If $\left|\begin{array}{rrr}x-4 & 2 x & 2 x \\ 2 x & x-4 & 2 x \\ 2 x & 2 x & x-4\end{array}\right|=(\mathrm{A}+\mathrm{B} x)(x-\mathrm{A})^{2}$, then the ordered pair $(\mathrm{A}, \mathrm{B})$ is equal to :

JEE Main · 2018 · 08 Apr 2018 · Offline · Determinants

x - 4 2 x 2 x 2 x x - 4 2 x 2 x 2 x x - 4 = (A + B x) (x - A)^{2}

, then the ordered pair

(A, B)

is equal to :

Answer: (D)

Adding all three rows into the first makes each first-row entry

5 x - 4

, so the determinant is

(5 x - 4) 1 2 x 2 x 1 x - 4 2 x 1 2 x x - 4

. Subtracting the first column from the other two clears the top row and leaves a triangular block, giving

(5 x - 4) (x + 4)^{2}

. Matching this with

(A + B x) (x - A)^{2}

forces

x - A = x + 4

, so

A = - 4

, and then

A + B x = 5 x - 4

gives

B = 5

, so the pair is

(- 4, 5)

JEE Main · 2018 · 08 Apr 2018 · Offline · Single correct option