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UPSC NDA Math Model Paper 3

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Question : 78 of 120

Marks: +1, -0

The roots of the equation

\begin{vmatrix} 1 & t-1 & 1 \\ t-1 & 1 & 1 \\ 1 & 1 & t-1 \end{vmatrix}

= 0

are

1, 2
-1, 2
1, -2
-1, -2

Solution:

Given,

\begin{vmatrix} 1 & t-1 & 1 \\ t-1 & 1 & 1 \\ 1 & 1 & t-1 \end{vmatrix}

= 0

\Rightarrow 1[1\cdot(t-1)-1] - (t-1)[(t-1)^2-1]

+ 1[(t-1)-1] = 0

\Rightarrow (t-2) - (t-1)(t^2-2t) + (t-2) = 0

\Rightarrow 2(t-2) - t(t-1)(t-2) = 0

\Rightarrow (t-2)(2 - t^2 + t) = 0

\Rightarrow (t-2)(t^2 - t - 2) = 0

\Rightarrow (t-2)(t-2)(t+1) = 0

\Rightarrow (t-2)^2(t+1) = 0

\Rightarrow t = 2, -1

The roots of the equation

\begin{vmatrix} 1 & t-1 & 1 \\ t-1 & 1 & 1 \\ 1 & 1 & t-1 \end{vmatrix}

= 0 are -1, 2

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