If `a lt b lt c lt d`, then for any real non-zero `lambda`, the quadratic equation `(x-a)(x-c)+lambda(x-b)(x-d)=0`,has real roots for

If a, b, c in R and the quadratic equation x^2 + (a + b) x + c = 0 has no real roots then

The least positive integeral value of real lambda so that the equation (x-a)(x-c)(x-e)+lambda (x-b)(x-d)=0, (a gt b gt c gt d gt e) has distinct real roots is __________.

If a, b, c, d in R such that a lt b lt c lt d, then roots of the equation (x-a)(x-c)+2(x-b)(x-d) = 0

lf 0 < a < b < c < d , then the quadratic equation ax^2 + [1-a(b+c)]x+abc-d=0 A) Real and distinct roots out of which one lies between c and d B) Real and distinct roots out of which one lies between a and b C) Real and distinct roots out of which one lies between b and c (D) non -real roots

Show that for any real numbers lambda , the polynomial P(x)=x^7+x^3+lambda , has exactly one real root.

If a ,b ,c ,d are four consecutive terms of an increasing A.P., then the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are a. non-real complex b. real and equal c. integers d. real and distinct

If a ,b ,c ,d are four consecutive terms of an increasing A.P., then the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are a. non-real complex b. real and equal c. integers d. real and distinct

If a , b , c , are real number such that a c!=0, then show that at least one of the equations a x^2+b x+c=0 and -a x^2+b x+c=0 has real roots.

Let a,b,c be the sides of a triangle. No two of them are equal and lambda in R If the roots of the equation x^2+2(a+b+c)x+3lambda(ab+bc+ca)=0 are real, then (a) lambda 5/3 (c) lambda in (1/5,5/3) (d) lambda in (4/3,5/3)

If a lt b lt c , then find the range of f(x)="|x-a|+|x-b|+|x-c|