Let be the roots of the equation and be the roots of the equation ; . If are in G.P., then |p + q| equals [2026]

Question

Let $α, β$ be the roots of the equation $x^{2} - x + p = 0$ and $γ, δ$ be the roots of the equation $x^{2} - 4 x + q = 0$ ; $p, q \in Z$ . If $α, β, γ, δ$ are in G.P., then |p + q| equals [2026]

Smart Achievers · Accepted Answer

(3)

Given, $x^{2} - x + p = 0$

$α + β = 1 and α β = p$

Given, $x^{2} - 4 x + q = 0$

$γ + δ = 4 and γ δ = q$

Given, $α, β, γ, δ$ are in G.P.

Let $a, a r, a r^{2}, a r^{3}$ are in G.P.

Now, $α + β = a + a r = a (1 + r)$

But $α + β = 1 \Rightarrow a (1 + r) = 1$ ... (i)

Also, $γ + δ = a r^{2} + a r^{3} = a r^{2} (1 + r)$

But $γ + δ = 4 a r^{2} (1 + r) = 4$ ... (ii)

From(i) and (ii), $r^{2} = 4 \Rightarrow r = \pm 2$

If r = 2 then $a = \frac{1}{2} \notin I$ $∴ r \neq 2$

If r = –2 then a = –1, $a \in I$

$∴$ Roots are –1, 2, –4, 8

$∴ p = α β = - 2 and q = γ δ = - 32$

$∴$ |p + q| = |–2 –32| = 34

Solution

Q.
Let $α, β$ be the roots of the equation $x^{2} - x + p = 0$ and $γ, δ$ be the roots of the equation $x^{2} - 4 x + q = 0$ ; $p, q \in Z$ . If $α, β, γ, δ$ are in G.P., then |p + q| equals [2026]

1 16

2 32

3 34

4 38

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