Question

Statement-I : If sin x + cos x = then

Statement-II : AM ≥ GM

- Statement-I is True, Statement-II is True; Statement-II is a correct explanation for Statement-I.
- Statement-I is True, Statement-II is True; Statement-II is NOT a correct explanation for Statement-I
- Statement-I is True, Statement-II is False
- Statement-I is False, Statement-II is True

Hint:

### In this question, given two statements. It is like assertion and reason. Statement1 is assertion and statement 2 is reason, Find the statement 1 is correct or not and the statement 2 correct or not if correct then is its correct explanation.

## The correct answer is: Statement-I is True, Statement-II is True; Statement-II is a correct explanation for Statement-I.

### Here, we have to find the which statement is correct and if its correct explanation or not.

Firstly,

Statement-I: If sinx + cosx = , x ∈ [ 0, π ] then x = , y = 1

Here, we have,

sinx + cosx =

we know

cosx + sinx ≤ 2 and ,

assuming y>0

and in x∈[0,π] equality only exit

if x= and y=1 as

sin +cos =

and =2 … (1)

when substituted in given equation

sinx + cosx =

⇒ √2 = √2 for these values it satisfies the equation

thus, solution is, x=, y=1

Therefore, statement-I is correct.

Now,

Statement-II: AM ≥ GM

For all,

We know that the inequality relation between AM and GM is

Let a1, a2…, can be n positive real numbers. The Arithmetic Mean and Geometric Mean defined as:

AM =

(a1+a2+⋯+an)/n

GM =

The AM–GM inequality states that

AM ≥ GM

Therefore, Statement-II is correct, but it is not explanation of statement-I.

The correct answer is, Statement-I is true, Statement-II is true; Statement-II is NOT a correct explanation for Statement-I

In this question, we have to find the statements are the correct or not and statement 2 is correct explanation or not, is same as assertion and reason. Here, Start solving first Statement and try to prove it. Then solve the Statement-II. Always, the AM–GM inequality states that AM ≥ GM.

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