Let α1, β1 are the roots of x2-6 x+p=0 and α2, β2 are the roots of x2-54 x+q=0. If α1, β1, α2, β2 form an increasing G.P., then find the value of (q-p).

Question

Let α1, β1 are the roots of x2-6 x+p=0 and α2, β2 are the roots of x2-54 x+q=0. If α1, β1, α2, β2 form an increasing G.P., then find the value of (q-p). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

text Let α1= A , β1= AR , α2= AR 2, β2= AR 3 text we have α1+β1=6 ⇒ A (1+ R )=6....(1) α1 β1=p ⇒ A2 R=p....(2) Also α2+β2=54 ⇒ operatornameAR2(1+ R )=54 ....(3) α2 β2= q A 2 R 5= q....(4) Now, on dividing equation (3) by equation (1), we get ( AR 2(1+ R )/ A (1+ R ))=(54/6)=9 ⇒ R 2=9 ∴ R =3 text (As it is an increasing G.P.) ∴ text On putting R =3 text in equation (1) A =(6/4)=(3/2) ∴ p = A 2 R =(9/4) × 3=(27/4) text and q = A 2 R 5=(9/4) × 243=(2187/4) text Hence q - p =(2187-27/4)=(2160/4)=540

Q. Let α1​,β1​ are the roots of x2−6x+p=0 and α2​,β2​ are the roots of x2−54x+q=0. If α1​,β1​,α2​,β2​ form an increasing G.P., then find the value of (q−p).

Q. Let $α_{1}, β_{1}$ are the roots of $x^{2} - 6 x + p = 0$ and $α_{2}, β_{2}$ are the roots of $x^{2} - 54 x + q = 0$ . If $α_{1}, β_{1}, α_{2}, β_{2}$ form an increasing G.P., then find the value of $(q - p)$ .