Paper F Sec 3 IP Solutions for students

1 Paper F Sec 3 IP Solutions for students Qn Solutions 1(a) [4]   3 323 523 x x   = 2(2 3)(4 6 9) 523 x x x x     = 24 6 9 5xx   = 24 6 4xx 1(b) [2] 9 5 2 4 2 5   =    9 5 2 4 5 2   =  5 5 2  or 5 5 10 2(a) [4] 3 2 0xx   232 xx 2 2 3 0xx   1 or -3x 2(b) [5] To solve 322 7 2 1 0x x x    Let 322 7 2 1f(x)= x x x   . 1 1 7f( ) 1 1 02 4 4=      321 2 7 2 12x is a factor of f(x)= x x x       21 2 8 22f(x)= x+ x x  or    22 1 4 1x x x   2 1 0x or 2 4 1 0xx   2( 4) ( 4) 41 22x or x        1 4 12 22x or x    1 232x or x    3(a) [3] 25 5 x 3 0xx      2 6 8 0xx     4 2 0xx    24 x   3(b) [4] Equal roots Discriminant=0 2[ ( 2)] 4(4)( 2) 0cc     2 12 20 0cc    ( 2)( 10) 0cc    2c or 10c 4 [3] Let 32ax x bx be f(x). f(x) = 2()x ax x b f( 1) 3 2ab   ……….(1) f(2) 6 14ba   ……….(2) Sub (2) into (1), 1, 3ab 

2 2f ( ) ( 3)x x x x     f (3) 27 5(a) [4]   4 lg 1.5 lg xxy y   4 lg lg 1.5 xxy y   42 lg 1.5xy x y   33lg 1.5xy 3lg 1.5xy lg 0.5xy 0.510 xy 10y x 5(b) [4] 22log 7 log (11 2 ) 1qp    2 7log 111 2qp   72 11 2qp   22 4 7qp ……….(1) 3 9(27)pq  233 3 (3 )pq  23pq ……….(2) Sub (2) into (1): 22 4(2 3 ) 7qq   3 2q . Sub 3 2q into (2): 13 2p . 6 (a) [4] 22f( ) 2 6 7 2( 3 ) 7x x x x x      = 22 2 3 9 3 92( 3 ) 7 2 7 2 4 2 2x x x                  = 2 3 9 142 2 2 2x    = 2 352 22x  Turning point = 35,22  f(0) 7 & f(5) 27 5 f( )<272 x 6 (b) [3] Let 1g( ) 2 xx x   be y. 21xy y x    21xy x y    ( 1) 2 1x y y    21 1 yx y    1 21( ) , 1 1 xg x x x    7(i) [3] 2y ax bx c    9c 22 ( 1)( 3) ( 4 3)y ax bx c a x x a x x         Comparing constant term, 3a  Comparing coefficients of x, 12b 7(ii) [2] When 2x , y = -12+24-9 = 3.  D (2,3)

3 8(i) [3] 8(iia) [1] 1.5 3.5x 8(iib) [1] 1 2 or 3 4xx    9(i) [1] 500 350 400 700 500 850   9(ii) [2] 6.8500 350 400 5220 3.6700 500 850 77501.4               5220 kg of stainless steel, 7750 kg of aluminum. 9(iii) [2]    52202.2 2.5 308597750   $30,859 10(i) [2] Gradient of BD = 1 3 10(ii) [2] Midpoint of BD = 17 15,22  , Eqn AC: 3y x c  Gradient of AC = -3 When 17 15,22xy , 33c ,  Equation of AC is 3 33yx  10(iii) [3] When 0y , 0 3 33x   11x  (11,0)A Solve 2 – 2 yx and 3 33yx  gives 7, 12xy  (7,12)C 10(iv) [2] area of the kite, units2 = 7 4 11 13 71 12 6 0 9 122 = 1 (42 99 156 48 66 63) 602       Check BD = 90 units, AC = 160 units 11(i) [2] 2 2 2( 3) (y 4) 6x   

4 11(ii) [2] When 8x , 2 2 2(8 3) (y 4) 6     2(y 4) 11  4 11y 1 11tan BCA 5    1 11BCD 2 tan 1.171 5 rad     11(iii) [3] Area of sector , units2 = 211 116 2 tan 21.0825     Area of shaded sector = 2(6 ) 21.08 92.0  units2 or Area of sector , units2 = 211 116 (2 2 tan ) 92.025        units2 12(ai) 12(aii) [1] range of y is 24 y   12(b) [3]  sec 18 2z     1cos 18 2z   18 120 ,240z  138 ,258z Note: 18 18 342z    13 (i) 2ye ax b Y aX b 13 (ii) On graph paper 13 (iiia) 3.99a 7b 13 (iiib) When 2.5x , 2 6.25x   32ye  from the graph.  ln32 3.47y 14 [6] 2 2 2 87 50 2(87)(50)cos55 71.3AD AD m     2 2 2 26 84 2(26)(84)cos80 83.5AC AC m     2 2 2 ˆ104 2(AD)(AC)cosDACAD AC   ˆˆcosDAC 0.10388 DAC 84.0   Area of the lake, m2

5 = 11(87)(50)sin55 (26)(84)sin8022 1 ( 5078.9 6973.5)sin84.02   = 5816 15(i). Answer for (i) 1 f ( )2yx G1 - correct vertical stretch by factor 1 2 G1 - correct reflection about x-axis −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 14 16 18 20 x y 15(ii)Answer for (ii) f (2 ) 6yx G1 - correct horizontal stretch by factor 1 2 G1 - correct vertical translation of 6 units down −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 14 16 18 20 x y

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