3 Solve the equation cot x 2 3 3 1 2 r - = b l , where x is in radians and x 0 1 G r. [5]

3 π tan 2 3 3   − = ±     x soi or π 3 sin 2 3 2   − = ±     x soi B1 B0 if negative root is rejected Allow truncated decimals May be implied by subsequent work From use of 2 2 π π cosec 2 1 cot 2 3 3     − − = −         x x π π π 2π 4π 2 , , , 3 3 3 3 3 x − = − 2π 5π 2 0, , π, 3 3 x = π π 5π 0, , , 3 2 6 x = or 0, 1.05, 1.57, 2.62 or greater accuracy 4 M1 for correct order of operations to obtain one solution in the range using π tan 2 3   − =     x k or π sin 2 , 1 3   − = <     x m m Dep M1 for correct order of operations to obtain a second solution in the range using ( ) 1 π 2 tan π 3 −   − = ±     x k or ( ) 1 π 2 π sin , 1 3 −   − = − <     x m m oe ( ) 1 π 2 sin , 1 3 −   − = − <     x m m oe A1 for any pair of correct solutions A1 for remaining pair of solutions, with no extra solutions within the range To earn marks, you must first correctly transform the equation using cot(x) = 1/tan(x) or a similar identity, allowing for a positive or negative root. Then, apply the correct order of operations to find at least one solution within the given range, followed by finding a second solution using the periodic nature of the trigonometric function. Finally, provide all correct solutions within the range, ensuring no extra solutions are included.

4 Solve the equation , cot x 2 3 3 0 r + - = b l where x 1 1 r r - radians. Give your answers in terms of r. [4]

4 π 1 tan 2 3 3   + =     x B1 7 5 11 , , , 12 12 12 12 π π π π = − − x 3 M1 for using correct order of operations A1 for two correct solutions A1 for two further correct solutions and no other solutions in range To earn these marks, you must first correctly rearrange the equation to isolate tan(x + pi/3). Then, you need to correctly find the principal value and use the periodicity of the tangent function to find all four correct solutions within the given range, with no extra solutions.

5 Find the possible values of the constant c for which the line y c = is a tangent to the curve . sin y x 5 3 4 = + [3]

5 Either Maximum when sin 1 3 = x or minimum when sin 1 3 = − x M1 For recognition that value of maximum or minimum is necessary 9 = c A1 1 = − c A1 or d 5cos d 3 3 = y x x When d 0 d = y x , sin 1 3 = + x or 1 − (M1) For differentiation, equating to zero to obtain values for sin 3 x 9 = c (A1) 1 = − c (A1)

9 (a) (i) Write xy y x 6 3 4 2 + + + in the form ( )( ) ax b cy d + + , where a, b, c and d are positive integers. [1] (ii) Hence solve the equation sin cos cos sin 6 3 4 2 0 i i i i + + + = for ° ° 0 360 1 1 i . [4] (b) Solve the equation r sec 2 1 4 2 3 1 z+ = b l for r r 1 1 z - , where z is in radians. Give your answers in terms of r. [5]

9(a)(i) ( )( ) 3 2 2 1 + + y x B1 9(a)(ii) ( )( ) 3cos 2 2sin 1 0 θ θ + + = 2 cos 3 θ = − , 1 sin 2 θ = − M1 For relating to part (i) and a correct attempt to obtain cos ... θ = or sin ... θ = o o 131.8 , 228.2 θ = o o 210 , 330 θ = 3 M1 for solving one of the equations to obtain one correct solution A1 for any two correct solutions A1 for a further two correct solutions with no extra solutions within the range 9(b) π 3 cos 2 4 2 φ   + =     oe B1 5π π 19π 23π , , , 24 24 24 24 φ = − − 4 M1 for solving to obtain one correct positive solution M1 for solving to obtain one correct negative solution A1 for any two correct solutions A1 for a further two correct solutions with no extra solutions within the range (a)(i) To earn this mark, you must correctly factorize the given expression into the form (ax+b)(cy+d). (a)(ii) To earn the first mark, you must correctly relate the given trigonometric equation to the factorization from part (i) and attempt to solve for cos(theta) or sin(theta). The second mark is for finding at least one correct solution for theta. The third mark is for finding any two correct solutions, and the final mark is for finding all four correct solutions within the given range, with no extra solutions. (b) To earn the first mark, you must correctly rewrite the equation in terms of cos(2phi + pi/4). The next two marks are for finding one correct positive solution and one correct negative solution for phi. The final two marks are for finding any two correct solutions and then all four correct solutions within the specified range, with no extra solutions.

9 M A O B D C N 6 cm 4 cm 12 cm The diagram shows a circle, centre O, radius 12 cm, and a rectangle ABCD. The diagonals AC and BD intersect at O. The sides AB and AD of the rectangle have lengths 6 cm and 4 cm respectively. The points M and N lie on the circumference of the circle such that MAC and NDB are straight lines. (a) Show that angle AOD is 1.176 radians correct to 3 decimal places. [2] (b) Find the perimeter of the shaded region. [4] (c) Find the area of the shaded region. [3]

9(a) 1 2 2 tan 3 −  = ×     AOD M1 For correct method to find AOD 1.1760... = AOD 1.176 = AOD [to 3dp] A1 Need to see 4 dp or more to justify 3 dp answer 9(b) Major arc ( ) 2 1.176 12 π = − MN B1 ND or MA =12 13 − B1 Perimeter = major arc MN + MA + ND + 16 oe B1 For their values in a correct plan, may be implied by a correct answer Perimeter = 94.1 B1 9(c) Minor sector area = 2 1 1.176 12 2 × × or Major sector area = ( ) 2 1 2 1.176 12 2 π × − × B1 Area = major sector area – remainder of rectangle or Area = area of circle – minor sector area – remainder of rectangle or Area = circle – rectangle – minor sector + triangle AOD B1 For their values in a correct plan, may be implied by a correct answer Area = 350 B1 Allow greater accuracy

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Trigonometry past paper questions

Revise O-Level Additional Mathematics (4037) — Trigonometry with real Cambridge past-paper questions grouped by topic, each shown with its marking scheme. Working one topic at a time targets exactly the skill each question tests.

72 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.