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Column-I
Column-II
(I) The number of real solution(s) of the equation $x^{2} - | x | - 2 = 0$ is/are (P) 0
(II) If one of the real roots is the square of the other for the equation $3 x^{2} + p x + 3 = 0, p < 0$ then $| p | =$ (Q) 1
(III) The number of real value(s) of $k$ for which the system of equations $(k + 1) x + 8 y = 4 k$ and $k x + (k + 3) y = 3 k - 1$ have infinitely many solutions is/are (R) 2
(IV) Number of real root(s) of $x^{2} - \frac{2}{x - 2} = 4 - \frac{2}{x - 2}$ is/are (S) 6

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(I)-(R); (II)-(S); (III)-(Q); (IV)-(R)

(I)-(R); (II)-(S); (III)-(Q); (IV)-(Q)

(I)-(R); (II)-(S); (III)-(S); (IV)-(R)

(I)-(R); (II)-(S); (III)-(S); (IV)-(Q)

answer is B.

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Detailed Solution

I) $| x | = 2, - 1$ $\Rightarrow x = 2, - 2$
(II) $ω, ω^{2}$ are the roots which gives $p = 3$
(III) $\frac{k + 1}{k} = \frac{8}{k + 3} = \frac{4 k}{3 k - 1}$
$\frac{k + 1}{k} = \frac{8}{k + 3} \Rightarrow k = 1, 3$
$\frac{8}{k + 3} = \frac{4 k}{3 k - 1} \Rightarrow k = 1, 2$
So, we get $k = 1$
(IV) Domain will be $x \neq 2$

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