If (1+x+x2)n=a0+a1x+a2x2+.....+a2nx2n,then Column – I Column – IIA)a02−a12+a22−a32+.........+a2n2 is equalsP)12(3n+an)B)a0a2−a1a3+a2a4......+a2n−2a2n is equalsQ)12(3n−1)C)a0+a1+a2+......+an is equalsR)an+1D)a2+a4+......+a2n is equalsS)an

A) a02−a12+a22..........+a2n2=coefficient x2nin(1+x+x2)n(x2−x+1)nB) a0a2−a1a3+.......+a2n−2a2n= coefficient x−2 in (1+x+x2)n(1−1x+1x2)nC) ar=a2n−rD) let fx=(1+x+x2)na0+a2+a4......+a2n=f(1)+f(−1)2a0=1

If (1+x+x2)n=a0+a1x+a2x2+.....+a2nx2n,then Column – I Column – IIA)a02−a12+a22−a32+.........+a2n2 is equalsP)12(3n+an)B)a0a2−a1a3+a2a4......+a2n−2a2n is equalsQ)12(3n−1)C)a0+a1+a2+......+an is equalsR)an+1D)a2+a4+......+a2n is equalsS)an

A) a02−a12+a22..........+a2n2=coefficient x2nin(1+x+x2)n(x2−x+1)nB) a0a2−a1a3+.......+a2n−2a2n= coefficient x−2 in (1+x+x2)n(1−1x+1x2)nC) ar=a2n−rD) let fx=(1+x+x2)na0+a2+a4......+a2n=f(1)+f(−1)2a0=1

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If ${(1 + x + x^{2})}^{n} = a_{0} + a_{1} x + a_{2} x^{2} + ..... + a_{2 n} x^{2 n}, t h e n$
Column – I Column – II
A) $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + ......... + a_{2 n}^{2} is$ equals P) $\frac{1}{2} (3^{n} + a_{n})$
B) $a_{0} a_{2} - a_{1} a_{3} + a_{2} a_{4} ...... + a_{2 n - 2} a_{2 n}$ is equals Q) $\frac{1}{2} (3^{n} - 1)$
C) $a_{0} + a_{1} + a_{2} + ...... + a_{n}$ is equals R) $a_{n + 1}$
D) $a_{2} + a_{4} + ...... + a_{2 n}$ is equals S) $a_{n}$

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$A - Q; B - P; C - R; D - S$

$A - S; B - R; C - P; D - Q$

$A - P; B - Q; C - S; D - R$

$A - P; B - S; C - Q; D - R$

answer is B.

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Detailed Solution

$A) a_{0}^{2} - a_{1}^{2} + a_{2}^{2} .......... + a_{2 n}^{2} =$ coefficient $x^{2 n} i n {(1 + x + x^{2})}^{n} {(x^{2} - x + 1)}^{n}$

$B) a_{0} a_{2} - a_{1} a_{3} + ....... + a_{2 n - 2} a_{2 n} =$ coefficient $x^{- 2}$ in ${(1 + x + x^{2})}^{n} {(1 - \frac{1}{x} + \frac{1}{x^{2}})}^{n}$

$C) a_{r} = a_{2 n - r}$

$D) l e t f (x) = {(1 + x + x^{2})}^{n}$

$a_{0} + a_{2} + a_{4} ...... + a_{2 n} = \frac{f (1) + f (- 1)}{2}$

$a_{0} = 1$

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	Column – I		Column – II
A)	$a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + ......... + a_{2 n}^{2} is$ equals	P)	$\frac{1}{2} (3^{n} + a_{n})$
B)	$a_{0} a_{2} - a_{1} a_{3} + a_{2} a_{4} ...... + a_{2 n - 2} a_{2 n}$ is equals	Q)	$\frac{1}{2} (3^{n} - 1)$
C)	$a_{0} + a_{1} + a_{2} + ...... + a_{n}$ is equals	R)	$a_{n + 1}$
D)	$a_{2} + a_{4} + ...... + a_{2 n}$ is equals	S)	$a_{n}$

If (1+x+x2)n=a0+a1x+a2x2+.....+a2nx2n,then Column – I Column – IIA)a02−a12+a22−a32+.........+a2n2 is equalsP)12(3n+an)B)a0a2−a1a3+a2a4......+a2n−2a2n is equalsQ)12(3n−1)C)a0+a1+a2+......+an is equalsR)an+1D)a2+a4+......+a2n is equalsS)an

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