3 and 4 .Determinants and Matrices
hard

यदि $x$ एक धनात्मक पूर्णांक हो, तो $\Delta  = \left| {\,\begin{array}{*{20}{c}}{x!}&{(x + 1)!}&{(x + 2)!}\\{(x + 1)!}&{(x + 2)!}&{(x + 3)!}\\{(x + 2)!}&{(x + 3)!}&{(x + 4)!}\end{array}\,} \right|$ का मान है

A

$2(x!)(x + 1)!$

B

$2(x!)(x + 1)!(x + 2)!$

C

$2(x!)(x + 3)!$

D

इनमें से कोई नहीं

Solution

$\Delta  = \left| {\begin{array}{*{20}{c}}{x!}&{(x + 1)!}&{(x + 2)\,!}\\{(x + 1)!}&{(x + 2)!}&{(x + 3)\,!}\\{(x + 2)!}&{(x + 3)!}&{(x + 4)\,!}\end{array}} \right|$

 = $x\,!\,(x + 1)\,!\,(x + 2)\,!\,\left| {\,\begin{array}{*{20}{c}}1&{(x + 1)}&{(x + 2)\,(x + 1)}\\1&{(x + 2)}&{(x + 3)\,(x + 2)}\\1&{(x + 3)}&{(x + 4)\,(x + 3)}\end{array}\,} \right|$

संक्रिया ${R_1} \to {R_2} – {R_1},{R_2} \to ({R_3} – {R_2})$ के प्रयोग से,

$ = x\,!(x + 1)\,!(x + 2)\,!$ $\,\left| {\,\begin{array}{*{20}{c}}0&1&{2(x + 2)}\\0&1&{2(x + 3)}\\1&{(x + 3)}&{(x + 4)\,(x + 3)}\end{array}\,} \right|$

$ = 2\,x!(x + 1)!(x + 2)!$,(हल करने पर).

ट्रिक  :  $x = 1$ रखकर विकल्पों का परीक्षण कीजिए।

Standard 12
Mathematics

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