Verified Solution • 2025–26 Session moderate

QJEEM25Ma5b7e: JEE Main 2025 Mathematics - Class 11 JEEM25Ma5b7e Solution: If in the expansion of , the coefficients of and are 1 and -2 , …

If in the expansion of (1+x)p(1−x)q(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}} , the coefficients of xx and x2x^2 are 1 and -2 , respectively, then p2+q2\mathrm{p}^2+\mathrm{q}^2 is equal to :

Multiple Choice Options

  • A 8
  • B 20
  • C 13
  • D 18

Step-by-Step Verified Solution

amp;(1+x)p(1−x)q=(pC0+pC1x+pC2x2+…)(qC0−qC1x+qC2x2+…)amp; coff of x≡pC0qC1−pC1qC0=1amp;p−q=1amp; coff of x2≡pC0qC2−pC1qC1+pC2qC0=−2amp;q(q−1)2−pq+p(p−1)2=−2amp;q2−q−2pq+p2−p=−4amp;(p−q)2−(p+q)=−4amp;p+q=5amp;p=3amp;q=2amp; so p2+q2=13\begin{aligned}& (1+\mathrm{x})^{\mathrm{p}}(1-\mathrm{x})^{\mathrm{q}}=\left({ }^{\mathrm{p}} \mathrm{C}_0+{ }^{\mathrm{p}} \mathrm{C}_1 \mathrm{x}+{ }^{\mathrm{p}} \mathrm{C}_2 \mathrm{x}^2+\ldots\right)\left({ }^q \mathrm{C}_0-{ }^q \mathrm{C}_1 \mathrm{x}+{ }^q \mathrm{C}_2 \mathrm{x}^2+\ldots\right) \\& \text { coff of } \mathrm{x} \equiv{ }^{\mathrm{p}} \mathrm{C}_0{ }^{\mathrm{q}} \mathrm{C}_1-{ }^{\mathrm{p}} \mathrm{C}_1{ }^q \mathrm{C}_0=1 \\& \mathrm{p}-\mathrm{q}=1 \\& \text { coff of } \mathrm{x}^2 \equiv{ }^{\mathrm{p}} \mathrm{C}_0{ }^q \mathrm{C}_2-{ }^{\mathrm{p}} \mathrm{C}_1{ }^q \mathrm{C}_1+{ }^{\mathrm{p}} \mathrm{C}_2{ }^{\mathrm{q}} \mathrm{C}_0=-2 \\& \frac{\mathrm{q}(\mathrm{q}-1)}{2}-\mathrm{pq}+\frac{\mathrm{p}(\mathrm{p}-1)}{2}=-2 \\& \mathrm{q}^2-\mathrm{q}-2 \mathrm{pq}+\mathrm{p}^2-\mathrm{p}=-4 \\& (\mathrm{p}-\mathrm{q})^2-(\mathrm{p}+\mathrm{q})=-4 \\& \mathrm{p}+\mathrm{q}=5 \\& \mathrm{p}=3 \\& \mathrm{q}=2 \\& \text { so } \mathrm{p}^2+\mathrm{q}^2=13\end{aligned}

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