O-LevelAdditional MathematicsLogarithmic and exponential functions4037/12 · May/June 2022 · Q1011 Marks

Given the simultaneous equations lg x + 2 lg y = 1, x - 3y² = 13, show that x² - 13x - 30 = 0.

(a)(i)Given the simultaneous equations lg x + 2 lg y = 1, x - 3y² = 13, show that x² - 13x - 30 = 0.[4]
(a)(ii)Solve these simultaneous equations, giving your answers in exact form.[2]
(b)Solve the equation logₐ x + 3 logₓ a = 4, where a is a positive constant, giving x in terms of a.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[B1] Simplification of the left-hand side of the first equation (lg xy²)
[B1] Simplification of right-hand side of equation (1 = lg 10)
[M1] For substitution of y² into linear equation oe and attempt to simplify
[A1] AG – must see sufficient detail to justify the given answer
[B1] x = 15 only
[B1] y = √2/√3 or √6/3 or exact equivalent only (isw once exact value seen)
[B1] For an appropriate change of base
[M1] For an attempt to obtain a 3-term quadratic equation in terms of logₐ x or logₓ a, equated to zero.
[M1 Dep] on previous M mark for correct solution of their quadratic equation
[A1] x = a³
[A1] x = a

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