Concept of Quadratic equation

Question

If !!a, b!! are roots of equation !!p(x^2-x)+x+5=0!! and !!p_1!! and !!p_2!! are two values of !!p!! obtained from relation

!!a/b+ b/a = 4/5!!. Then, !!p_1/p_2^2 + p_2/p_1^2= ?!!

a) !!1000!!

b) !!2000!!

c) !!4000!!

d) None

Solution

Equation: !!p(x^2-x)+x+5=0=> px^2-(p-1)x+5=0!!, Roots: !!a,b!!

!!=> a+b = (p-1)/p and ab = 5/p......(1)!!

!!a/b+b/a = 4/5!!

!!=> a^2+b^2 = 4/5 ab!!

!!=> (a+b)^2 - 2ab = 4/5 ab!!

From equation (1). we will get

!!=>((p-1)/p)^2 = 14/5 xx 5/p!!

!!=> (p-1)^2 = 14p!!

!!=> p^2- 16p+1=0!!

!! p_1 + p_2= 16!! and !!p_1p_2=1 ...(2)!!

We have to find,

!!p_1/p_2^2 + p_2/p_1^2= (p_1^3+p_2^2)/ (p_1p_2)^2!!

!!=> ((p_1+p_2)^3-3p_1p_2(p_1+p_2))/(p_1p_2)^2!!

!!=> ((16)^3- 3xx1xx16)/(1)^2!!

!!=> 4048!!

Therefore, option (d) is correct.