jee-advanced 2009 Q23

jee-advanced · India · paper1 Differential equations Integral Equations Reducible to DEs

Let $f$ be a non-negative function defined on the interval $[ 0,1 ]$. If
$$\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } d t = \int _ { 0 } ^ { x } f ( t ) d t , \quad 0 \leq x \leq 1 ,$$
and $f ( 0 ) = 0$, then
(A) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(B) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(C) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$
(D) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$

Let $f$ be a non-negative function defined on the interval $[ 0,1 ]$. If

$$\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } d t = \int _ { 0 } ^ { x } f ( t ) d t , \quad 0 \leq x \leq 1 ,$$

and $f ( 0 ) = 0$, then\\
(A) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$\\
(B) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$\\
(C) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$\\
(D) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$

Paper Questions

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