Moderate -0.8 Part (i) is a straightforward one-step natural logarithm equation requiring only the definition of ln (exponentiating both sides). Part (ii) asks for a counter-example to disprove a statement, which requires understanding that ln(y) ≥ 0 when y ≥ 1, but this is a routine C3 skill. Both parts are simpler than average A-level questions, requiring minimal problem-solving.
2. (i) Solve the equation
$$\ln ( 3 x + 1 ) = 2$$
giving your answer in terms of e.
(ii) Prove, by counter-example, that the statement
$$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$
is false.
2. (i) Solve the equation
$$\ln ( 3 x + 1 ) = 2$$
giving your answer in terms of e.\\
(ii) Prove, by counter-example, that the statement
$$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$
is false.\\
\hfill \mbox{\textit{OCR C3 Q2 [7]}}