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Problem 12. If each pair of the following three equations x2 + ax + b= 0, x2 + cx + d= 0, x2+ ex+f=0 has exactly one root in common, then show that (a+ c + e)= = 4 (ac + ce + ea-b-d-f)Problem 13. If the three equation x2 + ax + 12 =0, x2 + bx + 15=0 and x2 + (a+ b) x + 36 =0 have a common possible root then find a and b and the roots.Problem 14. If a is a root of the equation ax2 + bx + c=0, B is a root of the equation - ax2 + bx + c=0 a and y is a root of the equation x' + bx + c=0 then NI prove that y lies between a and B.Problem 15. Find the integral solutions of the equation x2 + x = y4 + y3 + y2+yProblem 16. A quadratic trinomial p(x) = ax2 +bx + c is such that Ip(x)| 0 is valid for atleast one real value of 'x' .Problem 19. Find the values of 'p' for which the inequality , |2 - log2 P X 2 p+ 1 P P + 2x |1 + 1082 p + 1 - 2 1 + 1082 > 0 p+1 is valid for all real x .> Problem 20. If ax2 - bx + c =0 have two distinct roots lying in the interval (0, 1), a, b, c E N, then prove that log, abc 2 2