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.`

In a square matrix A of order 3 the elements a_(ii) 's are the sum of the roots of the equation x^(2) - (a+b) x + ab =0, a_(i,i+1) 's are the product of the roots, a_(i,i-1) 's are all unity and the rest of the elements are all zero. The value of the det (A) is equal to

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.

Cnstruct a_(2xx2) matrix, where a_(ij) = |-2i+3j|

Write the first six terms of the following sequence : a_(1)=a_(2)=1, a_(n)=a_(n-1)+a_(n-2)(n ge 3) .

Find the argument s of z_(1)=5+5i,z_(2)=-4+4i,z_(3)=-3-3i and z_(4)=2-2i, where i=sqrt(-1).

Differentiate |x|+a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+...+a_(n-1)x+a_(n)

Let A be nxxn matrix given by A=[(a_(11),a_(12),a_(13)……a_(1n)),(a_(21),a_(22),a_(23)…a_(2n)),(vdots, vdots, vdots),(a_(n1),a_(n2),a_(n3).a_("nn"))] Such that each horizontal row is arithmetic progression and each vertical column is a geometrical progression. It is known that each column in geometric progression have the same common ratio. Given that a_(24)=1,a_(42)=1/8 and a_(43)=3/16 Let d_(i) be the common difference of the elements in with row then sum_(i=1)^(n)d_(i) is