O-LevelAdditional MathematicsLogarithmic and exponential functions4037/12 · Oct/Nov 2014 · Q46 Marks
Using the substitution y = 5ˣ, show that the equation 5²ˣ⁺¹ −5ˣ⁺¹ + 2 = 2(5ˣ) can be written in the form ay² + by + 2 = 0, where a and b are constants to be found.
- (i)Using the substitution y = 5ˣ, show that the equation 5²ˣ⁺¹ −5ˣ⁺¹ + 2 = 2(5ˣ) can be written in the form ay² + by + 2 = 0, where a and b are constants to be found.[2]
- (ii)Hence solve the equation 5²ˣ⁺¹ −5ˣ⁺¹ + 2 = 2(5ˣ).[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.
- [2] B1 for 5, B1 for -7 (5y² - 7y + 2 = 0)
- [1] M1 for solution of quadratic equation from (i)
- [1] M1 for use of logarithms to solve equation of the type 5ˣ = k
- [1] A1 must be evaluated to 3sf or better x = -0.569
- [1] B1 y = 1, x = 0
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