JEE Main Maths: Functions / Special Functions questions with solutions

Q1. Find the number of integral values of x satisfying sgn([15/(1+x²)]) = [1 + {2x}], where sgn(y), [y], {y} denote the signum function, greatest integer function, and fractional part function respectively.

1. 5
2. 7
3. 15
4. 16

Answer: 7

For any integer x, {2x}=0, so [1+{2x}]=1. We need sgn([15/(1+x²)])=1, which requires [15/(1+x²)]>=1, i.e., 15/(1+x²)>=1, i.e., x²<=14. Integer x with |x|<=3 (since 3²=9<=14 but 4²=16>14) gives x in {-3,-2,-1,0,1,2,3}: 7 values.