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Maths Question 1 – JEE-MAIN 2026

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

Recall the relationships between the roots and coefficients of a quadratic equation.

Step 1: Relate roots to coefficients and G.P. terms✦ Active

Let the four terms in G.P. be α=a, β=ar, γ=ar2, δ=ar3. For the first equation x2−x+p=0, the sum of roots is α+β=1⟹a(1+r)=1 and the product of roots is αβ=p⟹a2r=p. For the second equation x2−4x+q=0, the sum of roots is γ+δ=4⟹ar2(1+r)=4 and the product of roots is γδ=q⟹a2r5=q.

Step 2: Solve for common ratio r and first term a○ Expand

Divide the sum of roots equation for the second quadratic by that of the first: ar2(1+r)a(1+r)=41⟹r2=4. This gives r=2 or r=−2. If r=2, then a(1+2)=1⟹3a=1⟹a=1/3. This would make p=a2r=(1/3)2⋅2=2/9, which is not an integer. Thus, r≠2. If r=−2, then a(1−2)=1⟹−a=1⟹a=−1.

💡 Teacher's Secret Hint

Remember that p and q must be integers, which helps eliminate one possible value for r.

Step 3: Calculate p, q, and |p+q|○ Expand

Using a=−1 and r=−2: p=a2r=(−1)2(−2)=−2. q=a2r5=(−1)2(−2)5=1(−32)=−32. Both p=−2 and q=−32 are integers. Finally, calculate |p+q|=|−2+(−32)|=|−34|=34.

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