Question

Let y = f(x) defined on R satisfies $(1+x^2)\frac{dy}{dx}$ = 2x - 2xy and f(0) = 2, then

• A. f(x) is neither even nor odd function
• B. f(x) is increasing on ($-\infty$, 0) and decreases on (0, $\infty$)
• C. the x-intercept of normal on graph of y = f(x) at x = 1 equals $\frac{1}{4}$
• D. the area bounded by y = f(x) with x-axis between at x = 0 and x = 1 equals $(1+\frac{\pi }{4})sq.units$

Answer 1

Answer: (B), (C), (D)

The correct options are
B

f(x) is increasing on ($-\infty$, 0) and decreases on (0, $\infty$)

C

the x-intercept of normal on graph of y = f(x) at x = 1 equals $\frac{1}{4}$

D

the area bounded by y = f(x) with x-axis between at x = 0 and x = 1 equals $(1+\frac{\pi }{4})sq.units$

$f\left(x\right)=\frac{x^2+2}{x^2+1}=1+\frac{1}{x^2+1}$ (even functions)
Also,

$Area={\int }_0^1(1+\frac{1}{1+x^2})dx=1+\frac{\pi }{4}$