| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve using substitution or auxiliary variable |
| Difficulty | Moderate -0.3 Part (i) is a standard quadratic-in-disguise problem requiring substitution y=3^x, solving a simple quadratic, then applying logarithms—routine A-level technique. Part (ii) is a straightforward extension using symmetry of absolute value. The question requires multiple steps but uses only standard methods with no novel insight, making it slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks |
|---|---|
| (ii) | Carry out method for solving quadratic equation in 3x |
| Answer | Marks |
|---|---|
| State ±1.77, following positive answer from part (i) | M1 |
| Answer | Marks |
|---|---|
| B1 | [4] |
Question 1:
--- 1 (i)
(ii) ---
1 (i)
(ii) | Carry out method for solving quadratic equation in 3x
Obtain at least 3x =7
Use logarithms to solve an equation of the form 3x =k where k >0
Obtain 1.77
State ±1.77, following positive answer from part (i) | M1
A1
M1
A1
B1 | [4]
[1]\begin{enumerate}[label=(\roman*)]
\item It is given that $x$ satisfies the equation $3^{2x} = 5(3^x) + 14$. Find the value of $3^x$ and, using logarithms, find the value of $x$ correct to 3 significant figures. [4]
\item Hence state the values of $x$ satisfying the equation $3^{2|x|} = 5(3^{|x|}) + 14$. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q1 [5]}}