| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 9 |
| Topic | Laws of Logarithms |
| Type | Combine logs into single logarithm |
| Difficulty | Easy -1.3 This is a straightforward multi-part question testing basic logarithm laws, index laws, and surds rationalization. All parts are routine textbook exercises requiring direct application of standard rules with no problem-solving or insight needed. Part (a) uses basic log addition/subtraction rules, part (b) is simple index manipulation, and part (c) is standard rationalization by multiplying by conjugate. Significantly easier than average A-level questions. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.02b Surds: manipulation and rationalising denominators1.06f Laws of logarithms: addition, subtraction, power rules |
(a)(i) $\log_a 15$ [B1]
(a)(ii) Use $b \log a = \log a^b$ at least once [M1]
Use $\log a - \log b = \log \frac{a}{b}$ [M1]
Obtain $\log_b \frac{1}{2}$ [A1]
(b) $\frac{1}{3}$ [B1]
$\frac{1}{3a^2}$ o.e. [B1]
(c) Attempt to multiply numerator and denominator by $2\sqrt{3} + 3$ [M1]
Obtain $\frac{18 + 7\sqrt{3} - 3}{12 - 9}$ [A1]
Obtain given answer [A1]1
\begin{enumerate}[label=(\alph*)]
\item Express each of the following as a single logarithm.
\begin{enumerate}[label=(\roman*)]
\item $\log _ { a } 5 + \log _ { a } 3$
\item $5 \log _ { b } 2 - 3 \log _ { b } 4$
\end{enumerate}\item Express $\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }$ as an algebraic fraction in its simplest form.
\item Show that $\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q1 [9]}}