Paragraph: Consider two quadratic polynomials as P(x)=x^2-2 x+ (a^2-11 a+11 ) and Q(x)=x^2+(3-b) x+b , where a, b R . Question: If the graph of P(x) lies above and below the x -axis then the largest integral value of a , is

Step-by-step solution

(i) Given, P(x)=x^2-2 x+ (a^2-11 a+11 ) As, graph of P(x) lies above and below the x -axis so D 0 aligned & 4-4 (a^2-11 a+11 ) 0 & (a^2-11 a+11 )-1 0 & a^2-11 a+10 0 & (a-1)(a-10) 0 & 1 a 10 aligned Hence, the largest integral value of a equals 9. Ans. (ii) Given, Q(x)=x^2+(3-b) x+b As, minimum value of Q(x) is positive, so aligned & D 0 & (3-b)^2-4 b 0 & b^2-10 b+9 0 & (b-9)(b-1) 0 & 1 b 9 aligned So, possible integral values of b=2,3,4,5,6,7,8 Hence, the sum of possible integral values of b is 35 . (iii) (A) Obvi