SATHEE: Theory of Equations 1 Question 30

Theory of Equations 1 Question 30

31. The equation $x^{\frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4}} = \sqrt{2}$ has

(1989; 2M)

(a) atleast one real solution

(b) exactly three real solutions

(d) complex roots

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Solution:

Given, $x^{\frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4}} = \sqrt{2}$

$\Rightarrow \frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4} = \log_{x} \sqrt{2}$

$\Rightarrow \frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4} = \frac{1}{2 \log_{2} x}$

$\Rightarrow 3 {(\log_{2} x)}^{3} + 4 {(\log_{2} x)}^{2} - 5 (\log_{2} x) - 2 = 0$

Put $\log_{2} x = y$

$∴ 3 y^{3} + 4 y^{2} - 5 y - 2 = 0$

$\Rightarrow (y - 1) (y + 2) (3 y + 1) = 0$

$\Rightarrow y = 1, - 2, - 1 / 3$

$\Rightarrow \log_{2} x = 1, - 2, - 1 / 3$

$\Rightarrow x = 2, \frac{1}{2^{1 / 3}}, \frac{1}{4}$

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Theory of Equations 1 Question 30

31. The equation x34(log2⁡x)2+log2⁡x−54=2 has

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31. The equation $x^{\frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4}} = \sqrt{2}$ has