Maths-
General
Easy

Question

if sin space theta plus cos space theta equals p and sin cubed space theta plus cos cubed space theta equals q then straight p open parentheses straight p squared minus 3 close parentheses equals

  1. q
  2. 2 q
  3. -q
  4. negative 2 q

The correct answer is: negative 2 q


    Step by step solution:
    Given,
    sin space theta plus cos space theta equals p
    sin cubed space theta plus cos cubed space theta equals q
    Now, p squared equals open parentheses sin theta plus cos theta close parentheses squared
    rightwards double arrow p squared equals sin squared theta plus cos squared
theta plus 2 sin theta cos theta
    rightwards double arrow p squared equals 1 plus 2 sin theta cos theta
    rightwards double arrow p squared minus 1 equals 2 sin theta cos theta
    rightwards double arrow sin theta cos theta equals fraction numerator p squared minus 1 over denominator 2 end fraction  -(i)
    q equals sin cubed theta plus cos cubed thetaequals left parenthesis sin theta plus cos theta right parenthesis left parenthesis sin squared theta pluscos squared theta minus sin theta cos theta right parenthesis
    equals p left parenthesis 1 minus sin theta cos theta right parenthesis
    equals p open parentheses 1 minus fraction numerator p squared minus 1 over denominator 2 end fraction close parentheses
    equals p open parentheses fraction numerator 2 minus p squared plus 1 over denominator 2 end fraction close parentheses
    rightwards double arrow 2 q equals p left parenthesis 3 minus p squared right parenthesis
    rightwards double arrow p left parenthesis p squared minus 3 right parenthesis equals negative 2 q
    Hence, option(d) is the correct option.

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