Question

find the total resistance in picture

Answer 1

total resistance :-

b and c are in series then their resistance is

R + R
5 + 5
= 10 ohm

similarly for d and e
= 10 ohm

now a , ( b+c) , ( d + e) are in parallel connection

1 / R + 1 / R + 1 / R
1/ 5 + 1/ 10 + 1 /10
(2 + 1 + 1) / 10
= 5/10 ohm

total resistance = 5/ 10 = 1 / 2 ohm

note : please see the attachment.

Answer 2

f(x) = kx³ – 8x² + 5

Roots are α – β , α & α +β

Sum of roots = – (-8)/k

Sum of roots = α – β + α + α +β = 3α

= 3α = 8/k

= k = 8/3α

or we can solve as below

f(x) = (x – (α – β)(x – α)(x – (α +β))

= (x – α)(x² – x(α+β + α – β) + (α² – β²))

= (x – α)(x² – 2xα + (α² – β²))

= x³ – 2x²α + x(α² – β²) – αx² +2α²x – α³ + αβ²

= x³ – 3αx² + x(3α² – β²) + αβ² – α³

= kx³ – 3αkx² + xk(3α² – β²) + k(αβ² – α³)

comparing with

kx³ – 8x² + 5

k(3α² – β²) = 0 => 3α² = β²

k(αβ² – α³) = 5

=k(3α³ – α³) = 5

= k2α³ = 5

3αk = 8 => k = 8/3α

(8/3α)2α³ = 5

=> α² = 15/16

=> α = √15 / 4