ASRJC Prelims 2025 H2 Math P1 Solutions

Paper 1 Solutions 1 A cubic curve passes through the points (2, 3) and ( 3, 22)−− . Find the equation of the curve if it has a stationary point at (1, 6)− . [4] [Solution] Let 32y ax bx cx d= + ++ 32(2) (2) (2) 3a b cd+ + += ⇒ 8a + 4 b + 2c + d = 3 …Eq(1) 32( 3) ( 3) ( 3) 22a b cd− +− +−+= − ⇒ –27a + 9b – 3c + d = –22 …Eq(2) 32(1) (1) (1) 6a b cd+ + += − ⇒ a + b + c + d = –6 …Eq(3) 2d 32d y ax bx cx = ++ 23 (1) 2 (1) 0a bc + += ⇒ 3a + 2b + c = 0 …Eq(4) From GC, 2, 1, 8, 1a bc d=== −= − ∴ Equation of curve is 322 81y xx x= +−−

2 (a) Show that 3cos3 4cos 3cosθ θθ= − . [3] (b) Hence, or otherwise, evaluate 6 0 sin 2 cos3 d π θ θθ∫ exactly. [4] Solutions (i) ( )cos3 cos 2θ θθ= + cos 2 cos sin 2 sinθθ θθ= − ( ) ( ) 22cos 1 cos 2sin cos sinθ θ θθθ= −− ( ) 32 32 33 3 2cos cos 2sin cos 2cos cos 2 1 cos cos 2cos cos 2cos 2cos 4cos 3cos (shown) θ θ θθ θ θ θθ θθ θ θ θθ = −− = − −− = −− + = − (ii) ( ) ( ) ( ) 6 0 36 0 426 0 sin2 cos3 d 2sin cos 4cos 3cos d 8sin cos 6sin cos d π π π θ θθ θ θ θ θθ θθ θθ θ = − = − ∫ ∫ ∫ 53 6 0 8cos 6cos 53 π θθ= −+ 5 3 53 53 88cos 2cos cos 0 2cos 056 6 5 83 3 2 252 2 5 89 3 3 3 2 25 1 62 425 ππ  =− + −− +       = −+ −        = − +−         33 2 10 5= −

B O C A r O B C A r 3 (a) Given that θ is small, show that 21sec 1 2θθ≈+ [2] (b) The diagram below shows a circle, centre O and radius r , with points A and B on the circumference such that radians AOB θ∠= , where θ is small. AC is a tangent to the circle at A and OBC is a straight line. (i) Show that the length of chord AB can be approximated by rθ . [2] (ii) H ence show that the perimeter of triangle ABC can be approximated by ( )r abθθ + , where a and b are constants to be determined. [4] [Solution] (a) 12 1sec cos 1 since is small2 θ θ θ θ − = ≈− = ( ) 2 11 2 θ+− − +  211 2θ≈+ (shown) (b)(i) ( )( ) 222 22 2 cos 2 2 cos AB r r r r AB r r θ θ =+− ⇒= − Since θ is a small angle,

( ) 1 22 22 2 cosAB r r θ= − 1 2 2 21 1 2r θ ≈ −−   rθ= b(ii) ( ) 222 22 tan sec OC r r r θ θ = + = secBC OC OB r r θ=−= − 2 2 111 2 1 2 r r θ θ ≈+ − = Perimeter of triangle ABC = AB + BC + AC 2 2 1 tan2 1 2 r rr r rr θθ θ θ θθ = ++ ≈ ++ = 12 2rθθ + 12, 2ab∴= =

4 The Folium of Descartes is a curve, defined by the equation x³ + y³ = 3axy where a is a real constant. It is given that a ≠ 0 for this question. (a) Show that 2 2 d d y ay x x y ax −= − . [2] (b) The point 33 ,22 aa  lies on this curve. Show that the equation of the normal at this point on the curve is in