AP EAMCET · Maths · Basic of Mathematics
The equation \(\mathrm{x}^{\frac{3}{4}\left(\log _2 \mathrm{x}\right)^2+\log _2 \mathrm{x}-\frac{5}{4}}=\sqrt{2}\) has
- A no real roots
- B only one real solution
- C exactly two real solutions
- D exactly three real solutions
Answer & Solution
Correct Answer
(D) exactly three real solutions
Step-by-step Solution
Detailed explanation
Let `\( y = \log_2 x \)`. `\( (2^y)^{\frac{3}{4}y^2+y-\frac{5}{4}} = 2^{\frac{1}{2}} \)` `\( y\left(\frac{3}{4}y^2+y-\frac{5}{4}\right) = \frac{1}{2} \)` `\( 3y^3+4y^2-5y-2 = 0 \)` `\( (y-1)(3y^2+7y+2) = 0 \)` `\( (y-1)(3y+1)(y+2) = 0 \)` `\( y = 1, y = -\frac{1}{3}, y = -2 \)`…
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