Let be a positive integer and _r= vmatrix 2r-1 & ^n C_r & 1 n^2-1 & 2^n & n+1 ^2 n^2…

MATHEMATICS · Subjective · CBSE · Mathematics · Determinants · Determinants

Problem

Let nn be a positive integer and Δr=∣2r−1nCr1n2−12nn+1cos⁡2n2cos⁡2ncos⁡2(n+1)∣,\Delta_r=\begin{vmatrix} 2r-1 & {}^n C_r & 1\\ n^2-1 & 2^n & n+1\\ \cos^2 n^2 & \cos^2 n & \cos^2(n+1) \end{vmatrix}, prove that Δr=∣r2+rr+1r−2r2+3r−13r3r−3r2+2r+32r−12r−1∣.\Delta_r=\begin{vmatrix} r^2+r & r+1 & r-2\\ r^2+3r-1 & 3r & 3r-3\\ r^2+2r+3 & 2r-1 & 2r-1 \end{vmatrix}.

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