CAIE P2 2011 November — Question 4 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2011
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Equations & Modelling
TypeQuadratic in exponential form
DifficultyModerate -0.3 This is a standard quadratic-in-disguise problem requiring substitution u=3^x, solving the resulting quadratic, then taking logarithms. It's a routine textbook exercise with a well-known technique, making it slightly easier than average, though the logarithm step adds minor computational complexity beyond pure algebraic manipulation.
Spec1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b

4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.

AnswerMarks Guidance
Carry out recognizable solution method for quadratic in \(3^x\)M1
Obtain \(3^x = 5\) and \(3^x = 2\)A1
Use logarithmic method to solve an equation of the form \(3^x = k\), where \(k > 0\)M1
State answer 1.46A1
State answer 0.631A1 [5]
Carry out recognizable solution method for quadratic in $3^x$ | M1 |

Obtain $3^x = 5$ and $3^x = 2$ | A1 |

Use logarithmic method to solve an equation of the form $3^x = k$, where $k > 0$ | M1 |

State answer 1.46 | A1 |

State answer 0.631 | A1 | [5]
4 Solve the equation $3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0$, giving your answers correct to 3 significant figures.

\hfill \mbox{\textit{CAIE P2 2011 Q4 [5]}}