જો left| begin{array}{*{20}{c}}
{ - 2a}& | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

Let $\Delta  = \left| {\begin{array}{*{20}{c}}
{ - 2a}&{a + b}&{a + c}\
{b + a}&{ - 2b}&{b + c}\
{c + a}&{b + c}&am

Vedclass · Accepted Answer

$4$

Vedclass · Answer

Let $\Delta  = \left| {\begin{array}{*{20}{c}}
{ - 2a}&{a + b}&{a + c}\
{b + a}&{ - 2b}&{b + c}\
{c + a}&{b + c}&{ - 2c}
\end{array}} ight|$

Applying ${C_1} + {C_3}$ and ${C_2} + {C_3}$

$\Delta  = \left| {\begin{array}{*{20}{c}}
{ - a + c}&{2a + b + c}&{a + c}\
{2b + a + c}&{ - b + c}&{b + c}\
{a - c}&{b - c}&{ - 2c}
\end{array}} ight|$

Now, applying ${R_1} + {R_3}$ and ${R_2} + {R_3}$

$\Delta  = \left| {\begin{array}{*{20}{c}}
0&{2\left( {a + b} ight)}&{a - c}\
{2\left( {a + b} ight)}&0&{b - c}\
{a - c}&{b - c}&{ - 2c}
\end{array}} ight|$

On expanding, we get

$\Delta  =  - 2\left( {a + b} ight)\left\{ { - 2c\left[ {2\left( {a + b} ight)} ight] - \left( {a - c} ight)\left( {b - c} ight)} ight\}$ $ + \left( {a - c} ight)\left[ {2\left( {a + b} ight)\left( {b - c} ight)} ight]$

$\Delta  = 8c\left( {a + b} ight)\left( {a + b} ight) + 4\left( {a + b} ight)\left( {a - c} ight)\left( {b - c} ight)$

$ = 4\left( {a + b} ight)\left[ {2ac + 2bc + ab - bc - ac + {c^2}} ight]$

$ = 4\left( {a + b} ight)\left[ {ac + bc + ab + {c^2}} ight]$

$ = 4\left( {a + b} ight)\left[ {c\left( {a + c} ight) + b\left( {a + c} ight)} ight]$

$ = 4\left( {a + b} ight)\left( {b + c} ight)\left( {c + a} ight)$

$ = \alpha \left( {a + b} ight)\left( {b + c} ight)\left( {c + a} ight)$

Hence, $\alpha  = 4$

જો $\left| \begin{array}{*{20}{c}}
{ - 2a}&{a + b}&{a + c}\\
{b + a}&{ - 2b}&{b + c}\\
{c + a}&{b + c}&{ - 2c}
\end{array}\right|$ $ = \alpha \left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right) \ne 0$ તો $\alpha $ મેળવો.

Similar Questions

નિશ્ચાયકનું મૂલ્ય શોધો : $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

જો સમીકરણ સંહતિ

$2 x+y-z=5$

$2 x-5 y+\lambda z=\mu$

$x+2 y-5 z=7$

ને અસંખ્ય ઉકેલો હોય,તો

$(\lambda+\mu)^2+(\lambda-\mu)^2=........$

જો સમીકરણો ની જોડ $2x + 3y =\, -1; 3x + y = 2; \lambda x + 2y = \mu $ એ સુસંગત હોય તો . . . ..

સમીકરણ $\left| {\,\begin{array}{*{20}{c}}{3 - x}&{ - 6}&3\\{ - 6}&{3 - x}&3\\3&3&{ - 6 - x}\end{array}\,} \right| = 0$ ના બીજ મેળવો.

જો $\left| \begin{array}{*{20}{c}} { - 2a}&{a + b}&{a + c}\\ {b + a}&{ - 2b}&{b + c}\\ {c + a}&{b + c}&{ - 2c} \end{array}\right|$ $ = \alpha \left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right) \ne 0$ તો $\alpha $ મેળવો.