IJC_H2_MATH_P2_(9758)_Solutions

1 (i) (ii) Since OACB is a parallelogram with 4 equal sides, it is a rhombus. (iii)     2 2 22 cos isin cos isin cos isin cos 2i cos sin sin cos cos 2 i sin sin 2 332cos cos 2isin cos22 22 332c os cos isin22 2 zz                      Alternative  2 3arg 22zz     2 22 c o s 2zz O M    2 332cos cos isin22 2zz              132, , 22pq k   2 (i) 1f : 3 2x x  , ,2xx Let f( )yx . x y O

13 2 12 3 12 3 y x x y x y       1 1 f ( ) 2 3x x    , ,3xx (ii)    f f D2 , R3 ,   Since ffRD , the composite function 2f exists. (iii)  2 2 f 1f3 2 13 132 2 13 1 2 3( 1) ( 2) 1 45 (1 ) 55 0 xx xx x x xx x xx xx xx x xx                 Using GC, f 1.38 (rej 1.38 D ) or 3.62xx        11 1 ff ff f f ff xx x x xx      Therefore 3.62x satisfies     1ff x x .

3 When x = 0,  10 0 o r 1 1 1 or 2 tt t t yy      Coordinates are (0,1) and 10, 2   . (ii)     22 22 22 dd 2 21 , dd 1 d2 1 d2 1 1 2 12 1 xy tttt t yt xt t t tt         When tangent is parallel to y-axis,    2222 112 10 1 0 2tt t t         Equation of tangent: 1 4x (iii) Area of the required region   0 1/4 1 21/2 1 221/2 121 1/ 2 1 1 d 1 21 d 1 21 d 11 ln 1 tan 51ln 2 ln tan44 2 81ln tan54 2 yx tt t t t tt tt                            x y O 11When , 42 When 0, 1 xt xt  

4 (a)(i) Area of unsown ploughed land  2 0.4 0.4 300 100 88 m    (a)(ii) n Beginning of week End of week 1 300  0.4 300 2 0.4 300 100    2 0.4 0.4 300 100 0.4 300 0.4 100     3  20.4 300 0.4 100 100        2 32 0.4 0.4 300 0.4 100 100 0.4 300 0.4 100 0.4 100    .. … … n …       1 21 0.4 300 0.4 100 ... 0.4 100 0.4 100 nn    Area of land unsown ploughed land at the end of nth week     1 12 0.4 1 0.4 0.4 300 100 10 . 4 2000.4 300 1 0.4 m3 n n nn          the value of k is 200 3 . (a)(iii) Method 1       1 1 1 40 2000.4 300 1 0.4 703 200 2000.4 300 (0.4) 0.4 7033 400 100.433 10.4 40 ln ln 0.4 4.02588 nn nn n n n n           Hence the number of complete weeks required is 5.

Method 2   12000.4 300 1 0.4 703 nn   Using GC, when n = 4, unsown ploughed land = 70.08 (> 70) when n = 5, unsown ploughed land = 68.032 (< 70) when n = 6, unsown ploughed land = 67.213 (< 70) Hence the number of complete weeks required is 5. (b)(i) n Beginning of week End of week 1 300 300 80 2 300 100 80  300 100 80 100  3 300 2 100 80 100   300 2 100 80 100 120   ..