2026 HGV P1 MS

HILLGROVE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2026 SECONDARY 4G3/5G2 CANDIDATE NAME MARK SCHEME ( ) CLASS - CENTRE NUMBER S INDEX NUMBER ADDITIONAL MATHEMATICS Paper 1 Candidates answer on the Question Paper. No Additional Materials are required. 4049/01 27 Aug 2026 2 hours 15 minutes 11.40 AM to 1.55 PM READ THESE INSTRUCTIONS FIRST Write your class, index number and name in the spaces at the top of this page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE ON ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 90. For Examiner’s Use TOTAL 90 Parent’s/ Guardian’s Signature: ____________________ Setter: Mr Ng Choon Chooi This document consists of 14 printed pages.

Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 2 4 2 b b acx a−  −= Binomial expansion ( ) 1 2 2 ... ... ,12n n n n n r r nn n na b a a b a b a b b r− − −     + = + + + + + +           where n is a positive integer and ( ) ! ( 1)...( 1) ! ! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B=  tan tantan( ) 1 tan tanABAB AB=  sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for ABC sin sin sin a b c A B C== Abccba cos 2222 −+= 1 sin2 ab C=

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 3 1. Find the coordinates of the points of intersection between the line yx 253 =+ and the curve 143 =− xyx . [4] yx 253 =+ ……………………………(1) 143 =− xyx ……………………………(2) Subst. 2 3 5 into equation (2),yx=+ 3 2 (3 5) 1x x x− + = [M1] 20 6 7 1xx= + + (6 1)( 1) 0xx+ + = [M1] 1 or 16xx=− =− 12 or 14yy== ( )1,1 [A1] 11, 2 [A1]24 − −

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 4 2. Given that 4cos 5A=− and tan 0A , find the exact value of the following. (a) tan A [1] 3tan 4A= [B1] -4 -3 5 (b) sin 2A [2] sin 2 2sin cosA A A= 34sin 2 2( )( ) 55A= − − [M1] 24 25= [A1] (c) A21cos [3] 2cos 2cos 1 2AA=− 24( ) 2cos 152 A− = − [M1] 2 42cos 125A =− 2 1cos 2 10A = [M1] 1 1 10cos accept -2 10 10A=− [A1]

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 5 3. The equation of a curve is kkxxy +−= 23 , where k is a constant. (a) Find the smallest value of the integer k for which the curve is always positive for all values of x. [4] 2Subst. 0 into 3 ,y y x kx k= = − + 230x kx k− + = 2( ) 4(3)( ) 0kk− −  [M1] 2 12 0kk− ( 12) 0kk − [M1] 0 12 0 12k [A1] Smallest integer 1k = [A1] (b) Find the set of values of k for which the line 22 −= xy intersects the curve at two distinct points. [4] 2Subst. 2 2 into 3 ,y x y x kx k= − = − + 22 2 3x x kx k− = − + [M1] 23 ( 2) ( 2) 0x k x k− + + + = 2( ( 2)) 4(3)( 2) 0kk− + − +  [M1] 2 8 20 0kk− −  ( 2)( 10) 0kk+ −  [M1] -2 10 2 or 10kk−  [A1]

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 6 4. The area of a rectangle plot of land is (5 2 14)+ m2 and its breadth is (3 2 14)− m. Find the length of the plot of land in the form 7ab+ m, where 𝑎 and 𝑏 are integers to be found. [4] Length (3 2 14) (5 2 14) − = + (5 2 14)Length (3 2 14) += − [M1] (5 2 14) (3 2 14) (3 2 14) (3 2 14) ++= −+ [M1] 30 5 28 3 28 14 18 14 + + += − 44 16 7 4 += 11 4 7=+ [A1] 11 , 4ab== [A1]

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 7 5. It is given that 6cos 4 1yx=− . (a) State the amplitude and period of y. [2] Amplitude 6= [B1] 2Period 42== (accept 90o) [B1] (b) State the maximum and minimum values of y. [2] Maximum 6 1 5= − = [B1] Minimum 6 1 7=− − =− [B1] (c) Sketch the graph of 6cos 4 1yx=− for 0 x  . [3] Curve--------- 1 mark Max points--- 1 mark Min points---- 1 mark

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 8 6. Solve 06)3(153 22 =−++ xx . [6] 06)3(153 22 =−++ xx 223 3 15(3 ) 6 0xx + − = [M1] let 3 , xu= 29 15 6 0uu+ − = [M1] 23 5 2 0uu+ − = (3 1)( 2) 0uu− + = [M1] 1 or 23uu= =− [A1] 13 or 3 23xx= =− [M1] 1 or . .x N A=− [A1]

HGV 4G3/5G2 A MATH PRELIM EXAM P1 2026 9 7. (a) Show that 2 21 3sin 1 tan1 cos 2 2A AA− =−+ . [3] 21 3sin. . . 1 cos 2 AL H S A−= + 2 2 2 2 (sin cos ) 3sin 1 (2cos 1) A A A A +−= +− [M1]