Which of the following values of α satisfy the equation 1+α^2 1+2α^2 1+3α^2 2+α^2 2+2α^2 2+3α^2 3+α^2 3+2α^2 3+3α^2 = -648 α ?

Correct answer: (B) 9; (C) -9. 1+α^2 1+2α^2 1+3α^2 2+α^2 2+2α^2 2+3α^2 3+α^2 3+2α^2 3+3α^2 R_2→R_2-R_1 (Take common) R_3→R_3-R_1 2 1+α^2 1+2α^2 1+3α^2 3+2α 3+4α 3+6α 4+2α 4+4α 4+6α R_3→R_3-R_2 2 1+α^2 1+2α^2 1+3α^2 3+2α 3+4α 3+6α 1 1 1 C_2→C_2-C_1 (Take common) C_3→C_3-C_1 ∴2α2.α 1+α^2 2+3α 2+4α 3+2α 2 2 1 0 0 4α^2-2α=-8α^3=-648α α=0, α^2=81, α=±9

Q30 / 60 · JEE Advanced 2015 (Paper 1) · Mathematics

Which of the following values of $α$ satisfy the equation $|\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix}| = - 648 α$ ?

A
$- 4$
Correct answer
B
$9$
Correct answer
C
$- 9$
D
$4$

Solution

|\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix}|

R_{2} \to R_{2} - R_{1}

(Take common)

R_{3} \to R_{3} - R_{1}

2 |\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ 3 + 2 α & (3 + 4 α) & (3 + 6 α) \\ (4 + 2 α) & (4 + 4 α) & (4 + 6 α) \end{matrix}|

R_{3} \to R_{3} - R_{2}

2 |\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ 3 + 2 α & 3 + 4 α & 3 + 6 α \\ 1 & 1 & 1 \end{matrix}|

C_{2} \to C_{2} - C_{1}

(Take common)

C_{3} \to C_{3} - C_{1}

∴ 2 α 2 . α |\begin{matrix} {(1 + α)}^{2} & (2 + 3 α) & (2 + 4 α) \\ 3 + 2 α & 2 & 2 \\ 1 & 0 & 0 \end{matrix}|

4 α^{2} |- 2 α| = - 8 α^{3} = - 648 α

α = 0, α^{2} = 81, α = \pm 9

Hence, the correct answer is options: 'B', 'C'.

Which of the following values of α satisfy the equation 1+α21+2α21+3α22+α22+2α22+3α23+α23+2α23+3α2= -648 α ?

Solution

More Determinants Questions

Which of the following values of $α$ satisfy the equation $|\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix}| = - 648 α$ ?