Verified Solution • 2025–26 Session
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QJEEM07Maa3c5: JEE Main 2007 Mathematics - Class 12 JEEM07Maa3c5 Solution: Let and Then which one of the following is true?
Let and Then which one of the following is true?
Multiple Choice Options
- A and
- B and
- C and
- D and
Step-by-Step Verified Solution
We know that {{\sin x} \over x} < 1, for
\Rightarrow {{\sin x} \over {\sqrt x }} < \sqrt x on
\Rightarrow \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx < \int\limits_0^1 {\sqrt x dx} = \left[ {{{2{x^{3/2}}} \over 3}} \right]} _0^1
\Rightarrow \int\limits_0^1 {{{\sin x} \over {\sqrt x }}} dx < {2 \over 3} \Rightarrow I < {2 \over 3}
Also {{\cos x} \over {\sqrt x }} < {1 \over {\sqrt x }} for
\Rightarrow \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx < \int\limits_0^1 {{x^{ - 1/2}}dx} }
\Rightarrow \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx < 2}
\Rightarrow J < 2
\Rightarrow {{\sin x} \over {\sqrt x }} < \sqrt x on
\Rightarrow \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx < \int\limits_0^1 {\sqrt x dx} = \left[ {{{2{x^{3/2}}} \over 3}} \right]} _0^1
\Rightarrow \int\limits_0^1 {{{\sin x} \over {\sqrt x }}} dx < {2 \over 3} \Rightarrow I < {2 \over 3}
Also {{\cos x} \over {\sqrt x }} < {1 \over {\sqrt x }} for
\Rightarrow \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx < \int\limits_0^1 {{x^{ - 1/2}}dx} }
\Rightarrow \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx < 2}
\Rightarrow J < 2