IPMAT Indore 2019 (MCQ) - Let , be the roots of x^2 - x + p = 0 and , be the roots of x^2 - 4x + q = 0 where p and q are integers. If , , , are in geometric progression then p + q is | PYQs + Solutions

IPMAT Indore 2019

Algebra

Progression & Series

Hard

Let $\alpha, \beta$ be the roots of $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of $x^2 - 4x + q = 0$ where p and q are integers. If $\alpha, \beta, \gamma, \delta$ are in geometric progression then $p + q$ is

-34

-38

✅ Correct Option: 1

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Alternate solution

α, β

are roots of

x² - x + p = 0

, so

α + β = 1, αβ = p

γ, δ

are roots of

x² - 4x + q = 0, so γ + δ = 4, γδ = q

α, β, γ, δ

are in

GP

Let the

GP

be:

a, ar, ar², ar³

Then:

a + ar = 1

and

ar² + ar³ = 4

From these:

a(1 + r) = 1

and

ar²(1 + r) = 4

Dividing:

r² = 4

, so

r = ±2

Since we need integer

p, q, try r = -2:

a(1 - 2) = 1 ⟹ a = -1

GP: -1, 2, -4, 8

Therefore:

p = (-1)(2) = -2

q = (-4)(8) = -32

Answer:

p + q = -34