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(B) Given equation: $f(x) = x^2 + 2(a - 1)x + a + 5 = 0$.$A$. For imaginary roots,discriminant $D < 0$:$D = [2(a - 1)]^2 - 4(1)(a + 5) < 0$$4(a^2 - 2a

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$A 	o P, B 	o Q, C 	o R$

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(B) Given equation: $f(x) = x^2 + 2(a - 1)x + a + 5 = 0$.$A$. For imaginary roots,discriminant $D $D = [2(a - 1)]^2 - 4(1)(a + 5) $4(a^2 - 2a + 1) - 4a - 20 $a^2 - 2a + 1 - a - 5 $a^2 - 3a - 4 $B$. One root less than $3$ and other greater than $3$:This implies $f(3) $f(3) = 3^2 + 2(a - 1)(3) + a + 5 $9 + 6a - 6 + a + 5 $7a + 8 Actually,for $B$,$f(3) $C$. One root less than $1$ and other greater than $3$:This requires $f(1) $f(1) = 1 + 2a - 2 + a + 5 = 3a + 4 $f(3) = 7a + 8 Intersection: $a Correct matching: $A 	o P, B 	o Q, C 	o R$.

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

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