Question

The equation of a curve passing through $(2,\frac{7}{2})$ and having gradient $1-\frac{1}{x^2}$ at $(x,y)$ is:

• A. $y=x^2+x+1$
• B. $xy=x^2+x+1$
• C. $xy=x+1$
• D. None of these

Answer 1

Answer: (B)

$xy=x^2+x+1$

We have, $\frac{dy}{dx}=1-\frac{1}{x^2}\Rightarrow y=x+\frac{1}{x}+c$
This passes through $(2,\frac{7}{2})$, therefore
$\frac{7}{2}=2+\frac{1}{2}+c\Rightarrow c=1$
Thus the equation of the curve is
$y=x+\frac{1}{x}+1\Rightarrow xy=x^2+x+1$