CAIE P2 2009 November — Question 2 4 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2009
SessionNovember
Marks4
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TopicLaws of Logarithms
TypeSolve ln equation using power law
DifficultyModerate -0.3 This is a straightforward logarithm equation requiring application of the power law (2ln x = ln x²), forming a quadratic equation, and solving with domain restrictions. It's slightly easier than average as it follows a standard template with clear steps, though students must remember to check validity of solutions in the original logarithmic domain.
Spec1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b
2 Solve the equation \(\ln \left( 3 - x ^ { 2 } \right) = 2 \ln x\), giving your answer correct to 3 significant figures.
AnswerMarks Guidance
Use \(\ln x^2 = 2\ln x\)B1
Obtain \(3 - x^2 = x^3\), or equivalentB1
Solve for \(x\)M1
Obtain answer \(x = 1.22\), having rejected \(x = -1.22\)A1 [4 marks] 
Use $\ln x^2 = 2\ln x$ | B1 |
Obtain $3 - x^2 = x^3$, or equivalent | B1 |
Solve for $x$ | M1 |
Obtain answer $x = 1.22$, having rejected $x = -1.22$ | A1 | [4 marks]
2 Solve the equation $\ln \left( 3 - x ^ { 2 } \right) = 2 \ln x$, giving your answer correct to 3 significant figures.

\hfill \mbox{\textit{CAIE P2 2009 Q2 [4]}}
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