SATHEE: Theory of Equations 1 Question 51

Theory of Equations 1 Question 51

52. Solve for $x$ : $(5 + 2 \sqrt{6})^{x^{2} - 3} + (5 - 2 \sqrt{6})^{x^{2} - 3} = 10 (1985, 5 M)$

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Answer:

Correct Answer: 52. ( $\pm 2, \pm \sqrt{2}$ )

Solution:

Given, $(5 + 2 \sqrt{6})^{x^{2} - 3} + (5 - 2 \sqrt{6})^{x^{2} - 3} = 10$

Put $y = (5 + 2 \sqrt{6})^{x^{2} - 3} \Rightarrow (5 - 2 \sqrt{6})^{x^{2} - 3} = \frac{1}{y}$

From Eq. (i), $y + \frac{1}{y} = 10$

$\Rightarrow y^{2} - 10 y + 1 = 0 \Rightarrow y = 5 \pm 2 \sqrt{6}$

$\Rightarrow (5 + 2 \sqrt{6})^{x^{2} - 3} = 5 + 2 \sqrt{6}$

or $(5 + 2 \sqrt{6})^{x^{2} - 3} = 5 - 2 \sqrt{6}$

$\Rightarrow x^{2} - 3 = 1$ or $x^{2} - 3 = - 1$

$\Rightarrow x = \pm 2$ or $x = \pm \sqrt{2}$

$\Rightarrow x = \pm 2, \pm \sqrt{2}$

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Theory of Equations 1 Question 51

52. Solve for x : (5+26)x2−3+(5−26)x2−3=10(1985,5M)

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52. Solve for $x$ : $(5 + 2 \sqrt{6})^{x^{2} - 3} + (5 - 2 \sqrt{6})^{x^{2} - 3} = 10 (1985, 5 M)$