RI 9758 2023 Prelim P2 Solution

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ 2023 Year 6 H2 Mathematics Preliminary Examination Paper 2: Solutions 1 Solutions (a) [2] 2 22 2 2 11 22 11 24 xt t tt t = −  = −+ −   = −− Since 22 1 111 0, .2 244tt − ≥ − − ≥−   Hence 1 4x≥− for all values of t. (b) [3] d 3d y t = , d 21d x tt = − d dd 3 d d d 21 y yt x txt∴ =×= − When 1t =− , 2x= , 1y=− , d 1d y x =− & gradient of normal = 1. Equation of normal at the point where 1t =− is 12yx+=− 3yx⇒= − (c) [4] The curve and the line 1 4x=− intersect when 1 2t = . Required Area 11 44 11 24 22 12 d ( 3) d (3 2)(2 1) d ( 3) d 5.90625 5.906 (3.d.p) yx x x t tt xx −− − − = −− = + −− − = = ∫∫ ∫∫ Alternative Solution 1

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ Required Area 13 4 13 4 7 12 1 1 122 1 11( )d ( 3 )d44 113( ) d ( 3 ) d24 5.90625 5.906 (3.d.p) xy y y t ty y − − − − − − = + + ++ = − + ++ = = ∫∫ ∫∫ Note that 13 4 1 1 199( 3 ) d can be seen as .4 244yy − − ++ ××∫ Alternative Solution 2 Required Area ( ) ( ) 1 4 1 2 10 2 9 10 9 2 1 3 2 3 22 1 13 9d 1d 24 4 1 13 96 2 dt 1 6 2 dt24 4 yx yx tt tt − − − −  = + × +×−   = +− + × + × − +−  ∫∫ ∫∫

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ 2 Solutions (a) [2] 1 , 3 , 5 , 7 , . . . Number of terms in each bracket follows an AP with first term 1 and common difference 2. ( ) 2 Number of integers in the first sets 2(1) 1 22 n n nn ∴ = +− = (b) [4] Last integer in the nth set is the ( ) 2n th term of the AP 1, 4, 7, 10, 13, 16, ... which has first term 1 and common difference 3. ( ) 22Last integer of the th set 1 1 3 3 2n nn = +−=− From GC, n 232n − 25 1873 26 2026 ∴ 26k = OR: Given that 2023 occurs in the kth set, ( ) ( ) 2 2 2 2 first term in the th set 2023 last term in the th set 3 1 2 3 2023 3 2 2022 2025 1 and 33 24.961 26.961 and 25.980 or 25.980 25.980 26.961 kk kk kk k kk k ≤≤  − − +≤ ≤ − −≤ ≥ − ≤ ≤ ≤− ≥ ∴ ≤≤ Since k∈ + , 26k = OR: ( ) 2 23 1 2 3 2023 3 2kk − − +≤ ≤ − From GC, 24.961 26.961 and 25.980 or 25 .980k kk− ≤ ≤ ≤− ≥ 25.980 26.961k∴ ≤≤ Since k∈ + , 26k = (c) [3] Required sum

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ ( ) ( ) ( ) ( ) 22 22 22 Sum of first 10 th terms Sum of first 4 th terms 10 42(1) 10 1 3 2(1) 4 1 322 14 574 = −   = + − − +−   = OR: Last term in the 10th set 23(10) 2 298= −= Last term in the 4th set 23(4) 2 46= −= First term in t