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3 and 4 .Determinants and Matrices
hard
જો $\Delta=\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ 2 x-3 & 3 x-4 & 4 x-5 \\ 3 x-5 & 5 x-8 & 10 x-17\end{array}\right|=$ $Ax ^{3}+ Bx ^{2}+ Cx + D ,$ હોય તો $B + C$ ની કિમત શોધો
A
$-1$
B
$1$
C
$-3$
D
$9$
(JEE MAIN-2020)
Solution
$\Delta= \left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ 2 x-3 & 3 x-4 & 4 x-5 \\ 3 x-5 & 5 x-8 & 10 x-17\end{array}\right|$
$=A x^{3}+B x^{2}+C x+D$
$R_{2} \rightarrow R_{2}-R_{1}$
$R_{3} \rightarrow R_{3}-R_{2}$
$\Delta=\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ x-1 & x-1 & x-1 \\ x-2 & 2(x-2) & 6(x-2)\end{array}\right|$
$=(x-1)(x-2)\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ 1 & 1 & 1 \\ 1 & 2 & 6\end{array}\right|$
$=-3(x-1)^{2}(x-2)=-3 x^{3}+12 x^{2}-15 x+6$
$\therefore \quad B+C=12-15=-3$
Standard 12
Mathematics
