MATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12


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1 Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please Tur Over
2 Mathematics/P1 Jue 014 Commo Test INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 8 questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. Aswers oly will ot ecessarily be awarded full marks. A approved scietific calculator (oprogrammable ad ographical) 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. 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
3 Mathematics/P1 3 Jue 014 Commo Test QUESTION Solve for : (3) (3) (3) (5) 1. Simplify, without the use of a calculator: (3) 1.3 Solve for ad y where: ad (6) [3] QUESTION.1 Give the sequece: ; 5 ; 8 ;.1.1 If the patter cotiues, the write dow the et two terms. (1).1. Prove that oe of the terms of this sequece are perfect squares. (5). 1; 3; 5 are the first three terms of the first differeces of a quadratic sequece. The 7 th term of the quadratic sequece is Determie the 6 th ad 5 th terms of the quadratic sequece. (4).. Determie the th term of the quadratic sequece. (5).3 Prove that the sum to terms of a geometric sequece is give by: a( r 1) S ; r 1 (4) r 1.4 Calculate the value of if: k 1 k (5) [4] Please Tur Over
4 Mathematics/P1 4 Jue 014 Commo Test QUESTION 3 Give ad. 3.1 Draw graphs of ad o the same set of aes. Clearly show the itercepts with both aes, as well as the asymptote(s) where applicable. (8) 3. Write dow the value(s) of t if f ( ) t has: 3..1 equal roots. () 3.. oe root equal to 0. () 3.3 Write dow the equatio of the asymptote of h if () [14] QUESTION 4 y A(4 ; 6) a The diagram above shows the graph of f ( ) q. A(4 ; 6) is a poit o the graph. p 4.1 Determie the value(s) of a, p, ad q. (4) 4. Write dow the rage of g if g() = f(). () 4.3 If the graph of f is symmetrical with respect to the lie y = + c, determie the value of c. (3) [9] Please Tur Over
5 Mathematics/P1 5 Jue 014 Commo Test QUESTION Give: Determie. () 5. Give 5..1 Determie the iverse of i the form.. () 5.. Give a reaso why the iverse of is ot a fuctio. () 5..3 Write dow TWO ways i which you ca restrict the domai of so that its iverse is a fuctio. () 5..4 Hece, sketch the graphs of the fuctio. (4) 5..5 Determie the value(s) of for which h 1 ( ). () [14] QUESTION Determie the derivative of 3 f ( ) from first priciples. (5) 6. Calculate the derivative of the followig: (4) h( ) (4) [13] Please Tur Over
6 Mathematics/P1 6 Jue 014 Commo Test QUESTION 7 3 The graph below represets the fuctios f ad g with f ( ) a c ad g( ). A ad D( 1; 0) are the itercepts of f. The graphs of f ad g itersect at A ad C. D 7.1 Determie the coordiates of A. () 7. Show by calculatio that a = 1 ad c = 3. (5) 7.3 Determie the coordiates of B, a turig poit of f. (4) 7.4 Determie the coordiate of the poit of iflectio of f. () 7.5 Write dow the values of k for which f ( ) k will have oly ONE root. (3) 7.6 Write dow the values of for which f '( ) < 0. () [18] Please Tur Over
7 Mathematics/P1 7 Jue 014 Commo Test QUESTION 8 h A crate used o vegetable farms i the Pooo Area is i the form of a rectagular prism which is ope o top. It has a volume of 1 cubic metre. The legth ad the breadth of its base is, ad metres respectively. The height is h metres. The material used to maufacture the base of this cotaier costs R00 per square metre ad for the sides, R10 per square metre. 8.1 Epress h i terms of. () 8. Show that the cost, C, of the material is give by: C() = (3) 8.3 Calculate the value of for which the cost of the material will be a miimum ad hece the miimum cost of the material. (5) [10] Please Tur Over
8 Mathematics/P1 8 Jue 014 Commo Test INFORMATION SHEET: MATHEMATICS b b 4 ac a A P( 1 i) A P( 1 i) A P( 1 i) T a ( 1) d S a ( 1) d 1 T ar ar 1 S Please Tur Over ; r 1 r 1 1 i 1 [1 (1 i) ] F P i i f ( h) f ( ) f '( ) lim h 0 h ( ) ( ) d 1 y y1 M y ; y m c y y m ) a y b r 1 ( y S A P( 1 i) a ; 1 r 1 1 r y y1 m m ta a b c IABC: a b c 1 bc. cos A area ABC ab. si C si A si B sic si si.cos cos. si si si.cos cos. si cos cos.cos si. si cos cos.cos si. si cos si cos 1 si si si. cos cos 1 i f i1 ( A) P( A) P(A or B) = P(A) + P(B) P(A ad B) S yˆ a b b ( ) 1 ( y y)
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