MI 9758 2023 Prelim P2 Solution

1 © Millennia Institute 9758/02/PU3/Prelim/23 Solutions PU3 MATHEMATICS Paper 9758/02 Section A: Pure Mathematics Qn Solution 1(i) [2] ( ) ( ) ( ) ( ) 2 2 2 3 2 32 2 23 3 3 2 3 33 (1) dd 2dd dd 2 (2)dd dSubst (1) and (2) into 2 ,d d22 d d22 d d d d 1 (shown)d y vx yv vx xxx yv vx xxx yx x xy yx vx vx x x x vx vxx vvx x x vx vxx vx x vxx v vx = −−−−−− = + = + −−−−−− =++  += ++ + = ++ = + = + 1(ii) [3] Method 1 2 2 2 1 2 d 1d 1 d 1 d1 ln 1 1e 1e 1 e where e 1e Subst 1 and 1, 11e 1 2 e 21e e 1 2e xc xc xc x x x v vx vxv v xc v v vA A y Ax yx A A y x y x + + − = + =+ +=+ += += ± += = ± += = = += = +=  += ⌠⌡ ∫

2 © Millennia Institute 9758/02/PU3/Prelim/23 Solutions Qn Solution Method 2 2 2 2 d 1d 1 d 1 d1 ln 1 ln 1 Subst 1 and 1, 1ln 1 11 ln 2 1 ln 1 ln 2 1 v vx vxv v xc y xcx yx c c y xx = + =+ +=+ += + = = += + = − += +− ⌠⌡ ∫ Qn Solution 2(a) [5] ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 4 2 22 f sec f 2sec sec tan 2 f 2 sec 2sec s s ta ec tan 2sec 4 n tan s ec se e c c tan x x x xx x x xx x xx x xx x xx = ′ = = ′′ = + = + When 0x= , ( ) ( ) ( ) f0 1 f0 f0 2 x = ′ = ′′ = ( ) 2 2 2f 1 0 ... 2! 1 ... x xx x ∴ = ++ + = ++

3 © Millennia Institute 9758/02/PU3/Prelim/23 Solutions 2(b) [3] ( ) ( )( ) ( )( )( ) 42 22 451 1 4 ...2! 1 4 10 ... ax ax ax ax a x − −−+ =+− + + = −+ + Since the coefficients of the x and 2x terms in the expansion are equal, ( ) 2 2 4 10 10 4 0 25 2 0 aa aa aa −= += += 20 (rejected since 0) or 5a aa= ≠= − Qn Solution 3(a) [3] Method 1 (most used this) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 234 1 2 2 2 ... 1ln 1 ln 1 ln 1 ln1 ln 3 ln 2 ln 4 ln 3 ln 5 ln 3 ln 1 ln 2 ln ln 1 ln 1 ln1 ln 2 ln ln 1 ln 2 ln 1 N NN N n n N n N n S uuu u u u n n nn NN NN NN NN NN − = = = =++ + + = −=  + = −− + = − +− +− + −− − + −− + −− + =+− − + = −+ ∑ ∑ ∑  Method 2 ( )( ) 234 1 ... 123 2 1ln ln ln ... ln ln345 1 1 2 3 ... ( 2) ( 1)ln 3 4 5 ... ( 1) ( ) ( 1) 2ln ( 1) ln 2 ln 1 N NNS uuu u u NN NN NN N NN NN NN −=++ + + −−      = + + ++ +       +        × × × ×− ×−=  × × × ×− × ×+ =  + = −+

4 © Millennia Institute 9758/02/PU3/Prelim/23 Solutions Qn Solution 3(b) (i) [3] 123 1 3 333 ...1 111 n n x xxx x xxx ∞ =    =+++   ++++   ∑ is a geometric series with 3 1 xr x= + . If the series has a finite sum, S∞ exists and 1r < . 1 3 11 r x x < <+ Method 1 (Graphical): 0.25 0.5x− << Method 2 (Algebraic): ( ) ( ) ( )( ) 22 22 2 3 11 31 31 9 21 8 2 10 4 12 1 0 0.25 0.5 x x xx xx xx x xx xx x <+ <+ <+ <++ − −< + −< − <<

5 © Millennia Institute 9758/02/PU3/Prelim/23 Solutions Qn Solution 3(b) (ii) [2] Method 1 ( ) 11 1 123 3 0.253 1 0.25 1 3 5 333 ...555 3 5 31 5 3 2 nn nn n n x x ∞∞ = = ∞ =  =  ++  =    =+++     = − = ∑∑ ∑ Method 2 ( ) ( ) 1 3 3 1 31 1 1 3 1 12 1 3 12 3 0.25 since 0.251 2 0.25 1.5 n n x x x xx x x x x x x x x ∞ = 