CAIE P3 2015 June — Question 2 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2015
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Equations & Modelling
TypeQuadratic in exponential form
DifficultyModerate -0.8 This is a straightforward substitution question requiring recognition that 4^(x+2) = 16ยท4^x, leading to a simple linear equation 16u - u = 16, so u = 16/15. The substitution is explicitly given, and solving for x requires only basic logarithms. Below average difficulty due to the guided approach and minimal problem-solving required.
Spec1.02b Surds: manipulation and rationalising denominators1.06g Equations with exponentials: solve a^x = b

2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.

AnswerMarks Guidance
Use laws of indices correctly and solve for \(u\)M1
Obtain \(u\) in any correct form, e.g. \(u = \frac{16}{16-1}\)A1
Use correct method for solving an equation of the form \(4^x = a\), where \(a > 0\)M1
Obtain answer \(x = 0.0466\)A1 [4]
Use laws of indices correctly and solve for $u$ | M1 |
Obtain $u$ in any correct form, e.g. $u = \frac{16}{16-1}$ | A1 |
Use correct method for solving an equation of the form $4^x = a$, where $a > 0$ | M1 |
Obtain answer $x = 0.0466$ | A1 | [4]
2 Using the substitution $u = 4 ^ { x }$, solve the equation $4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }$, giving your answer correct to 3 significant figures.

\hfill \mbox{\textit{CAIE P3 2015 Q2 [4]}}