x 5 4 or invalid/ no solution SEPTEMBER 2015 MATHEMATICS P1 MEMORANDUM

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Britney Singleton
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1 SEPTEMBER 05 MATHEMATICS P MEMORANDUM GENERAL GUIDELINES FOR MARKING If a learner makes more than one attempt at answering a question and does not cancel any of them out, only the first attempt will be marked irrespective of which of the attempt(s) may be the correct answer. Consistent Accuracy(CA) marking regarding calculations will be followed in the following cases: - Sub-question to sub-question: When a certain variable is incorrectly calculated in one subquestion and needs to be substituted into another sub-question full marks can be awarded for the subsequent sub-questions provided the methods used are correct and the calculations are correct. Assuming values/answers in order to solve a problem is unacceptable. QUESTION/VRAAG Q# SUGGESTED ANSWER(S) DESCRIPTOR(S).. x(x + ) = 0 x = 0 OR/OF x = x = 0 x ().. x + x 0 x + x 0 graph/grafiek/ critical x(x + ) 0 pts x > 0 If x 0 only mark - 0 OR/OF Indien x 0 slegs punt - 0 x > 0. ( 3) ( 3) 4()( 7) x () x,77 OR/ OF x,7 If left in surd form only marks. Indien in wortelvorm, slegs punte..3. k 5k 4 0 ( k )( k 7) 0 k or k 7.3. x 5 or x 5 7 x 5 4 or invalid/ no solution x substitution into correct formula/ substitusie in korrekte formule simplify,77 -,7 multiplying factorising both solutions substitution identifying invalid value of x

2 .4 3 () x y x y () From (): x y.. Substituting in () yields: 3 y y y y 3y y 4y 3y(y ) 6y 3y 6y y 0 ( y )(3y ) 0 y or y 3 x or or 6 ; i.e. ; x 3 or.5 For real roots : 4 0k 0 0k 4 k equation substitution simplification standard form factors both y values both x-values (7) 4 0k 0 Answer () [4]

3 3 QUESTION/ VRAAG (n )() = 7 n = 68 n = 70 n = 35 The 35 th term is equal to 7/Die 35ste term is gelyk aan 7.. a = 3 en/and d = 35 S 40 = 40 [ + 39()] = 680 substitusie/ substitution terme/terms 8 vorming van reeks/generating series () () MR/GS: a = ; r = ; n = 9 S 9 = ( 9 ) = JA/YES all will fit into st tank.3 x +. x + 3. x + 4. x. +5. x a = x en/and d = x S n = 5 [x + 5. x ] S n = 5 [6. x ] S n = 5[8. x ] S n = 5[ 3. x ] S n = 5. x+3 ( 5488 ) = < a = ; r = ; n = 9 substitusie/substitution < JA/YES vorming van reeks/ generating series a = x en/and d = x S n = 5 [x + 5. x ] S n = 5. x+3 (5) [3]

4 4 QUESTION/ VRAAG ; 4; x; 30; 7; x 4; 30 x (Eerste verskil/first difference) x ; 34 x (Tweede verskil/second difference) x = 34 x 3x = 45 x = r = x < x < < x < < x < 3 Eerste verskil/first difference Tweede verskil/second difference x = 34 x x = 5 r = x < x < < x < a = 3 3 x ; r = r = 4 r = 4 8 S = 3 4 ( 8 ) = 3 a = 3 4 r = 8 S = 3 4 ( 8 ) S = 3 []

5 5 QUESTION/ VRAAG 4 4. R(-; 4) - 4 () 4. B(-4; 0) through symmetry AB = 4 units OR roots: (x+) = 4 OR -x - 4x = 0 x + = x(x+ 4) = 0 x = 0 or -4 AB = 4 units 4.3 m = - eqn: y = -x -4 4 units () m = - eqn () 4.4 x < OR x > 0 x < x > 0 () 4.5 h(x) = f( x) = ( x + ) + 4 sym- axis: x = h(x) x = Answer Only = FULL () marks 4.6 p(x) = f(x) = (x + ) 4 range: y 4 ; y R OR [ 4; ) p(x) y 4 Answer Only = FULL marks () []

6 QUESTION/ VRAAG 5 5. x = ; y = 5. y-int: y = 0; x-int: = x x = x = 0 6 x = y = () y = 0 = x x = asymptotes y x/y intercept shape x 5.4 x R; x x R x () 5.5. Graph shifts(translates) 3 units to the left 3 units to the left () 5.5. Graph shifts(translates) units down units down () [4]

7 7 QUESTION/ VRAAG A(0; ) Answer () 6.. f - : y = log 3 x y = log 3 x () < x endpoints notation () 6..4 y = 0 y = 0 () 6.. f(x) = x a OR f (x) = ax (8; ): = 8 a a = (; 8): 8 = a() a = a = eqn () f(x) = x x f (x)= f(x): x = y y = x 6.. f (x) = ax (; 8): 8 = a() a = eqn f (x)= x () 6..3 ( 3; ) each value () [0]

8 8 QUESTION/VRAAG 7 7. A = P( + i) n = 500 ( + 0,09 4k 4 ) 5 = 65 ( )4k OR,435 = (,05) 4k 4k = log,05,435 4k = 40,000 k = 0 years n eff i n i ( ) n eff 0,8 i ( ), , eff i 9,56% 7.. x i 7..3 P i n 60 0,8 x = 0,8 x R 696,7 0,8 700[( ) 0,8 4 balance / saldo ( ) 0, , ,74 OR/OF x i P i R3505,69 n ( i) = ( i) = 0,6094 i = % 700 0,8 R35 90,69 0, ] substitution into correct formula/substitusie in korrekte formule( A and/en P) 0,09 9 i = OR k = log,05,435 answer/antwoord substitution into correct formula/substitusie in korrekte formule 0,8 i = answer/antwoord substitution into correct formula/substitusie in korrekte formule P = n = 60 answer/antwoord substitute into correct formula/substitusie in korrekte formule n = 4 or/of n = -36 0,8 answer/antwoord substitute into correct formula/substitusie in korrekte formule answer/antwoord () [7]

9 9 QUESTION/VRAAG 8 8. f(x) = x 8.. f(x + h) = (x + h) f(x + h) f(x) = (x + h) ( x ) = 4xh h 4xh h h h(x h) h 4x + h f (x) = 4x f(x + h) f(x) h OR (x + h) ( x ) h 4xh h h 4x + h f (x) = 4x y = x x - x y = x 3 x dy dx = 3x + x D x [(3x ) ] OR chain rule: D x = (3x ). 3 = D x [9x x + 4] = 8x = 8x 8.3 y = x dy dx = x 3 for x > 0 x 3 > 0 x 3 < 0 gradient of tangent is < 0 if x > 0 note: notation error penalise mark candidates do NOT have to give their answers with positive exponents substitution (x + h) simplification subst into formula simplification answer substitution (x + h) subst into formula simplification simplification answer y = x 3 x 3x + x 3 simplify 8x x 3 x 3 > 0 x 3 < 0 (5) [4]

10 0 k 0 QUESTION 9 9. ( ; 0) (; 0) (0;-) (;-4) shape (5) 9. < x < endpoints notation () 9.3 Pt. of inflection: halfway between turning points(x-values) x = + x = f (x) > 0 x > 0 QUESTION/ VRAAG 0 x = + x = 0 () f (x) > 0 x > 0 () [] P A (0 k) Q k k B R k 0. QR = k Pythagoras SR = (0 k) Pythagoras Area PQRS = k[ (0 k)] = k(0 k) = 40k k 0. A = k + 40k da = 4k + 40 = 0 dk k = 0 D S 0 k C QR = k SR = (0 k) Area PQRS = k[ (0 k)] 4k + 40 = 0 0 [8]

11 QUESTION/ VRAAG.. P(A or B) = = P(A and B) = = 3 3 P(A or B) = P(A) + P(B) P(A and B) = = VW VW;VW P(A B) or P(A and B) = 3 3 P(A B) = P(A) + P(B) P(A B) 7 3 () VW BMW 7 3 BMW VW BMW P(both BMW) = = ,3 VW;BMW BMW;VW BMW;BMW Boomdiagram/Tree diagram (takke/branches) ,3.. P(BMW VW) = = , n(e) = 7.3!.6 n(s) = 5! P(E) = n(e) 7.3!.6 = = n(s) 5! 5.3. n(e) = 8! 7! n(s) = 5! P(E) = n(e) 8! 7! = n(s) 5! = ,5 () n(e) = 7.3!.6 n(s) = 5! 5 n(e) = 8! 7! P(E) = n(e) = 8!7! n(s) 5! 6435 [6] TOTAL: 50

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 Mathematics P LimpopoDoE/September 08 NATIONAL SENI CERTIFICATE GRADE MATHEMATICS P MEMANDUM SEPTEMBER 08 MARKS: 50 TIME: hours This memorandum consists of 6 pages. Mathematics P LimpopoDoE/September 08