Exams Joint Entrance Examination Mathematics Binomial Theorem Question 30 Question 30 of 347 Medium MCQ
If in the expansion of ( 1 + x ) p ( 1 − x ) q (1+x{)}^{p}(1-x{)}^{q} ( 1 + x ) p ( 1 − x ) q , the coefficients of x x x and x 2 {x}^{2} x 2 are 1 and -2 , respectively, then p 2 + q 2 {p}^{2}+{q}^{2} p 2 + q 2 is equal to :
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\[ ( 1 + x ) p ( 1 − x ) q = ( p C 0 + p C 1 x + p C 2 x 2 + … ) ( q C 0 − q C 1 x + q C 2 x 2 + … ) coff of x ≡ p C 0 q C 1 − p C 1 q C 0 = 1 p − q = 1 coff of x 2 ≡ p C 0 q C 2 − p C 1 q C 1 + p C 2 q C 0 = − 2 q ( q − 1 ) 2 − p q + p ( p − 1 ) 2 = − 2 q 2 − q − 2 p q + p 2 − p = − 4 ( p − q ) 2 − ( p + q ) = − 4 p + q = 5 p = 3 q = 2 so p 2 + q 2 = 13 \begin{aligned}
& (1+\mathrm{x})^{\mathrm{p}}(1-\mathrm{x})^{\mathrm{q}}=\left({ }^{\mathrm{p}} \mathrm{C}_0+{ }^{\mathrm{p}} \mathrm{C}_1 \mathrm{x}+{ }^{\mathrm{p}} \mathrm{C}_2 \mathrm{x}^2+\ldots\right)\left({ }^q \mathrm{C}_0-{ }^q \mathrm{C}_1 \mathrm{x}+{ }^q \mathrm{C}_2 \mathrm{x}^2+\ldots\right) \\
& \text { coff of } \mathrm{x} \equiv{ }^{\mathrm{p}} \mathrm{C}_0{ }^{\mathrm{q}} \mathrm{C}_1-{ }^{\mathrm{p}} \mathrm{C}_1{ }^q \mathrm{C}_0=1 \\
& \mathrm{p}-\mathrm{q}=1 \\
& \text { coff of } \mathrm{x}^2 \equiv{ }^{\mathrm{p}} \mathrm{C}_0{ }^q \mathrm{C}_2-{ }^{\mathrm{p}} \mathrm{C}_1{ }^q \mathrm{C}_1+{ }^{\mathrm{p}} \mathrm{C}_2{ }^{\mathrm{q}} \mathrm{C}_0=-2 \\
& \frac{\mathrm{q}(\mathrm{q}-1)}{2}-\mathrm{pq}+\frac{\mathrm{p}(\mathrm{p}-1)}{2}=-2 \\
& \mathrm{q}^2-\mathrm{q}-2 \mathrm{pq}+\mathrm{p}^2-\mathrm{p}=-4 \\
& (\mathrm{p}-\mathrm{q})^2-(\mathrm{p}+\mathrm{q})=-4 \\
& \mathrm{p}+\mathrm{q}=5 \\
& \mathrm{p}=3 \\
& \mathrm{q}=2 \\
& \text { so } \mathrm{p}^2+\mathrm{q}^2=13
\end{aligned} ( 1 + x ) p ( 1 − x ) q = ( p C 0 + p C 1 x + p C 2 x 2 + … ) ( q C 0 − q C 1 x + q C 2 x 2 + … ) coff of x ≡ p C 0 q C 1 − p C 1 q C 0 = 1 p − q = 1 coff of x 2 ≡ p C 0 q C 2 − p C 1 q C 1 + p C 2 q C 0 = − 2 2 q ( q − 1 ) − pq + 2 p ( p − 1 ) = − 2 q 2 − q − 2 pq + p 2 − p = − 4 ( p − q ) 2 − ( p + q ) = − 4 p + q = 5 p = 3 q = 2 so p 2 + q 2 = 13 \] Answer the question to unlock