Question

Two points $A \left( x _{1}, y _{1}\right)$ and $B \left( x _{2}, y _{2}\right)$ are chosen on the graph of $f(x)=\ell n x$ with $0 < x_{1} < x_{2}$. The points $C$ and $D$ trisect line segment $A B$ with $A C < C B$. Through $C$ a horizontal line is drawn to cut the curve at $E \left( x _{3}, y _{3}\right)$. If $x _{1}=1$ and $x _{2}=1000$ then the value of $x_{3}$ equals

• A. 10
• B. $\sqrt{10}$
• C. $(10)^{2 / 3}$
• D. $(10)^{1 / 3}$

Answer 1

Answer: (A)

Using section formula
$a =\frac{2 \cdot 1+1 \cdot 1000}{3}=334 $
$b =\frac{2 \cdot 0+1 \cdot \ell n 1000}{3} $
$\Rightarrow b =\frac{\ell n 1000}{3}\,\,\,...(i)$
Now line $y=b$ intersects the curve $y=\ln x$
$\therefore b=\ln x \,\,\,\,...(ii)$

from (i) and (ii)
$\frac{\ell n 1000}{3}=\ln x$
$\Rightarrow {\ell n}(1000)^{1 / 3}=\ell {n} x$
$\therefore x=(1000)^{1 / 3}=10$