Find the exact root of the equation $$\(9 \mathrm{e}^{9 y}-6 \mathrm{e}^{6 y}-20 \mathrm{e}^{3 y}-8=0\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s20_qp_21 Year：2020 Question No：7(c)
Answer:
State or imply \(9 x^{3}-6 x^{2}-20 x-8=(3 x+2)\left(3 x^{2}-4 x-4\right)\)
(FT from quotient in part (a))
Attempt to solve cubic eqn to find positive value of \(x\) (or of \(\left.\mathrm{e}^{3 y}\right)\)
Use logarithms to solve equation of form \(\mathrm{e}^{3 y}=k\) where \(k>0\)
Obtain \(\frac{1}{3} \ln 2\) or exact equivalent
(FT from quotient in part (a))
Attempt to solve cubic eqn to find positive value of \(x\) (or of \(\left.\mathrm{e}^{3 y}\right)\)
Use logarithms to solve equation of form \(\mathrm{e}^{3 y}=k\) where \(k>0\)
Obtain \(\frac{1}{3} \ln 2\) or exact equivalent
Knowledge points:
2.1.3 use the factor theorem and the remainder theorem.
2.2.3 use logarithms to solve equations and inequalities in which the unknown appears in indices
Solution:
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