Show that all the roots of the equation `a_(1)z^(3)+a_(2)z^(2)+a_(3)z+a_(4)=3,`
`(where|a_(i)|le1,i=1,2,3,4,)` lie
outside the circle with centre at origin and radius `2//3.`

The number of real roots of the equation 1+a_(1)x+a_(2)x^(2)+………..a_(n)x^(n)=0 , where |x| lt (1)/(3) and |a_(n)| lt 2 , is

The number of all three element subsets of the set {a_(1),a_(2),a_(3)......a_(n)} which contain a_(3) , is

If one quarter of all three element subsete of the set A={a_(1),a_(2),a_(3),......,a_(n)} contains the element a_(3) , then n=

If A_(1),A_(2),A_(3),...,A_(n),a_(1),a_(2),a_(3),...a_(n),a,b,c in R show that the roots of the equation (A_(1)^(2))/(x-a_(1))+(A_(2)^(2))/(x-a_(2))+(A_(3)^(2))/(x-a_(3))+…+(A_(n)^(2))/(x-a_(n)) =ab^(2)+c^(2) x+ac are real.

Show that for any real numbers a_(3),a_(4),a_(5),……….a_(85) , the roots of the equation a_(85)x^(85)+a_(84)x^(84)+……….+a_(3)x^(3)+3x^(2)+2x+1=0 are not real.

If the quadratic equation a_(1)x^(2)-a-(2)x+a_(3)=0 where a_(1),a_(2),a_(3) in N has two distinct real roots belonging to the interval (1,2) then least value of a_(1) is_______

find all the possible triplets (a_(1), a_(2), a_(3)) such that a_(1)+a_(2) cos (2x)+a_(3) sin^(2) (x)=0 for all real x.

If a_(n)=3-4n , then show that a_(1),a_(2),a_(3), … form an AP. Also, find S_(20) .

If alpha and beta are roots of the equation x^(2)-3x+1=0 and a_(n)=alpha^(n)+beta^(n)-1 then find the value of (a_(5)-a_(1))/(a_(3)-a_(1))

The number of increasing function from f : AtoB where A in {a_(1),a_(2),a_(3),a_(4),a_(5),a_(6)} , B in {1,2,3,….,9} such that a_(i+1) gt a_(i) AA I in N and a_(i) ne i is