TJC_JC2_H2_Maths_2012_Prelim_Paper_1_Solutions

TJC_JC2_H2_Maths_2012_Prelim_Paper_1_Solution 1 An ellipse E has equation 229 16 144xy . (i) Find d d y x in term of x and y. [1] (ii) Show that the point P with coordinates ( 4cos, 3sin ) lies on E. [1] (iii) Find the equation of the normal to the ellipse E at the poin t P, in the form of y mx c where m and c are single trigonometric expressions of . [3] Solution: (i) 229 16 144xy  18 32 0 dyxy dx  9 16 dy x dx y (ii) Since 2 2 2 29 16 9(4cos ) 16(3sin )xy    22144(cos sin ) 144   Hence, point P lies on C. (iii) Equation of normal at point P: 16(3sin )3sin ( 4cos )9(4cos )yx     3(cos ) 9cos sin 4(sin ) 16cos sinyx         74 tan sin33yx   

TJC_JC2_H2_Maths_2012_Prelim_Paper_1_Solution 2 Consider the equation     322 1 2i i 2 2i 0z z a b z       , where a and b are real. Given that 2 is a root of the equation, find the values of a and b. [3] Given also that 1i is another root, find the third root of the equation. [3] Solution: Since 2 is a root of the equation,         32 2 2 1 2i 2 i 2 2 2i 0 ab           16 4 8i 2 2 i 2 2i 0ab         10 6i 2 2 iab    by comparing coeff., 5, 3ab   Let z  be the third root.         322 1 2i 5 3i 2 2i 2 2 1 iz z z z z z             By comparing constant term, we have   2 2 1 i 2 2i       2 2 1 i 2 2i 1= 2       

TJC_JC2_H2_Maths_2012_Prelim_Paper_1_Solution 3 HIMHEYS' Confectionery recently created three types of chocolate: Organic white, Organic milk and Organic dark, available in 250g bars. Cocoa butter, an essential ingredient in chocolate bars, makes up 25%, 20% and 15% of the mass o f an Organic white, Organic milk and Organic dark chocolate bar respectively. To prepare for the official launch of the ir chocolate bars at an upcoming Food Expo, HIMHEYS' decides to manufacture a total of 300 bars for the event, with more than 70 bars of each type. The confectionary intends to use 14kg of cocoa butter in the production of t he above batch of chocolate bars. I f the number of milk chocolate bars is to be smaller than the number of white chocolat e bars, d etermine how many organic chocolate bars of each type can be produced. [6] Solution: Let D, M and W be the number of dark, white and milk chocolate bars (250g organic) respectively. 300D M W   -------- (1) 15 20 25250 250 250 14000100 100 100D M W                        ie. 3D + 4M +5W =1120 -------- (2) Solving (1) and (2): 80 , 220 2D W M W    Given 70, 70, 70D W M W    , 70 220 2 WW    73.3 75W  