2007 TJC H2 MA 9740 PRELIM P2 Sol

2007 TJC H2 MA 9740 PRELIM P2 Sol - Year 2007 Temasek...

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TJC/MA9740/P2/PRELIM2007/SOL 1 Year 2007 Temasek Junior College H2 Mathematics Preliminary Examination Paper 2 Solution Section A [40 marks] 1 [6 marks] The number of molecules of P that remains after t hours is n – x . The rate of formation of a molecule of R is proportional to the product of the number of molecules of P and number of molecules of Q at time t hours.   2 d d x k n x t    2 d d x kt nx    2 dd n x x k t  1 kt c  At time t = 0, x = 0 1 c n  2 11 1 n kt kt x n x n nkt 
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TJC/MA9740/P2/PRELIM2007/SOL 2 2 [9 marks] (i) w 7 = 1 w = (cos 0 + i sin 0) 1/7 = (2 /7) ik e k = 0, 1, 2, 3. (ii)     / 2 / 2 / 2 / 2 / 2 / 2 1 1 i i i i i i i i e e e e e e e e = 2 sin / 2 2 cos / 2 i = tan 2 i [AG] 2 (iii)   7 7 (1 ) 1 iz iz 7 1 1 1 iz iz    , 1 1 iz iz  (2 e k = 0, 1, 2, 3 = i e , where = 2 4 6 0, , , 7 7 7  1 + iz = i e iz i e iz (1 + i e ) = i e − 1 iz = 1 1 i i e e iz = i tan 2 z = tan 7 k where k = 0, 1, 2, 3
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TJC/MA9740/P2/PRELIM2007/SOL 3 3 [11 marks] 2 1 xt  , 1 y t     1 2 2 2 d1 12 d2 1 tt t t   and 2 d y t t 22 23 d d d 1 1 1 d d d y y x t t x t t t    At 2 1 1, Pt t    , Equation of tangent: 2 2 3 11 1 t y x t t t   Equation of normal: 3 2 2 1 1 1 t y x t t t    At 3 ,2 2 P , 2 2 yt t     Equation of tangent at P : 1 1 3 4 2 1 2 8 yx   4 3 4 When 0, 4 xy   . Therefore T has coordinates   0, 4 . Equation of normal at 3 2 P : 13 2 2 43    When 17 3 0, 2  . Therefore N has coordinates 17 3 ,0 2 . Since PT PN, area of triangle PTN = 1 2 PN PT  =       2 2 1 17 3 3 3 0 2 0 2 4 2 2 2 2   = 1 147 7 196 147 2 4 2 or 42.4 (to 3 s.f.)
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TJC/MA9740/P2/PRELIM2007/SOL 4 4 [14 marks] Given 1 2 4 2 : . 3 2 , 1 and 5 0 OA OB pq             r  (i) Equation of line AX : r = 42 13 0 p  where . X is the point of intersection between line AX and plane 1 , therefore 4 2 2 1 3 3 2 0 p              8 4 3 9 2       1 . Therefore 2 2 OX p    If OX makes an angle of 45 o with z -axis 2 20 1 cos45 8 p p    
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2007 TJC H2 MA 9740 PRELIM P2 Sol - Year 2007 Temasek...

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