The curve $C$ has equation $y = 2 - \ln(1 - x)$. Sketch the graph of $C$, giving the exact coo

Question

The curve $C$ has equation $y = 2 - \ln(1 - x)$. Sketch the graph of $C$, giving the exact coordinates of any points of intersection with the axes.[2] Find the equation of the tangent to $C$ at the point where $x = \frac{1}{2}$, giving your answer in the form of $y = mx + a + \ln b$ where $m$, $a$ and $b$ are integers to be determined.[3] By adding a suitable graph, solve the inequality $$3x^2 + \ln \sqrt{1 - x} \leq 1.$$[3] Hence solve the inequality $$3e^{2x} + \ln \sqrt{1 - e^x} \leq 1.$$[2]

Accepted Answer

(b) $y=2x+1+\ln2$ (c) $-0.514 \le x \le 0.752 $ or $0.976 \le x < 1 $ (d) $ x \le -0.285 $ or $-0.0248 \le x <0 $

2023 VJC H1 Q4

2023 VJC H1 Q4