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_(ij) '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

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

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.

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

If the expansion in powers of x of the function (1)/((1-ax)(1-bx)) is a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+ .. . . then a_(n) is:

Let ( a_(n) ) be a G.P. such that ( a_(4))/( a_(6)) =( 1)/( 4) and a_(2) + a_(5) = 216 . Then a_(1) =

Find the principal value of the arguments of z_(1)=2+2i,z_(2)=-3+3i,z_(3)=-4-4iand z_(4)=5-5i,where i=sqrt(-1).

If a_(1),a_(2),a_(3),a_(4) and a_(5) are in AP with common difference ne 0, find the value of sum_(i=1)^(5)a_(i) " when " a_(3)=2 .

If a_(1) , a_(2), "……………" a_(n) are in H.P., then the expression a_(1) a_(2) + a_(2) a_(3)+"…….."+ a_(n-1) a_(n) is equal to :