JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
If \({\Delta _r} = \left| {\begin{array}{*{20}{c}}
r&{2r - 1}&{3r - 2} \\
{\frac{n}{2}}&{n - 1}&a \\
{\frac{1}{2}n\left( {n - 1} \right)}&{{{\left( {n - 1} \right)}^2}}&{\frac{1}{2}\left( {n - 1} \right)\left( {3n - 4} \right)}
\end{array}} \right|\) then the value of \(\sum\limits_{r = 1}^{n - 1} {{\Delta _r}} \)
- A depends only on \(a\)
- B depends only on \(n\)
- C depends both on \(a\) and \(n\)
- D is independent of both \(a\) and \(n\)
Answer & Solution
Correct Answer
(D) is independent of both \(a\) and \(n\)
Step-by-step Solution
Detailed explanation
\(\sum\limits_{r = 1}^{n - 1} {r = 1 + 2 + 3 + ... + \left( {n - 1} \right)} = \frac{{n\left( {n - 1} \right)}}{2}\) \(\sum\limits_{r = 1}^{n - 1} {\left( {2r - 1} \right) = 1 + 3 + 5} \) \( + ... + \left[ {2\left( {n - 1} \right) - 2} \right] = {\left( {n - 1} \right)^2}\)…
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