If $k \in R$ is such that the equation $2 \cosh ^2 x=3 \sinh x+k$ has no real solution, then which of the…
If $k \in R$ is such that the equation $2 \cosh ^2 x=3 \sinh x+k$ has no real solution, then which of the following is correct?
- $k < \frac{1}{2}$
- $k < \frac{3}{8}$
- $k < \frac{7}{8}$
- $k < \frac{5}{8}$
Solution
Given, $2 \cosh ^2 x-3 \sinh x-k=0$ has no real solution.
$\because \quad \cosh ^2 x-\sinh ^2 x=1$
$\Rightarrow \quad \cosh ^2 x=1+\sinh ^2 x$
Then, $2\left(1+\sinh ^2 x\right)-3 \sinh x-k=0$
$\Rightarrow \quad 2 \sinh ^2 x-3 \sinh x+(2-k)=0$
Above equation has no real solution.
$\therefore D < 0$
$\Rightarrow \quad 9 \sinh ^2 x-4(2-k)\left(2 \sinh ^2 x\right) < 0$
$\Rightarrow \quad \sinh ^2 x(9-16+8 k) < 0 \Rightarrow 8 k-7 < 0$
$\Rightarrow \quad k < 7 / 8$
Asked in: AP EAMCET 2022 (08 Jul Shift 2)
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