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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

$S = \{1\}$
$S = Z$
$S = N$
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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