2 Solve the following simultaneous equations. Give your answers in the form a b 3 + , where a and b are rational. x y x y 3 2 3 5 + = - = [5]

2 Solves 2 2 6 + = x y and 2 3 5 − = x y oe by elimination as far as 2 3 1 + = y y or substitutes x = 3 – y into 2 3 5 − = x y oe OR solves 3 3 3 3 + = x y and 2 3 5 − = x y oe by elimination as far as 2 3 3 3 5 + = + x x or substitutes y = 3 – x into 2 3 5 − = x y oe M1 1 2 3 = + y or 3 3 5 2 3 + = + x A1 1 2 3 2 3 2 3 − = × + − y oe or 3 3 5 2 3 2 3 2 3 + − = × + − x oe M1 FT their value of x or y providing of equivalent difficulty 2 3 = − y and 1 3 = + x A2 A1 for either and no extra values

5 (a) Solve the following simultaneous equations. e e 2 3 8 - = e e 5 x y x y + = [5] (b) Solve the equation e e 5 ( ) ( ) t t 2 1 5 3 = - - . [4]

5(a) Solves 3e 3e 15 x y + = and 2e 3e 8 x y − = oe by elimination as far as 3 e 2e 23 x x + = or substitutes e 5 e y x = − into 2e 3e 8 x y − = oe OR Solves 2e 2e 10 x y + = and 2e 3e 8 x y − = oe by elimination as far as 2e 3e 2 y y + = or substitutes e 5 e x y = − into 2e 3e 8 x y − = oe M1 23 e 5 x = or 2 e 5 y = oe A1 [ ] ln4.6 1.53 x = = oe or [ ] ln0.4 0.916 y = = − oe A1 If M0 scored SC1 for using their expression of the form cex d = to give ln d x c = provided 0 d c > Finds the other value, e or e y x , by substituting their ex or ey M1 FT their ex or ey [ ] ln0.4 0.916 y = = − oe or [ ] ln4.6 1.53 x = = oe A1 5(b) ( ) 2 1 5 3 e 5 t t −− − = or ( ) 5 3 2 1 1 e 5 t t −− −= oe M1 2 3 e 5 t − = or 3 2 1 e 5 t−= A1 2 3 ln5 t − = or 1 3 2 ln 5 t − = M1 FT their e 5 a bt − = or their 1 e 5 ct d −= where a, b, c and d are positive integers 2 ln5 3 t − = or 2 ln0.2 3 t + = or 0.13[0] oe A1 Alternative method 2 1 5 3 ln e ln5 lne t t − − = + oe (M1) [ ] (2 1)[lne] ln5 (5 3) lne t t − = + − oe (A1) 5t – 2t = 3 – 1 – ln5 oe (M1) Dep on one correct log law applied with at most one sign error 2 ln5 3 t − = or 2 ln0.2 3 t + = or 0.13[0] oe (A1) (a) To earn marks, you must correctly eliminate one variable (e^x or e^y) from the simultaneous equations, solve for the remaining exponential term, and then use logarithms to find the exact values of x and y. (b) You earn marks by correctly manipulating the equation using logarithm properties to isolate the exponential term, then taking logarithms to solve for t.

9 Solve the following simultaneous equations. x xy y xy 4 3 8 4 0 2 2 + + = + = [6]

9 Correctly eliminates x or y e.g. 2 2 4 4 4 3 8     + − + − =         x x x x oe or 2 2 4 4 4 3 8     − + − + =         y y y y oe M1 Rearranges to a 3-term quadratic in x2 or y2 soi e.g. 4 2 4 20 16 0 − + = x x or 4 2 20 64 0 − + = y y A1 Factorises or solves their 3-term quadratic in x2 or y2 soi : (x2 – 1)(x2 − 4) or (y2 – 16)(y2 − 4) M1 2 1 = x , 2 4 = x oe, nfww or 2 16 = y , 2 4 = y oe, nfww A1 x = ±1 x = ± 2 y = ∓4 y = ∓2 oe, nfww A2 A1 for all 4 x values or all 4 y values

6 The line y x 5 6 = + meets the curve xy 8 = at the points A and B. (a) Find the coordinates of A and of B. [3] (b) Find the coordinates of the point where the perpendicular bisector of the line AB meets the line y x = . [5]

6(a) ( ) 5 6 8 x x + = 2 5 6 8 0 x x + − = M1 For attempt to equate and obtain a 3- term quadratic in either x or y 4 , 10 5       A1 Allow A1 if only x-coordinates or only y-coordinates are given (–2, –4) A1 6(b) Midpoint 3, 3 5   −     B1 Gradient 5 B1 1 3 3 5 5 y x   − = − +     M1 Attempt at perp bisector using their midpoint and perp gradient 1 3 3 5 5 x x   − = − +     M1 For use of y x = and attempt to solve 12 12 , 5 5       A1

2 Solve the simultaneous equations. y x x x y 3 4 2 5 4 2 + = + = [5]

2 3 4 x x − = + oe M1 0.5 x =− oe A1 2 2 4 2 3 4 5 x x x −   + =     M1 eliminate x or y ( ) 2 12 20 0 x x − + = A1 3 terms on one side if eliminating y ( ) 2 5 16 0 y y + = if eliminating x ( )( ) ( ) 2 10 0 x x − − = M1 or ( ) ( ) 5 16 0 y y + = 2 or 10 x x = = nfww A1 or correct pair 16 0 or 5 y y = = − nfww A1

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Simultaneous equations past paper questions

Revise O-Level Additional Mathematics (4037) — Simultaneous equations 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.

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

Simultaneous equations in O-Level Additional Mathematics 4037 exams, 2020–2025

Across 2020–2025, Simultaneous equations accounted for 23 questions worth 176 marks in Cambridge O Level Additional Mathematics (4037) papers.