VJC 2023 promo prac paper (A) solutions

2023 PROMO PRACTICE PAPER A Solutions Qn Solution Comments 1i 1n nnuSS −= − ( ) ( )ln 1 ln 1 1nn= + − −+ 11ln ln 1n nn + = = +  1ii Observe that as 1, 0 ln1 0. nnu n→∞ → ∴ → = The sequence decreases and converges to 0. 2 2 2 10 3 10 3 03 3 31 24 03 xx xx x x x ≥− −+ ≥−  −+ ≥− Since 2 3 31 024x − +> for all real x values, 1 03 30 x x ≥− −> 3x< If we conclude that the numerator has no real roots, (i.e. 2 40b ac−< , it is not sufficient to conclude .. 1 03 x ≥− ..). What else do we need? 210 3 xx−+ is an expression, not an equation. There are no roots for an expression. Is it correct to only state that 2 3 31 024x − +≥ ? Which real number will be a solution to 2 3 31 024x − += ? 3i ( ) 32f x x ax bx c=+ ++ ( ) ( ) ( ) ( ) 32 f1 1 1 1 8 7 (1) a bc abc = + + += ++= ( ) ( ) ( ) ( ) 32 f 2 2 2 2 12 4 2 4 (2) a bc a bc = + + += + += ( ) ( ) ( ) ( ) 32 f 3 3 3 3 25 9 3 2 (1) a bc a bc = + + += + += − By GC, a = – 1.5, b = 1.5 and c = 7 Recall factor / remainder theorem from O-level Additional Math. This is an assumed knowledge that you should have. 3ii ( ) ( ) ( ) ( ) ( ) 32 2 2 22 f 1.5 1.5 7 f ' 3 3 1.5 3 0.5 0.75 1.5 3 0.5 0.75 0 since 0.5 0 xx x x xxx x xx = − ++ = −+ =− −+ = − +> − ≥ Since ( )f' 0x > , graph of f is a strictly increasing cubic graph which will only cut the x-axis once. Hence, there is only 1 real root for ( )f0 x = . [In fact, since f(1) = 8 > 0 and f(–2) = –10 < 0, it suggests that there is a root between –2 and 1.] “Show that the gradient of the curve is always positive. Hence, explain ...” Your explanation should make mention to the fact that ( )f' 0x >

2023 PROMO PRACTICE PAPER A Solutions 4ai 1 44 4 144 32 4 x h hhx hrx = − −= = − =−=+ This is a show question. Clear working has to be shown. A clearly labelled diagram will aid in our explanation. Extra Note: There are no similar trapeziums in the figure. Having the same corresponding angles does not mean that the figures are similar. If the above statement is true, then all rectangles are similar to each other. Alternative 23 84 xxx= ⇒=+ 2 88 8 244 r h hhr =+ += = + Alternative 2 32 4 2 4 r h hr −− = = + 4aii ( ) 21Volume of water, π4 23V r rh= ++ . 2 2 32 2 1 π4 2 223 44 1 π 44 43 16 2 13π 123 16 2 d 13 π 3 12d 3 16 hhVh hhhh hh h Vh hh   = ++ + +     = +++ ++ = ++  = ++ . When h = 1, d dd d dd 13 d9 π 3 123 16 d 81πd 16 d d 16 d9 π V Vh t ht h t h t h t = × = ++ × = × = The rate of increase of the depth of water is 116 m min9π − . We can use the given formula for volume of frustum of right circular cone, with base radius 2 m, and radius r m. You can only substitute constant values before differentiation. ( ) ( ) 2 2 1 π4 23 d1 π4 2d3 V r rh V rrh = ++ ≠ ++ r i s a variable too . We have to use implicit differentiation and product rule if we wa