Q65 / 160 · AP EAMCET 2022 (08 Jul Shift 2) · Mathematics

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?

  • A
    $k < \frac{1}{2}$
  • B
    $k < \frac{3}{8}$
  • Correct answer
    C
    $k < \frac{7}{8}$
  • D
    $k < \frac{5}{8}$

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. $\therefore D < 0$ $\Rightarrow \quad 9 \sinh ^2 x-4(2-k)\left(2 \sinh ^2 x\right) < 0$ $\Rightarrow \quad \sinh ^2 x(9-16+8 k) < 0 \Rightarrow 8 k-7 < 0$ $\Rightarrow \quad k < 7 / 8$

Hence, the correct answer is option: 'C'.

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