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Algebra

Blazing goIITian

 Joined: 19 Nov 2007 Post: 375
28 Jun 2009 15:53:57 IST
0 People liked this
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let x1>x2>x3>x4> ...................>x6>0,A = x1 + ......... x6 .B = x1x3
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

1)let x1>x2>x3>x4> ...................>x6>0,A = x1 + ......... x6 .B = x1x3 + x1x5 + x3x5+ x2x4 + x2x6 +x4x6.C = x1x3x5 + x2x4x6 .then the eqnn 2x^3 - Ax^2 +Bx - C = o has a-excatly 1 real rootb-3 +ve rootsc- 3 -ve rootsd-none of these

2)let  A,B,C be the angles of triangle and tanA,tanB,tanC be the 3 roots of x- ax+bx2 - cx + d = o , then the fourth root is also root of -----------------

3)let p(x) be a polynomial with integral co-efficients such that p(0) and p(1) r both odd integers then the no of integral rootes of p(x) =0 is

Blazing goIITian

Joined: 24 Dec 2007
Posts: 1111
28 Jun 2009 16:10:23 IST
1 people liked this

1)Since A, B and C are all positive (coz of given conditions)...we have a cubic eqn of type Dx^3 - Ax^2 +Bx - C, where A,B,C,D are all positive.... so U can use Descartes' rule of signs

Scorching goIITian

Joined: 25 Jan 2009
Posts: 246
28 Jun 2009 17:41:08 IST
1 people liked this

Ans  2 )

in a traingle we know that

tan A +tan B + tan C = tan A .tan b .tan C

let the fourth root be ' y '

a - y = d / y

=> ay - y2 = d

=> y2 -ay  +d = 0

thus fourth root is also a root of  (x 2 -ax +d )

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