Correct option is D
Given:
5 + 12/(x + 1/x) = 10
Therefore,
12/(x + 1/x) = 5
x + 1/x = 12/5
Multiplying both sides by 5x:
5x² − 12x + 5 = 0
Using the quadratic formula:
x = (12 ± √(144 − 100))/10
x = (12 ± √44)/10
x = (6 ± √11)/5
Hence,
a = (6 − √11)/5
and
b = (6 + √11)/5
Now, a + b = 12/5.
Therefore,
5a + 5b = 5(a + b) = 5 × 12/5 = 12.
Hence, Statement A is correct.
Next,
a = (6 − √11)/5 ≈ 0.537
and
b = (6 + √11)/5 ≈ 1.863
Therefore,
a < 1 < b
Hence, Statement D is correct.
Also,
b − a = 2√11/5
Therefore,
(b − a)² = 44/25
So,
(b − a)² + 6/25 = 44/25 + 6/25 = 50/25 = 2
Hence, Statement E is correct.
For Statement B:
b − a = 2√11/5 ≈ 1.326
Since 1.326 is greater than 1/2, Statement B is incorrect.
For Statement C:
Both a and b are positive, as a ≈ 0.537 and b ≈ 1.863.
Therefore, a < 0 < b is incorrect.
Thus, the correct statements are A, D and E.
Correct Answer: (d) A, D, E only
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