SATHEE: Matrices and Determinants 2 Question 21

Matrices and Determinants 2 Question 21

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

======= ####23. Which of the following values of $α$ satisfy the equation

$| \begin{matrix} (1 + α)^{2} & (1 + 2 α)^{2} & (1 + 3 α)^{2} \\ (2 + α)^{2} & (2 + 2 α)^{2} & (2 + 3 α)^{2} \\ (3 + α)^{2} & (3 + 2 α)^{2} & (3 + 3 α)^{2} \end{matrix} | = - 648 α$ ?

3e0f7ab6f6a50373c3f2dbda6ca2533482a77bed

(a) -4

(b) 9

(d) 4

2015 Adv.

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Answer:

Correct Answer: 23. $(b, c)$

Solution:

Given determinant could be expressed as product of two determinants.

$i.e. | \begin{matrix} (1 + α)^{2} & (1 + 2 α)^{2} & (1 + 3 α)^{2} \\ (2 + α)^{2} & (2 + 2 α)^{2} & (2 + 3 α)^{2} \\ (3 + α)^{2} & (3 + 2 α)^{2} & (3 + 3 α)^{2} \end{matrix} | = - 648 α$

$\Rightarrow | \begin{matrix} 1 + 2 α + α^{2} & 1 + 4 α + 4 α^{2} & 1 + 6 α + 9 α^{2} \\ 4 + 4 α + α^{2} & 4 + 8 α + 4 α^{2} & 4 + 12 α + 9 α^{2} \\ 9 + 6 α + α^{2} & 9 + 12 α + 4 α^{2} & 9 + 18 α + 9 α^{2} \end{matrix} |$ $- 648 α$

$\Rightarrow | \begin{matrix} 1 & α & α^{2} \\ 4 & 2 α & α^{2} \\ 9 & 3 α & α^{2} \end{matrix} | \cdot | \begin{matrix} 1 & 1 & 1 \\ 2 & 4 & 6 \\ 1 & 4 & 9 \end{matrix} | = - 648 α$

$\Rightarrow α^{3} | \begin{matrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{matrix} | \cdot | \begin{matrix} 1 & 1 & 1 \\ 2 & 4 & 6 \\ 1 & 4 & 9 \end{matrix} | = - 648 α$