Write ‘True’ or ‘False’ and justify your answer.The value of 2

1.	Write ‘True’ or ‘False’ and justify your answer.The value of 2 sin θ can be a + 1/a, where a is a positive number and a ≠ 1.
Answer» False Given: ‘a’ is a positive number and a≠1 ⇒ AM > GM (Arithmetic Mean (AM) of a list of non- negative real numbers is greater than or equal to the Geometric mean (GM) of the same list) If a and b be such numbers, then AM = (a + b)/2 and GM = √ab By assuming that a + 1/2 = 2 sin θ statement is be true. Similarly, AM and GM of a and 1/a are (a+1/a)/2 and √(a.1/a) respectively. By property, (a+1/a)/2 > √(a.1/a) a + 1/a > 2 ⇒ 2 sin θ > 2 (By our assumption) ⇒ sin θ > 1 But -1 ≤ sin θ ≤ 1 ∴ Our assumption is wrong and that 2 sin θ cannot be equal to a + 1/a

Write ‘True’ or ‘False’ and justify your answer.The value of 2 sin θ can be a + 1/a, where a is a positive number and a ≠ 1.

Answer»

False

Given: ‘a’ is a positive number and a≠1

⇒ AM > GM

(Arithmetic Mean (AM) of a list of non- negative real numbers is greater than or equal to the Geometric mean (GM) of the same list)

If a and b be such numbers, then

AM = (a + b)/2 and GM = √ab

By assuming that a + 1/2 = 2 sin θ statement is be true.

Similarly, AM and GM of a and 1/a are (a+1/a)/2 and √(a.1/a) respectively.

By property, (a+1/a)/2 > √(a.1/a)

a + 1/a > 2

⇒ 2 sin θ > 2 (By our assumption)

⇒ sin θ > 1

But -1 ≤ sin θ ≤ 1

∴ Our assumption is wrong and that 2 sin θ cannot be equal to a + 1/a