Question

The curve $C$ has equation $y=\cos hx-3\sin hx$ The $x-$co-ordinate of the point where the curve $C$ intersects the $x-$axis is:

• A. $(\ln 2,-1)$
• B. $(-\ln 2,1)$
• C. $(-\ln 2,-1)$
• D. $(\ln 2,1)$

Answer 1

The correct option is A $(\ln 2,-1)$

$y=\cos hx-3\sin hx$ meets $y=-1$ at $(k,-1)$
This gives $-1=\cos hk-3\sin hk$
$\Rightarrow -1=\frac{1}{2}(e^k+e^{-k})-\frac{3}{2}(e^k-e^{-k})$
$=\frac{1}{2}[e^k+e^{-k}-3e^k+3e^{-k}]$
$=\frac{1}{2}[4e^{-k}-2e^k]$
$=2e^{-k}-e^k$
or $2e^{-k}-e^k+1=0$
Multiplying by $e^k$ we get
$2-e^{2k}+e^k=0$
Let $e^k=x$ then
$2-x^2+x=0$
Rearranging, we get
$x^2-x-2=0$
The factors: $(x-2)(x+1)=0$
Hence,$x=2,-1$
$\Rightarrow e^k=2$ or $-1$
$\therefore k=\ln 2$
$\because e^x$ is positive for all $x$
The $x-$coordinate is $\ln 2$