Q53 / 155 · WBJEE 2024 · Mathematics
All values of a for which the inequality $\frac{1}{\sqrt{a}} \int_1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) \mathrm{dx} \lt 4$ is satisfied, lie in the interval
- A(1, 2)
- B(0, 3)
- Correct answerC(0, 4)
- D(1, 4)
Solution
Hint : $\frac{1}{\sqrt{a}} \int_1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) \mathrm{dx} \lt 4$
$\begin{array}{ll}
\Rightarrow \quad \frac{1}{\sqrt{a}} \cdot\left[\frac{3}{2} \cdot \frac{x^{3 / 2}}{\frac{3}{2}}+x-\frac{x^{1 / 2}}{1 / 2}\right]_1^a \lt 4 & \Rightarrow \frac{1}{\sqrt{a}}\left(a^{3 / 2}+a-2 a^{1 / 2}-1-1+2\right)\gt4 \\
\Rightarrow \quad \frac{1}{a^{1 / 2}} \cdot\left(a^{3 / 2}+a-2 a^{1 / 2}\right) \lt 4 & \Rightarrow a+a^{1 / 2}-2 \lt 4 \\
\Rightarrow \quad\left(a^{1 / 2}\right)^2+a^{1 / 2}-6 \lt 0 & \Rightarrow\left(a^{1 / 2}+3\right)\left(a^{1 / 2}-2\right) \lt 0 \\
\text { but, } a^{1 / 2}+3 \nless 0 \quad \therefore a^{1 / 2}-2 \lt 0 \quad \Rightarrow a \lt 4 \text { and } a\gt0 \quad \therefore a \in(0,4)
\end{array}$
Hence, the correct answer is option: 'C'.