Let m be the positive integer a-b-c be three real numbers and - r-2r-1 mCr 1 m2-1 2m m-1 a b c 0 - r - m Then the value of -r-0m- r is given by

The correct option is A 0 #160; _ r = 2r 1 amp; ^ m C _ r amp; 1 m ^ 2 1 amp; 2 ^ m amp; m+1 a amp; b amp; c ∑ _ r=0 ...

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Properties of Determinants

Let m be th...

Question

Let $m$ be the positive integer $a, b, c$ be three real numbers and
$Δ_{r} = ∣ ∣ ∣ ∣ \begin{matrix} 2 r - 1 & ^{m} C_{r} & 1 m^{2} - 1 & 2^{m} & m + 1 a & b & c \end{matrix} ∣ ∣ ∣ ∣ (0 \leq r \leq m)$
Then the value of $m \sum r = 0 Δ_{r}$ is given by

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m^{2} - 1

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2^{m} a b c

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2^{m} (a + b + c)

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Solution

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m

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 r - 1 & ^{m} C_{r} & 1 m^{2} - 1 & 2^{m} & m + 1 {sin}^{2} (m^{2}) & {sin}^{2} (m) & {sin}^{2} (m + 1) \end{matrix} ∣ ∣ ∣ ∣ ∣ (0 \leq r \leq m) .

\sum_{r = 0}^{m} Δ_{r}

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 r - 1 & ^{m} C_{r} & 1 m^{2} - 1 & 2^{m} & m + 1 s i n^{2} (m^{2}) & s i n^{2} (m) & s i n^{2} (m + 1) \end{matrix} ∣ ∣ ∣ ∣ ∣

m \sum r = 0 Δ_{r}

m

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 r - 1 & _{m} C_{r} & 1 m^{2} - 1 & 2^{m} & m + 1 {sin}^{2} (2 m) & {sin}^{2} (m) & sin (m^{2}) \end{matrix} ∣ ∣ ∣ ∣ ∣

m \sum r = 0 Δ r

m

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 r - 1 & {^{m} C}_{r} & 1 m^{2} - 1 & 2^{m} & m + 1 {sin}^{2} (m^{2}) & {sin}^{2} (m) & {sin}^{2} (m + 1) \end{matrix} ∣ ∣ ∣ ∣ ∣ (0 \leq r \leq m)

\sum_{r = 0}^{m} Δ_{r}

Let a,b,c be non-zero real numbers such that a+b+c = 0; let $q = a^{2} + b^{2} + c^{2} a n d r = a^{4} + b^{4} + c^{4}$ Then

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