Question

Hi , can anyone solve question 70 ?

Answer 1

roots have opposite sign
it means between roots zero must be lies ,
condition for zero lying ---
(1) af(k)<0
(2)D≥0
now,
(1) 3f(0)<0
3{a^2-3a+2}<0
(a-2)(a-1)<0
1<a<2

(2) D≥ 0
b^2-4ac≥0
{2(a^2+1)}^2-4.3(a^2-3a+2)≥0

(a^2+1)^2-3(a^2-3a+2)≥0

a^4+2a^2+1-3a^2+9a-6≥0

a^4-a^2+9a-5≥0

f(a)=a^4-a^2+9a-5≥0
check decreasing or increasing function
put a=1
f(1)=1-1+9-5=4≥0
put a=2
f(2)=16-4+18-5≥0
hence there is no chance of negative between 1 and 2 so,
a^4-a^2+9a-5≥0 for all real numbers

now put a value from (1) and (2) number line then you find a€(1,2)
€ mean belongs to

Answer 2

Answer:

Sn = NP + 1/2n ( n - 1) Q

Sn = n ( P + 1/2 (n-1) Q)

Sn = n/2 ( P/2 + 1(n-1) Q)

So Common Difference is Q