Mathematics Grade 11 Paper 1 KZN June 2024 Memo

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. com NATIONAL SENIOR CERTIFICATE GRADE 11 MATHEMATICS P1 COMMON TEST JUNE 2024 MARKS: 100 TIME: 2 hours This question paper consists of 6 pages including cover page Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Mathematics P1 2 Common Test June 2024 Downloaded f rom St anmorephysics. NSC-Grade 11 com INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 5 questions. 2. Answer ALL the questions. 3. Number the answers correctly according to the numbering system used in this question paper. 4. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in determining your answers. 5. Answers only will NOT necessarily be awarded full marks. 6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 7. If necessary, round off answers correct to TWO decimal places, unless stated otherwise. 8. Diagrams are NOT necessarily drawn to scale. 9. Write neatly and legibly. Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Mathematics P1 3 Common Test June 2024 Downloaded f rom St anmorephysics. NSC-Grade 11 com QUESTION 1 1.1 Solve for x: 1.1.1 x(1 − 2 x) = 0 (2) 1.1.2 2 x2 − 5x + 3 = 0 (3) 1.1.3 x(3x − 5) = 7 (correct to TWO decimal places) (4) 1.1.4 (1 − x)( x + 3)  −5 (4) x2 + 7 2 1 1.1.5 + =− (5) x − 2x − 3 x +1 2 x −3 1.2 Given that: x − 3 = 2 x 1.2.1 Solve for x. (4) 1.2.2 Hence, if 3 3t − 2 3 3t = 3 , determine the value of t . (3) 1.3 Solve simultaneously for x and y: 3 y = x + 1 and ( x − y )( 5 y − 3x ) = 0 (5) 33 5 1.4 The roots of the equation x 2 − 3x − k = 0 are x = . Determine the value of k. (4) 2 [34] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Mathematics P1 4 Common Test June 2024 Downloaded f rom St anmorephysics. NSC-Grade 11 com QUESTION 2 2.1 Simplify the following fully, without using a calculator: 24 x+1.9 x.62 x −1 2.1.1 (4) 26 x.27 x.3x 32021 − 32017 2.1.2 ( 60. 3 36048 ) (4) 2.2 Solve for x, without using a calculator: 2.2.1 m8 x−4 = 1 (3) 44 2.2.2 4.3x − 3x − 2 + 3x = (4) 3 2.2.3 x x = 22048 (3) [18] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Mathematics P1 5 Common Test June 2024 Downloaded f rom St anmorephysics. NSC-Grade 11 com QUESTION 3 x 1 The function g is defined as g ( x) =   − 4 2 3.1 Write down the equation of asymptote of g . (1) 3.2 Calculate the y - intercept of g . (2) 3.3 Calculate the x - intercept of g . (2) 3.4 Draw a neat sketch of g. Clearly show all the intercepts with the axes and the asymptote. (3) 3.5 Calculate the average gradient of g between x = 0 and y = 0 . (2) 3.6 Write down the equation of k if it is given that k ( x) = g ( x) + 4 . (1) 3.7 It is further given that h ( x) = 2 x+3 − 4 . Explain, in words, how graph g must be transformed to obtain graph h. (3) [14] QUESTION 4 a The function f ( x) = + 1 has an axis of symmetry with an equation of x = 3 − y . x+ p 4.1 Write down the range of f . (1) 4.2 Determine the equation of the vertical asymptote of f . (2) 4.3 If f (0) = 2 , calculate the value of a . (2) 4.4 Calculate the x-intercept of f . (3) 4.5 Draw a neat sketch of f . Clearly show all intercepts with the axes and asymptotes. (3) 4.6 Write down the values of x for which f ( x)  0 . (2) 4.7 The straight line y = −2 x + 5 passes through the point of intersection of the asymptotes and intersects f at P(1 ; 3) and Q. Write down the coordinates of Q. (2) [15] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Mathematics P1 6 Common Test June 2024 Downloaded f rom St anmorephysics. NSC-Grade 11 com QUESTION 5 The graphs of h( x) = ax 2 + bx + c and p( x) = 8 − 2 x are sketched below. The x-intercepts of h are (−3 ; 0) and (1 ; 0) and the y-intercept of h is (0 ; 6). M is the turning point of h . D and E are the 𝑥- and y-intercepts of p respectively. p y M E 6 (–3 ; 0) (1 ; 0) D O x h 5.1 Write down the coordinates of D. (1) 5.2 Show that a = −2 , b = −4 and c = 6 . (4) 5.3 Calculate the coordinates of M, the turning point of h . (2) 5.4 Write down the range of h . (1) 5.5 Determine the values of x for which h( x). p( x) < 0 . (3) 5.6 If h( x) = k , determine the value(s) of k for which: 5.6.1 roots are non-real. (1) 5.6.2 roots have the same sign. (2) 5.7 Calculate how graph p must be translated so that it becomes a tangent to graph h. (5) [19] TOTAL: 100 Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. com FINAL MATHEMATICS P1 COMMON TEST JUNE 2024 MARKING GUIDELINES NATIONAL SENIOR CERTIFICATE GRADE 11 MARKS: 100 Copyright Reserved Please turn over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 2 com Common Test June 2024 GRADE 11 Marking Guideline These marking guidelines consist of 10 pages. QUESTION 1 1 1 1.1.1 x = 0 or x = A x = 0 A x = 2 2 (2) 1.1.2 2x − 5x + 3 = 0 2 (2 x − 3)( x − 1) = 0 A correct factors 3 x= or x = 1 CA answer CA answer 2 (3) 1.1.3 x ( 3x − 5) = 7 A standard form 3x 2 − 5 x − 7 = 0 −(−5)  (−5) 2 − 4(3)(−7) CA correct substitution x= 2(3) x = −0,91 or x = 2,57 CA answer CA answer (4) 1.1.4 (1 − x)( x + 3)  −5 x + 3 − x 2 − 3x + 5  0 − x2 − 2x + 8  0 x2 + 2x − 8  0 A standard form ( x + 4)( x − 2)  0 OR + − + −4 2 −4 2 CA critical values x  −4 or x  2 CA answer CA answer (4) Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 3 com Common Test June 2024 GRADE 11 Marking Guideline 1.1.5 x2 + 7 2 1 + =− x − 2x − 3 x +1 2 x−3 x +7 2 2 1 + + =0 ( x − 3)( x + 1) x + 1 x − 3 A correct factorisation LCD : ( x − 3)( x + 1) x 2 + 7 + 2( x − 3) + 1( x + 1) =0 ( x − 3)( x + 1) CA correct LCD x2 + 7 + 2 x − 6 + x + 1 =0 ( x − 3)( x + 1) x 2 + 3x + 2 = 0 CA standard form ( x + 1)( x + 2) = 0 CA factors x  −1 or x = −2 CA answers with rejection (5) 1.2.1 x−3 = 2 x ( x − 3) = ( 2 x ) 2 2 x2 − 6x + 9 = 4 x A squaring both sides x 2 − 10 x + 9 = 0 CA standard form ( x − 9)( x − 1) = 0 CA factors x = 9 or x  1 CA answers with rejection (4) OR x−3 = 2 x x −2 x −3 = 0 ( x) −2 x −3 = 0 2 A standard form ( x − 3)( x + 1) = 0 CA factors x = 3 or x  −1 CA both equations x = 9 CA answers with rejection (4) 1.2.2 3 3t = 9 CA 3 3t = x from 1.2.1 ( 3t ) = 9 3 3 3 3t 729 M cubing both sides = 3 3 t = 243 CA answer (3) Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 4 com Common Test June 2024 GRADE 11 Marking Guideline 1.3 3 y = x + 1...............................(1) ( x − y )( 5 y − 3x ) = 0................(2) x = 3 y − 1...............................(3) A rewriting in terms of x ( (3 y − 1) − y )( 5 y − 3(3 y − 1) )  = 0 CA subst. of (3) into (2) ( 2 y − 1)( 3 − 4 y ) = 0 CA simplification 1 3 y= or y = CA both values of 𝑦 2 4 1 5 x= or x = CA both values of 𝑥 2 4 (5) OR OR 3 y = x + 1..........................................................(1) ( x − y )( 5 y − 3x ) = 0...............................................................(2) from (2) 3 x= y or y= x.................(3) 5 A solving for y ito x in (2) sub (3) into (1) 3  3x = x + 1 or 3 x  = x +1 5  CA subst. both into eq. (1) 9 2x = 1 or x = x +1 5 CA simplification 1 5 x= or x= 2 4 CA both values of 𝑥 sub x − values into (3) 1 3 y= or y= 2 4 CA both values of 𝑦 (5) Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 5 com Common Test June 2024 GRADE 11 Marking Guideline 1.4 33 5 x= 2 3  45 A rewriting ∆ as 45 x= 2  = 45 and x 2 − 3x − k = 0  = b 2 − 4ac  = (−3) 2 − 4(1)(− k ) A substitution into ∆  = 9 + 4k  9 + 4k = 45 CA equating two ∆s 4k 45 − 9 = 4 4 k =9 CA answer (4) OR OR (2 x − 3 + 3 5)(2 x − 3 − 3 5) = 0 A rewriting roots as (2 x − 3) − (3 5) = 0 2 2 factors of equation 4 x 2 − 12 x + 9 − 9(5) = 0 4 x 2 − 12 x − 36 = 0 A standard form x 2 − 3x − 9 = 0 CA  equation by 4 k = 9 CA answer (4) [34] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 6 com Common Test June 2024 GRADE 11 Marking Guideline QUESTION 2 2.1.1 24 x +1.9 x.62 x −1 26 x.27 x.3x 24 x +1.( 32 ) .( 3  2 ) x 2 x −1 A rewriting with prime bases = 2 .( 3 ) .3 6x 3 x x 24 x +1.32 x.32 x −1.22 x −1 CA separate prime bases = 26 x.33 x.3x = 24 x +1+ 2 x −1−6 x.32 x + 2 x −1−3 x− x = 20.3−1 CA simplifying: exponential laws 1 = 1 3 1 = CA answer 3 (4) 2.1.2 32021 − 32017 ( 60. 3 36048 ) 32017 (34 − 1) A common factor in numerator 32017 = A 32016 in denominator 60.32016 32017 − 2016 (80) A 80 in numerator = 60 =4 CA answer (4) 2.2.1 m8 x − 4 = 1 m8 x − 4 = m 0 A m0 8x − 4 = 0 CA equating exponents 1 x= CA answer 2 (3) OR OR m8 x − 4 = 1 m8 x =1 m4 m8 x = m4 A m8 x = m4 8x = 4 CA equating exponents 1 x= CA answer 2 (3) Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 7 com Common Test June 2024 GRADE 11 Marking Guideline 44 2.2.2 4.3x − 3x − 2 + 3x = 3 44 4.3x − 3x.3−2 + 3x = 3 44 A common factor of 3x 3x (4 − 3−2 + 1) = 3  44  44 44 3x   = A factor of  9  3 9 44 9 3x =  3 44 3x = 31 CA simplification x =1 CA answer (4) 2.2.3 x x = 22048 x x = 22(1024) A rewriting the exp. as 2(1024) x x = 42(512) x x = 16(512) A rewriting as 16(512) x x = 162(256) x x = 256256  x = 256 A answer (3) OR x x = 22048 x x = 21024.21024 A rewriting the RHS as 21024.21024 x x = (2.2)1024 x x = 41024 x x = 4512.4512 x x = 16256.16256 A rewriting as 16256.16256 x x = (16.16) 256 x x = 256256  x = 256 A answer (3) [18] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 8 com Common Test June 2024 GRADE 11 Marking Guideline QUESTION 3 3.1 y = −4 A answer (1) x 1 3.2 g ( x) =   − 4 2 0 1 g (0) =   − 4 A g (0) OR substitute x = 0 2 g (0) = 1 − 4 y = −3 CA value of 𝑦 (2) x 1 3.3 g ( x) =   − 4 2 x 1 0=  −4 A g ( x) = 0 2 4 = 2− x 22 = 2− x x = −2 CA value of 𝑥 (2) 3.4. A shape CA intercepts A asymptote (3) y2 − y1 3.5 m= x2 − x1 (0) − ( −3) CA substitution of x- and y- intercepts m= from 3.2 and 3.3 (−2) − (0) 3 m=− CA answer 2 (2) x 1 A answer 3.6 k ( x) =   2 (1) 3.7 reflection about y -axis A reflection about y−axis and then translation/shift of 3 units A translation of 3 units to the left A left (3) OR OR translation/shift of 3 units A translation of 3 units to the right A right and then reflection about y-axis A reflection about y−axis (3) [14] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 9 com Common Test June 2024 GRADE 11 Marking Guideline QUESTION 4 4.1 y , y 1 or y  ( −  ; 1)  (1 ; ) A answer (1) 4.2 x = 3− y x = 3 −1 A substitution x=2 A answer (2) a 4.3 f ( x) = +1 x+ p a f ( x) = +1 x−2 a 2= +1 A substitution of a point (0; 2) 0−2 a 2 −1 = −2 a = −2 CA answer (2) −2 4.4 f ( x) = +1 x−2 −2 0= +1 A f ( x) = 0 x−2 −2 −1 = x−2 −1( x − 2) = −2 CA simplification − x + 2 = −2 − x = −4 x=4 CA answer (3) 4.5 CA shape based on a in 4.3 CA vertical asymptote from 4.2 but horizontal asymptote of y = 1 must be accurate CA x-intercept from 4.4 but y- intercept must be accurate (given in 4.3) f (3) 4.6 2< x  4 OR x  ( 2;4] CA answer (2) 4.7 Q(3 ; − 1) A x−value A 𝑦 −value (2) [15] Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 10 com Common Test June 2024 GRADE 11 Marking Guideline QUESTION 5 5.1 D(4 ; 0) A answer (1) 5.2 h( x) = a ( x − x1 )( x − x2 ) A defining equation h( x) = a ( x − ( −3))( x − (1)) A substituting x-intercepts 6 = a (0 + 3)(0 − 1) A substituting ( 0 ; 6 ) 6 = a ( −3) 6 =a −3 −2 = a h( x) = −2( x + 3)( x − 1) h( x) = −2( x 2 + 2 x − 3) A simplification h( x) = −2 x 2 − 4 x + 6 (4) a = −2 , b = −4 , c = 6 5.3 h( x) = −2 x 2 − 4 x + 6 b x=− 2a −4 x= 2(−2) x = −1 A x−value h(−1) = −2(−1) 2 − 4(−1) + 6 y =8 CA 𝑦 −value M(−1 ; 8) (2) OR OR −3 + 1 xM = 2 xM = −1 A x−value h(−1) = −2(−1) − 4(−1) + 6 2 y =8 CA 𝑦 −value M(−1 ; 8) (2) 5.4 y , y 8 CA answer (using xM in 5.3) OR OR y  (− ; 8] CA answer (using xM in 5.3) (1) 5.5 x < −3 or 1 < x < 4 A x < − 3 ACA 1 < x < 4 NOTE: OR OR CA is only on 4 i.e. xD in 5.1 x ( −  ; − 3)  (1 ; 4 )  x ( −  ; − 3) ACA x (1 ; 4 ) (3) Copyright Reserved Please Turn Over

Downloaded from hlayiso.com Downloaded f rom St anmorephysics. Mathematics P1 11 com Common Test June 2024 GRADE 11 Marking Guideline 5.6.1 k 8 CA answer (using xM in 5.3) OR k  (8 ;  ) CA answer (using xM in 5.3) (1) 5.6.2 6<k 8 ACA answer NOTE: OR CA is only on 8 i.e. xM in 5.3 k (6; 8] ACA answer (2) 5.7 If graph p is to now become a tangent, then this implies that the NEW 𝑦 −intercept is unknown h( x ) = p ( x ) −2 x 2 − 4 x + 6 = −2 x + k A equating equations −2 x 2 − 2 x + 6 − k = 0 2 x2 + 2 x − 6 + k = 0 CA standard form =0 b 2 − 4ac = 0 (2) 2 − 4(2)(−6 + k ) = 0 CA equating ∆= 0 4 − 8(−6 + k ) = 0 4 + 48 − 8k = 0 NOTE: Correct answer only: 52 = 8k award only 1 mark 52 13 1 k= = = 6 = 6,5 CA value of k 8 2 2 Graph p must shift/be translated 1,5 units downwards CA answer (5) OR h( x ) = p ( x ) + k −2 x 2 − 4 x + 6 = −2 x + 8 + k A equating equations −2 x 2 − 2 x − 2 − k = 0 2 x2 + 2 x + 2 + k = 0 CA standard form =0 b 2 − 4ac = 0 (2) 2 − 4(2)(2 + k ) = 0 CA equating ∆= 0 4 − 8(k + 2) = 0 4 = 8(k + 2) NOTE: Correct answer only: 1 =k+2 award only 1 mark 2 3 1 k = − = −1 = −1,5 CA value of k 2 2 Graph p must shift/be translated 1,5 units downwards CA answer (5) [19] TOTAL:100 Copyright Reserved Please Turn Over