If α and β are the roots of the equation [1 5][ 1 3 -4 7 ]2[ (7/19) -(3/19) (4/19) (1/19) ]4 [ 1 3 -4 7 ]2[ x2-5x+5 -3 ]=[- 4], then the value of (2 - α )(2 - β ) is

Question

If α and β are the roots of the equation [1 5][ 1 3 -4 7 ]2[ (7/19) -(3/19) (4/19) (1/19) ]4 [ 1 3 -4 7 ]2[ x2-5x+5 -3 ]=[- 4], then the value of (2 - α )(2 - β ) is (A) 51 (B) -12 (C) 12 (D) -7

Tardigrade Admin · Accepted Answer

Let, A=[ 1 -4 3 7 ] then A- 1=[ (7/19) -(3/19) (4/19) (1/19) ] Since, AA- 1=I=A- 1A Hence, [ 1 3 -4 7 ][ (7/19) -(3/19) (4/19) (1/19) ]=I=[ (7/19) -(3/19) (4/19) (1/19) ][ 1 3 -4 7 ] ⇒ [ 1 5 ][ x2-5x+5 -3 ]=[- 4] ⇒ x2+5x+5-15=-4 ⇒ x2-5x-6=0⇒ x=6,-1 Hence, (2 - α )(2 - β )=-4× 3=-12

Q. If $α$ and $β$ are the roots of the equation $[15] [1 - 4 37]^{2} [\frac{7}{19} \frac{4}{19} - \frac{3}{19} \frac{1}{19}]^{4}$ $[1 - 4 37]^{2} [x^{2} - 5 x + 5 - 3] = [- 4],$ then the value of $(2 - α) (2 - β)$ is

$51$

$- 12$

$12$

$- 7$

Q. If α and β are the roots of the equation [15][1−4​37​]2[197​194​​−193​191​​]4 [1−4​37​]2[x2−5x+5−3​]=[−4], then the value of (2−α)(2−β) is

51

−12

12

−7

Q. If $α$ and $β$ are the roots of the equation $[15] [1 - 4 37]^{2} [\frac{7}{19} \frac{4}{19} - \frac{3}{19} \frac{1}{19}]^{4}$ $[1 - 4 37]^{2} [x^{2} - 5 x + 5 - 3] = [- 4],$ then the value of $(2 - α) (2 - β)$ is

$51$

$- 12$

$12$

$- 7$