Question

If \( a \) is a real number and \( 2\sinh^2 x - 3\cosh x + a = 0 \) has a solution, then the range of \( a \) is:

• A. \([3, \infty) \)
• B. \((0, \frac{25}{8}) \)
• C. \([3, \frac{25}{8}] \)
• D. \((-\infty, 3] \)

Hint:

For hyperbolic equations, always combine domain condition (\(\cosh x \ge 1\)) with discriminant condition.

Answer 1

The Correct Option is C

Concept: We use \(\sinh^2 x = \cosh^2 x - 1\) and reduce the equation into a quadratic in \(\cosh x\).

Step 1: Convert the equation in terms of \(\cosh x\).
\[ 2\sinh^2 x - 3\cosh x + a = 0 \] Using \(\sinh^2 x = \cosh^2 x - 1\): \[ 2(\cosh^2 x - 1) - 3\cosh x + a = 0 \] \[ 2\cosh^2 x - 3\cosh x + (a - 2) = 0 \]

Step 2: Let \(u = \cosh x\), with \(u \ge 1\).
\[ 2u^2 - 3u + (a - 2) = 0 \] For real solution in \(x\), discriminant must satisfy: \[ D \ge 0 \]

Step 3: Apply discriminant condition.
\[ D = 9 - 8(a - 2) = 25 - 8a \ge 0 \] \[ a \le \frac{25}{8} \]

Step 4: Check boundary condition from \(u \ge 1\).
For \(u = 1\): \[ 2 - 3 + (a - 2) = 0 \Rightarrow a = 3 \] Thus: \[ 3 \le a \le \frac{25}{8} \] \[ \boxed{[3, \frac{25}{8}]} \]