Consider the matrices M=abccabbca such that det(M)=1, a,b,c∈R, then List-IList-II(I)There is only one matrix M(P)if a∈{−1,0,1},b∈{−2,−1,0}, c∈{1,2,3}(II)There are two distinct matrices M(Q)if a∈{−1,0,1},b∈{−2,0,1}, c∈{−2,−1,0}(III)There are more than 2, but finitely many matrices M(R)if a,b,c∈{0,1,2}(IV)There are infinitely many matrices M(S)if a,b,c all are negative integers (T)if a,b,c∈Q

(a+b+c)((a−b)2+(b−c)2+(c−a)2)=2If a,b,c∈Z⇒(1,0,0),(0,1,0),(0,0,1) are solutionsIf t∈θ⇒a=c, 2a+b=1t2,a−b=t⇒ infinite possibility

Consider the matrices M=abccabbca such that det(M)=1, a,b,c∈R, then List-IList-II(I)There is only one matrix M(P)if a∈{−1,0,1},b∈{−2,−1,0}, c∈{1,2,3}(II)There are two distinct matrices M(Q)if a∈{−1,0,1},b∈{−2,0,1}, c∈{−2,−1,0}(III)There are more than 2, but finitely many matrices M(R)if a,b,c∈{0,1,2}(IV)There are infinitely many matrices M(S)if a,b,c all are negative integers (T)if a,b,c∈Q

(a+b+c)((a−b)2+(b−c)2+(c−a)2)=2If a,b,c∈Z⇒(1,0,0),(0,1,0),(0,0,1) are solutionsIf t∈θ⇒a=c, 2a+b=1t2,a−b=t⇒ infinite possibility

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Consider the matrices $M = [\begin{matrix} a & b & c \\ c & a & b \\ b & c & a \end{matrix}]$ such that $\det (M) = 1, a, b, c \in R$ , then
List-I
List-II
(I)
There is only one matrix M
(P)
if $a \in {- 1, 0, 1}, b \in {- 2, - 1, 0}$ , $c \in {1, 2, 3}$
(II)
There are two distinct matrices M
(Q)
if $a \in {- 1, 0, 1}, b \in {- 2, 0, 1}$ , $c \in {- 2, - 1, 0}$
(III)
There are more than 2, but finitely many matrices M
(R)
if $a, b, c \in {0, 1, 2}$
(IV)
There are infinitely many matrices M
(S)
if $a, b, c$ all are negative integers

(T)
if $a, b, c \in Q$

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$(I) \to (P); (I I) \to (T); (I I I) \to (R); (I V) \to (S)$

$(I) \to (S); (I I) \to (Q); (I I I) \to (R); (I V) \to (T)$

$(I) \to (P); (I I) \to (R); (I I I) \to (S); (I V) \to (T)$

$(I) \to (P); (I I) \to (Q); (I I I) \to (R); (I V) \to (T)$

answer is D.

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Detailed Solution

$(a + b + c) ({(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}) = 2$

If $a, b, c \in Z \Rightarrow (1, 0, 0), (0, 1, 0), (0, 0, 1)$ are solutions
If $t \in θ \Rightarrow a = c, 2 a + b = \frac{1}{t^{2}}, a - b = t \Rightarrow$ infinite possibility

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