The sum of all the integral value(s) of `k in [0, 15]` for which the inequality `1 + log_5 (x^2+1) <= log_5 (kx^2 + 4x+ k)` is true for all `x in R`, is

The sum of all positive integral values of a , where a in[1,500] for which the equation [x]^(3)+x-a=0 has a solution is:

Sum of the squares of all integral values of a for which the inequality x^(2) + ax + a^(2) + 6a lt 0 is satisfied for all x in ( 1,2) must be equal to

Sum of the squares of all integral values of a for which the inequality x^(2) + ax + a^(2) + 6a lt 0 is satisfied for all x in ( 1,2) must be equal to

The number of integer values of x for which the inequality "log "_(10) ((2x-2007)/(x+1))ge0, is true, is

The sum of all integral values of 'a' for which the equation 2x ^(2) -(1+2a) x+1 +a=0 has a integral root.

The sum of all integral values of 'a' for which the equation 2x ^(2) -(1+2a) x+1 +a=0 has a integral root.

The number of integral values of k for which the inequality x^(2)-2(4k-1)x+15k^(2)-2k-7 ge0 hold for x in R are _____________

The number of integral values of k for which the inequality x^(2)-2(4k-1)x+15k^(2)-2k-7 ge0 hold for x in R are _____________

If the inequality (x^(2)-kx-2)/(x^(2)-3x+4)>-1 for every x in R and S is the sum of all integral values of k, If the inequalitythen find the value of S^(2)

The number of negative integral values of x satisfying the inequality log_((x+(5)/(2)))((x-5)/(2x-3))^(2)lt 0 is :