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Mathematics · Quadratic Equations

JEE Main 2026 — 5 April, Evening Shift — Question 25

Let α,βα, β be the roots of the equation x2−x+p=0x² - x + p = 0 and γ,δγ, δ be the roots the equation x2−4x+q=0,p,q∈Zx² - 4x + q = 0, p, q ∈ Z. If α,β,γ,δα, β, γ, δ are in G.P.G.P., then ∣p+q∣|p + q| equals :

  1. Option A:

    1616

  2. Option B:

    3232

  3. Option C:

    3434

    Correct
  4. Option D:

    3838

Answer: C

Step-by-step solution

Let α=a,β=ar,γ=ar2,δ=ar3\alpha=\mathrm{a}, \beta=\mathrm{ar}, \gamma=\mathrm{ar}^{2}, \delta=\mathrm{ar}^{3}

a+ar=1ar2+ar3=4⇒ar2(1+r)=4⇒r=2,a=13(reject asp∈z) or r=−2,a=−1 Now ∣p+q∣=∣a(ar)+ar2−ar3∣=∣a2r(1+r4)∣∣1×−2×17∣=34\begin{aligned} & \mathrm{a}+\mathrm{ar}=1 \\& \mathrm{ar}^{2}+\mathrm{ar}^{3}=4 \\& \Rightarrow \mathrm{ar}^{2}(1+\mathrm{r})=4 \\& \left.\Rightarrow \mathrm{r}=2, \mathrm{a}=\frac{1}{3} \text {(reject as} \mathrm{p} \in \mathrm{z}\right) \\& \text { or } \mathrm{r}=-2, \mathrm{a}=-1 \\& \text { Now }|\mathrm{p}+\mathrm{q}|=\left|\mathrm{a}(\mathrm{ar})+\mathrm{ar}^{2}-\mathrm{ar}^{3}\right| \\& =\left|\mathrm{a}^{2} \mathrm{r}\left(1+\mathrm{r}^{4}\right)\right| \\& |1 \times-2 \times 17|=34 \end{aligned}

Answer key and solution verified before publishing.

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Exam
JEE Main 2026
Paper
5 April, Evening Shift
Subject
Mathematics
Chapter
Quadratic Equations
Topic
Theory of Quadratic Equations