cosθ y siθ ; y cosθ + xsiθ ) ( xi x) = i= 1 fx x ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) y ˆ = a + bx ( S ) σ b = ( x x) ( x x) 1 ( y y) =