The value of the determinant nbeginvmatrix1 - 181 cr-n C-n-1 -n-1 C- n -n-2 C- 1 1-n C-n-2-n-1 C-n-1-n-2 C- 2lend vmatrix A- 2B- 1C- 0D- n C n -1

The correct option is B 1= 1 amp; 1 amp;1 ^nC_1 nbsp; nbsp; amp; ^n+1C_1 amp;^n+2C_1 ^nC_2 amp; ^n+1C_2 amp;^n+2C_2 = 1 amp; 1 amp;1 ^n ...

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Byju's Answer

Standard XII

Mathematics

Cofactor

The value of ...

Question

The value of the determinant $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1^{n} C_{n - 1} & ^{n + 1} C_{n} & ^{n + 2} C_{1}^{n} C_{n - 2} & ^{n + 1} C_{n - 1} & ^{n + 2} C_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$

2

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1

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0

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^{n} C_{n - 1}

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Solution

The correct option is B $1$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1^{n} C_{1} & ^{n + 1} C_{1} & ^{n + 2} C_{1}^{n} C_{2} & ^{n + 1} C_{2} & ^{n + 2} C_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1^{n} C_{1} & ^{n + 1} C_{1} & ^{n + 1} C_{0} +^{n + 1} C_{1}^{n} C_{2} & ^{n + 1} C_{2} & ^{n + 1} C_{1} +^{n + 1} C_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$C_{3} \to C_{3} - C_{2}$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 0^{n} C_{1} & ^{n + 1} C_{1} & ^{n + 1} C_{0}^{n} C_{2} & ^{n + 1} C_{2} & ^{n + 1} C_{1} \end{matrix} ∣ ∣ ∣ ∣ ∣$

Similarly,
$C_{2} \to C_{2} - C_{1}$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0^{n} C_{1} & ^{n} C_{0} & ^{n + 1} C_{0}^{n} C_{2} & ^{n} C_{1} & ^{n + 1} C_{1} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$=^{n} C_{0} \times^{n + 1} C_{1} -^{n + 1} C_{0} \times^{n} C_{1} = n + 1 - n = 1$

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Similar questions

Let $f (n) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} n & n + 1 & n + 2^{n} P_{n} & ^{n + 1} P_{n + 1} & ^{n + 1} P_{n + 2}^{n} C_{n} & ^{n + 1} C_{n + 1} & ^{n + 1} C_{n + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Then, f(n) is divisible by

The arithmetic mean of nC0,nC1,nC2....nCn is
(A)2^n/n
(B)2^n-1/n
(C)2^n/n+1
(D)2^n-1/n+1

$The value of l i m n \to \infty \frac{3^{n} + 2^{n}}{3^{n} - 2^{n}} is$

$r \cdot^{n} C_{r} =$

Q. The value of

\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}

equals

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The value of the determinant ∣∣ ∣ ∣∣111nCn−1n+1Cnn+2C1nCn−2n+1Cn−1n+2C2∣∣ ∣ ∣∣

The value of the determinant $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1^{n} C_{n - 1} & ^{n + 1} C_{n} & ^{n + 2} C_{1}^{n} C_{n - 2} & ^{n + 1} C_{n - 1} & ^{n + 2} C_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$