The value of - 10C4 10C5 11Cm 11C6 11C7 12Cm-2 12C8 12C9 13Cm-4 - is equal to zero when m is

The correct option is D 5 #10; #10; ^ 10 C _ 4 amp; ^ 10 C _ 5 #10; amp; ^ 11 C _ m ^ 11 C _ 6 amp; ^ 11 C _ 7 amp; #10 ...

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Standard XII

Mathematics

Properties of Determinants

The value of ...

Question

The value of $∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{10} C_{4} & ^{10} C_{5} & ^{11} C_{m}^{11} C_{6} & ^{11} C_{7} & ^{12} C_{m + 2}^{12} C_{8} & ^{12} C_{9} & ^{13} C_{m + 4} \end{matrix} ∣ ∣ ∣ ∣ ∣$ is equal to zero when $m$ is

6

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4

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5

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none of these

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Solution

The correct option is D $5$

$∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{10} C_{4} & ^{10} C_{5} & ^{11} C_{m}^{11} C_{6} & ^{11} C_{7} & ^{12} C_{m + 2}^{12} C_{8} & ^{12} C_{9} & ^{13} C_{m + 4} \end{matrix} ∣ ∣ ∣ ∣ ∣$

$C_{1} \to C_{1} + C_{2}^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}$

$∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{11} C_{5} & ^{10} C_{5} & ^{11} C_{m}^{12} C_{7} & ^{11} C_{7} & ^{12} C_{m + 2}^{13} C_{9} & ^{12} C_{9} & ^{13} C_{m + 4} \end{matrix} ∣ ∣ ∣ ∣ ∣$

$C_{1}$ and $C_{3}$ same then $d e t = 0$

$∴ m = 5$

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The value of ∣∣ ∣ ∣∣10C410C511Cm11C611C712Cm+212C812C913Cm+4∣∣ ∣ ∣∣ is equal to zero when m is

The correct option is D 5 ∣∣ ∣ ∣∣10C410C511Cm11C611C712Cm+212C812C913Cm+4∣∣ ∣ ∣∣ C1→C1+C2nCr+nCr+1=n+1Cr+1 ∣∣ ∣ ∣∣11C510C511Cm12C711C712Cm+213C912C913Cm+4∣∣ ∣ ∣∣ C1 andC3 same then det=0 ∴m=5

The value of $∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{10} C_{4} & ^{10} C_{5} & ^{11} C_{m}^{11} C_{6} & ^{11} C_{7} & ^{12} C_{m + 2}^{12} C_{8} & ^{12} C_{9} & ^{13} C_{m + 4} \end{matrix} ∣ ∣ ∣ ∣ ∣$ is equal to zero when $m$ is