if a,b,c are the roots of the equation 2x3-4x2+5x-7=0 Find the value of the determinant1+a1111+b1111+c - Maths - Determinants

Question

i f   a , b , c   a r e   t h e   r o o t s
  o f   t h e   e q u a t i o n   2 x 3 - 4 x 2
+ 5 x - 7 = 0 F i n d   t h e   v a l u e   o f
  t h e   d e t e r m i n a n t 1 + a 1 1 1 1 + b 1 1 1 1
+ c

Varun Rawat · Accepted Answer

The given equation is,2x3 - 4x2 + 5x - 7 = 0Now, a + b + c = --42 = 2Now, ab + bc + ca = 52Now, abc = 72Let ∆ = 1+a1111+b1111+c=abc1a+11a1a1b1b + 11b1c1c1c+1Applying R1 = R1 + R2 + R3, we get∆ = abc1a+1b+1c+11a+1b+1c+11a+1b+1c+11b1b+11b1c1c1c+1=abc1a+1b+1c+11111b1b+11b1c1c1c+1=abcbc + ca + ab + abcabc001-111b0-11c+1=ab+bc+ca+abc11-0=ab+bc+ca+abc=52+72=122=6

i f a , b , c a r e t h e r o o t s o f t h e e q u a t i o n 2 x 3 - 4 x 2 + 5 x - 7 = 0 . F i n d t h e v a l u e o f t h e d e t e r m i n a n t 1 + a 1 1 1 1 + b 1 1 1 1 + c

$i f a, b, c a r e t h e r o o t s o f t h e e q u a t i o n 2 x^{3} - 4 x^{2} + 5 x - 7 = 0 . F i n d t h e v a l u e o f t h e d e t e r m i n a n t |\begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix}|$