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    Maths-
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    Question

    Statement - I The value of x for which (sin x + cos x)1 + sin 2x = 2, when 0 ≤ x ≤ , is straight pi divided by 4 only.

    Statement - II The maximum value of sin x + cos x occurs when x =straight pi divided by 4

    1. Statement-I is true, Statement-II is true ; Statement-II is 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:

    Here two statements are given.  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 correct explanation for Statement-I.

      Here we have to find the which statement is correct and if its correct explanation or not.
      Firstly, statement 1: The value of x for which left parenthesis sin space x space plus space cos space x right parenthesis to the power of left parenthesis 1 plus sin 2 x right parenthesis end exponent = 2, when 0 ≤ x ≤, is straight pi over 4 only.
      So, we have,
      left parenthesis sin space x space plus space cos space x right parenthesis to the power of left parenthesis 1 plus sin 2 x right parenthesis end exponent 
      =left parenthesis s i n x plus c o s x right parenthesis to the power of left parenthesis s i n squared x plus c o s squared x plus 2 s i n x c o s x right parenthesis end exponent [ since, sin2x + cos2x = 1 and sin2x = 2sinx.cosx]
      =left parenthesis sin space x space plus space cos space x space right parenthesis to the power of left parenthesis sin x space plus space cos x right parenthesis squared end exponent [ a2 + b2 + 2ab = (a+b)2]
      Now , at x = straight pi over 4, we have,
      =open parentheses sin straight pi over 4 plus cos straight pi over 4 close parentheses to the power of open parentheses sin straight pi over 4 plus cos straight pi over 4 close parentheses squared end exponent
      =open parentheses fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses to the power of open parentheses fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared end exponent
      = open parentheses fraction numerator 2 over denominator square root of 2 end fraction close parentheses to the power of open parentheses fraction numerator 2 over denominator square root of 2 end fraction close parentheses squared end exponent
      = open parentheses square root of 2 close parentheses to the power of open parentheses square root of 2 close parentheses squared end exponent
      = open parentheses square root of 2 close parentheses squared
      = 2
      Therefore, left parenthesis s i n x plus c o s x right parenthesis to the power of left parenthesis 1 plus s i n 2 x right parenthesis end exponent = 2 is True.
      Now for statement 2 – The maximum value of sinx + cosx occur when x = straight pi over 4,
      Let y = sinx + cosx
      fraction numerator d y over denominator d x end fraction = cosx – sinx
      fraction numerator d y over denominator d x end fraction = 0
      cosx = sinx ,
      tanx = 1 = tan straight pi over 4
      we know, x = n π + straight pi over 4
      in 0 ≤ x ≤ straight pi over 4
      x = straight pi over 4
      fraction numerator d squared y over denominator d x squared end fraction= -sinx – cosx < 0
      therefore, sinx + cosx is maximum at π/4. And statement 2 is correct explanation because √2 is maximum value in π/4. And it is the only case which satisfies the statement 1 at π/4.
      The correct answer is Statement-I is true, Statement-II is true; Statement-II is 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.

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