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Easy

Question

If S equals open curly brackets x element of N blank colon 2 plus log subscript 2 end subscript invisible function application square root of x plus 1 end root greater than 1 minus log subscript 1 divided by 2 end subscript invisible function application square root of 4 minus x to the power of 2 end exponent end root close curly brackets comma then

  1. S equals open curly brackets 1 close curly brackets  
  2. S equals Z  
  3. S equals N  
  4. None of these  

The correct answer is: S equals open curly brackets 1 close curly brackets


    2 plus log subscript 2 end subscript invisible function application square root of x plus 1 end root greater than 1 minus log subscript 1 divided by 2 end subscript invisible function application square root of 4 minus x to the power of 2 end exponent end root
    rightwards double arrow 1 plus log subscript 2 end subscript invisible function application square root of x plus 1 end root minus log subscript 2 end subscript invisible function application square root of 4 minus x to the power of 2 end exponent end root greater than 0
    rightwards double arrow log subscript 2 end subscript invisible function application 2 plus log subscript 2 end subscript invisible function application square root of x plus 1 end root minus log subscript 2 end subscript invisible function application square root of 4 minus x to the power of 2 end exponent end root greater than 0
    rightwards double arrow log subscript 2 end subscript invisible function application fraction numerator 2 square root of x plus 1 end root over denominator square root of 4 minus x to the power of 2 end exponent end root end fraction greater than 0 rightwards double arrow fraction numerator 2 square root of x plus 1 end root over denominator square root of 4 minus x to the power of 2 end exponent end root end fraction greater than 1
    rightwards double arrow 4 open parentheses x plus 1 close parentheses to the power of 2 end exponent greater than 4 minus x to the power of 2 end exponent rightwards double arrow 4 x to the power of 2 end exponent plus 8 x plus 4 greater than 4 minus x to the power of 2 end exponent rightwards double arrow 5 x to the power of 2 end exponent plus 8 x greater than 0 rightwards double arrow x greater than 0 blank open parentheses i close parentheses
    Also x plus 1 greater than 0 blank a n d blank 4 minus x to the power of 2 end exponent greater than 0
    rightwards double arrow x greater than negative 1 blank a n d minus 2 less than x less than 2 blank open parentheses i i close parentheses
    From Eqs. (i) and (ii), 0 less than x less than 2
    rightwards double arrow x equals 1 blank a s blank x element of N

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