CJC 2020 Prelims P1 Ans

1 2020 JC2 H2 PRELIMS Paper 1’s Suggested Solutions 1: (i) 23f( ) 35 21 3 3(3 5) 1 2 3 33 5 xx x x x += + = − + − = + + (ii) y = 1 x → 11 1 99y xx = = : scale by a factor of 1 9 parallel to the y-axis Or scale by a factor of 1 9 parallel to the x-axis → 1 59 3 y x =  + : translate by 5 3 in the negative x-direction → 1 59 3 y x =−  + : reflect about the x-axis → 21 53 9 3 y x = −  + : translate by 2 3 in the positive y-direction

2 2. (i) i 41 i 2e w π =+= i 8 i 4 1 i 842 2e * 2e 2e n nn n n z w π π ππ   −   +−   = = (ii) For * nz w to be real and negative, arg 2 , where * nz kkw ππ= ±∈  2 84 1 1284 6 16 Smallest 6, 22 n k n k nk n ππ ππ+=± += ± = ± =

3 Q3: Since all coefficients are real, 1iz= −− is also a root. A quadratic factor ( ) ( ) [ ][ ] ( ) ( ) 2 1i 1i 1i 1i 1i zz zz z = −−+ −−−     = +− ++ =+− 2 22zz=++ ( )( ) 32 22 5 2 22z z p zq z z zd+ + += + + + ( ) ( ) 32 32 2 32 25 244 22 2 4 42 2 z z pz q z z z dz dz d z dz dz d + + += + + + + + = ++ ++ + Comparing the coefficients of 2z , z and constant terms, 45 1dd+=⇒= 42 6dp p+= ⇔ = 2 2dq q= ⇔= Alternative: ( ) ( ) ( ) ( ) ( ) 32 2 1i 5 1i 1i 0 4 6i i 0 4 i 6 0 i0 pq pp q pq p −+ + −+ + −+ + = −−+ += − ++ −= + Comparing the real and imaginary parts, 40 pq−+= 6p= 4qp= − 2q= Last factor is ( )2zd+ = ( )21z+ . Therefore, the real root is 1 2− . All the other roots are 11 i, 2z= −± − .

4 Q4: (i) Horizontal asymptote (y = 0) and y-intercept 50, 2   (ii) Since every horizontal line cuts the graph at most once. Therefore, f is a 1-1 function and hence f has an inverse function. When x < 2, let y = 5 2 x− 5 2xy= − 1 5f () 2xx− = − , 1 2x> When x ≥ 2, let 1y x= 1x y= 1 1f () x x − = , 10 2x<≤ Hence, 1 1 1, ,0 2f () 15 ,, 22 xxxx xxx −  ∈ <≤=  ∈>−   (iii) Since the range of f -1 =  or ( ),−∞ ∞ ⊄ of Domain of g =[2, )∞ , gf -1 does not exist.

5 Q5: 2 32 3 81002 2 16200 2 8100 3 1 8100 3 A r rh h r r V rr r r rr ππ π ππ π π = + = ⇒= − = +−  = −+ 2d 8100 d V rr π= −+ At maximum V, d 0d V r = . 2 2 2 8100