જો f(x) = left| {begin{array}{*{20}{c}}{x - 3}&{2{x^2} - 18}&{3{x^3} - 81}{x - 5}&{2{x^2} - 50}&{4{x^3} - 500}1&2&3end{array}} right| તો f(1).f(3) + f(3).f(5) + f(5).f(1)=

Q: જો $f(x) = \left| {\begin{array}{*{20}{c}}{x - 3}&{2{x^2} - 18}&{3{x^3} - 81}\\{x - 5}&{2{x^2} - 50}&{4{x^3} - 500}\\1&2&3\end{array}} \right|$ તો $f(1).f(3) + f(3).f(5) + f(5).f(1)=$

$f(x) = 2(x - 3)(x - 5)$; $\left| {\,\begin{array}{*{20}{c}}1&{x + 3}&{3({x^2} + 3x + 9)}\\1&{x + 5}&{4({x^2} + 5x + 25)}\\1&1&3\end{array}\,} \right|$ $($Taking out $(x - 3),(x - 5)$ and $2$ from $I^{st}$ row, IInd row and $II^{rd}$ column respectively$)$ $f(x) = 2(x - 3)(x - 5)$ $\left| {\,\begin{array}{*{20}{c}}0&{(x + 2)}&{3({x^2} + 3x + 8)}\\0&2&{{x^2} + 11x + 73}\\1&1&3\end{array}\,} \right|,\,\,$ $({R_1} \to {R_1} - {R_3}$ and ${R_2} \to {R_2} - {R_1}$) $ = 2(x - 3)(x - 5)[1(x + 2)$ $({x^2} + 11x + 73) - 6({x^2} + 3x + 8)]$ $ = 2({x^2} - 8x + 15)({x^3} + 13{x^2} + 95x \ + 146 - 6{x^2} - 18x - 48)$ $ = 2({x^2} - 8x + 15)({x^3} + 7{x^2} + 77x + 98)$ $ = 2({x^5} - {x^4} + 36{x^3} - 413{x^2} + 371x + 1470)$ $f(1) = 2928$, $f(3) = 0$,$f(5) = 0$ $f(1).f(3) + f(3).f(5) + f(5).f(1)$ $ = 0 + 0 + 0$ $ = 0 = f(3)$.

MCQ

ગુજરાતી માધ્યમ

જો $f(x) = \left| {\begin{array}{*{20}{c}}{x - 3}&{2{x^2} - 18}&{3{x^3} - 81}\\{x - 5}&{2{x^2} - 50}&{4{x^3} - 500}\\1&2&3\end{array}} \right|$ તો $f(1).f(3) + f(3).f(5) + f(5).f(1)=$

A$f(1)$
B$f (3)$
C$f(1) + f(3)$
D$f(1) + f(5)$

Difficult

ધોરણ 12 સાયન્સ
ગણિત
3 and 4 . determinant and metrices
JEE

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