TMJC_8865_2022_Prelim_Solutions_vetted

Page 1 of 11 2022 H1 MATH (8865/01) JC 2 PRELIMINARY EXAMINATION SOLUTIONS Qn Solution 1 System of Linear Equations Let x, y and z be the original selling price for a waffle, a scoop of ice cream and a cookie in dollars. 2 (0.6) (3 3) 20.2 1.2 3 23.2x y z x y z+ − + =  + + = --- (1) 0.6 (2 3) 2 12.6 0.6 2 2 15.6x y z x y z+ − + =  + + = --- (2) 3x(0.6)+(6y – 3(3)) + 5z = 36.8  1.8 6 5 45.8x y z+ + = --- (3) Using GC, 6, 5, 1x y z= = = the original selling price for a waffle is $6. Qn Solution 2 Techniques of Differentiation & Techniques of Integration (a) ( ) ( ) ( )( ) ( ) ( ) 2 2 3 3 d1 d 2 1 3 d1 13d2 1 2 1 3 32 3 13 x x xx x x − −   + =+  = − + =− + (b) 1 12 1 12 1 12 1 e d .45 1eln 4 5 + 15 2 1 ln 4 5 2e 5 x x x xx xC xC + + +  + − =− − + =− − +  , where C is an arbitrary constant

Page 2 of 11 Qn Solution 3 Equations and Inequalities (a) 2 2 2 2 (2 ) 3 5 2 (2 ) 3 5 0 2 (5 ) 5 0 x k x k x x k x x k x k x k + − − =− − + − + − + = + − + − = Since 22 (2 )y x k x k= + − − and 35yx=− − intersect at two distinct points, Discriminant 0 ( ) ( )( ) 2 2 2 (5 ) 4 2 (5 ) 0 25 10 40 8 0 2 15 0 5 3 0 3 or 5 kk k k k kk kk kk − − −  − + − +  − −  − +  −  Set of values =  : 3 or 5k k k −  (b)(i) (b)(ii) 21 ln( 1) 11 21 1 ln( 1)1 x xx x xx + − − − +  + −− Range of x values: 1 5.97x (3 s.f) x y x y

Page 3 of 11 Qn Solution 4 Applications of Differentiation (i) ( ) ( ) ( ) 2 2 2 ln 4 7 d 2 4 d 47 24 0 2, ln 3 47 y x x yx x xx x xy xx = − + −= −+ − =  = = −+ Coordinates of stationary point: ( )2,ln 3 (ii) (iii) When d40, ln 7, d7 yxy x= = =− Equation of tangent: ( )4ln 7 0 7yx− =− − 4 ln 77yx=− + (iv) 4 ln 77yx=− + When 0, ln 7xy== and 70, ln 74yx== ( )7 has coordinates ln 7,0 and has coordina tes 0,ln 74PQ   ( ) ( ) 2 21 7 7Area of triangle ln 7 ln 7 ln 7 units2 4 8OPQ ==  𝑦 (0, ln 7) (2, ln 3) 𝑂 𝑥 ( ) 2ln 4 7y x x= − +

Page 4 of 11 Qn Solution 5 Cross topical – curve sketching, differentiation, integration (i) ( )0.04 4 195 174e 46.727 A − =− = sales = $46727 (nearest dollar) (ii) Using GC, d 5.0539973d A t = Rate of change of weekly sales is $5054 per week (nearest dollar) (iii), (iv) (iv) ( ) ( ) 2 0.04 0 2 0.04 0 3 2 0.04 0 3 2 0.04 3 2 0.04 0.25 9 30 195 174e d 0.25 9 165 174e d 0.25 9 174 165 e3 2 0.04 0.25 9 174 174 165 e3 2 0.04 0.04 19 165 4350e 435012 2 k t k t k t k k t t t t t t t t t k k k k k k − − − − − − + + − − = − + − + −= + − − − = + − − − −  =− + − − +   The integral represents the increase in total sales in the first k weeks due to the advertising campaign. (v) For an eight-week advertising campaign, 8,k = increase in total sales ( ) ( ) ( ) ( )32 0.04 819 8 8 165 8 4350e 435012 2 − =− + − − + 116.585= thou