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Mathematics Question on Properties of Determinants

The value of Δ=∣10C410C511Cm\[0.3em]11C611C712Cm+2\[0.3em]12C812C913Cm+4∣\Delta =\begin{vmatrix} ^{10}C_4 & ^{10}C_5 & ^{11}C_m \\\[0.3em] ^{11}C_6 & ^{11}C_7 &^{12}C_{m+2} \\\[0.3em] ^{12}C_8 & ^{12}C_9 & ^{13}C_{m+4} \end{vmatrix} is equal to zero, where m is

A

6

B

4

C

5

D

none of these.

Answer

5

Explanation

Solution

Δ=∣10C410C511Cm\[0.3em]11C611C712Cm+2\[0.3em]12C812C913Cm+4∣=0\Delta =\begin{vmatrix} ^{10}C_4 & ^{10}C_5 & ^{11}C_m \\\[0.3em] ^{11}C_6 & ^{11}C_7 &^{12}C_{m+2} \\\[0.3em] ^{12}C_8 & ^{12}C_9 & ^{13}C_{m+4} \end{vmatrix} = 0 Operate C1+C2C_1 + C_2 Δ=∣10C410C4+10C511Cm\[0.3em]11C611C6+11C712Cm+2\[0.3em]12C812C8+12C913Cm+4∣=0\Delta =\begin{vmatrix} ^{10}C_4 & ^{10}C_4+ ^{10}C_5 & ^{11}C_m \\\[0.3em] ^{11}C_6 & ^{11}C_6+^{11}C_7 &^{12}C_{m+2} \\\[0.3em] ^{12}C_8 & ^{12}C_8 +^{12}C_9 & ^{13}C_{m+4} \end{vmatrix} = 0 =∣10C412C411Cm\[0.3em]11C612C712Cm+2\[0.3em]12C813C913Cm+4∣=0=\begin{vmatrix} ^{10}C_4 & ^{12}C_4& ^{11}C_m \\\[0.3em] ^{11}C_6 & ^{12}C_7 &^{12}C_{m+2} \\\[0.3em] ^{12}C_8 &^{13}C_9& ^{13}C_{m+4} \end{vmatrix} = 0 Clearly, m=5m = 5 satisfies the above result, [∵ C1,C3\because \, C_1, C_3 will be identical]