Maths-
General
Easy

Question

Statement-I : If sin x + cos x = square root of open parentheses y plus 1 over y close parentheses end root comma x element of left square bracket 0 comma pi right square bracket then x equals pi over 4 comma y equals 1

Statement-II : AM ≥ GM

  1. Statement-I is True, Statement-II is True; Statement-II is a correct explanation for Statement-I.
  2. Statement-I is True, Statement-II is True; Statement-II is NOT a correct explanation for Statement-I
  3. Statement-I is True, Statement-II is False
  4. Statement-I is False, Statement-II is True

hintHint:

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 = square root of open parentheses y plus 1 over y close parentheses end root, x ∈ [ 0, π ] then x = straight pi over 4, y = 1
    Here, we have,
    sinx + cosx = square root of open parentheses y plus 1 over y close parentheses end root
    we know
    cosx + sinx ≤ 2 and y plus 1 over y less or equal than 2,
    assuming y>0
    and in x∈[0,π] equality only exit
    if x=straight pi over 4 and y=1 as
    sin straight pi over 4 +cos straight pi over 4 =fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction equals square root of 2
    and fraction numerator 1 plus 1 over denominator 1 end fraction=2 … (1)
    when substituted in given equation
    sinx + cosx = square root of open parentheses y plus 1 over y close parentheses end root
    ⇒ √2 = √2 for these values it satisfies the equation
    thus, solution is, x=straight pi over 4, 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 =square root of left parenthesis n & a 1 a 2 horizontal ellipsis a n right parenthesis end root
    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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