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General
Easy

Question

text  If  end text 0 less or equal than x less or equal than pi comma 81 to the power of sin squared space x end exponent plus 81 to the power of cos squared space x end exponent equals 30 text  then  end text x equals

  1. pi over 6
  2. pi over 4
  3. pi over 15
  4. pi over 8

The correct answer is: pi over 6


    Step by step solution:
    Given, 81 to the power of sin squared x end exponent plus 81 to the power of cos squared x end exponent equals 30
    rightwards double arrow 81 to the power of 1 minus cos squared x end exponentplus 81 to the power of cos squared x end exponent equals 30
    rightwards double arrow 81 over 81 to the power of cos squared x end exponentplus 81 to the power of cos squared x end exponent equals 30
    Let 81 to the power of cos squared x end exponent equals z
    Now, 81 over z plus z equals 30
    rightwards double arrow z squared minus 30 z plus 81 equals 0
    rightwards double arrow z equals fraction numerator 30 plus-or-minus square root of 900 minus 324 end root over denominator 2 end fraction
    rightwards double arrow z equals fraction numerator 30 plus-or-minus 24 over denominator 2 end fraction=54 over 2 o r space 6 over 2equals 27 space o r space 3
    CASE 1: z= 27
    rightwards double arrow 81 to the power of cos squared x end exponent equals 27
    rightwards double arrow 3 to the power of 4 cos squared x end exponent equals 3 cubed
    rightwards double arrow 4 cos squared x equals 3
    rightwards double arrow cos squared x equals 3 over 4
    rightwards double arrow cos space x space equals plus-or-minus square root of 3 over 4 end root
equals plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction
    rightwards double arrow space x space equals space straight pi over 6 o r space fraction numerator 3 pi over denominator 4 end fraction
    CASE2: z = 3
    rightwards double arrow 81 to the power of cos squared x end exponent equals 3
    rightwards double arrow 3 to the power of 4 cos squared x end exponent equals 3 to the power of 1
    rightwards double arrow 4 cos squared x equals 1
    rightwards double arrow cos squared x equals 1 fourth
rightwards double arrow cos x equals square root of 1 fourth end root
    rightwards double arrow cos x space equals 1 half
    rightwards double arrow x equals pi over 3
    Thus, from the options pi over 6 is the correct option.
    Hence, option(a) is the correct option.

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