Match the following:
Column – I Column – II
i) If $| | x^{2} | - 2 | x | - 3 | = | 2 k + 1 |$ has exactly five solutions then the number of possible values of k is p) 0
ii) The sum of the possible values of x satisfying the equation ${(31 + 8 \sqrt{15})}^{x^{2} - 3} + 1 = {(32 + 8 \sqrt{15})}^{x^{2} - 3}$ is q) 8
iii) The set of values of real parameter a for which the equation $x^{4} - 2 a x^{2} + x + a^{2} - a = 0$ has all real solutions is given by $[\frac{m}{n}, \infty), m, n$ are relatively prime integers then $1 + m + n$ is r) 2
iv) The total number of solutions of the equation ${(x^{2} + x - 57)}^{3 x^{2} + 3} = {(x^{2} + x - 57)}^{10 x}$ is s) 7
t) 9

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i – s, ii – p, iii – r, iv – q

i – q, ii – s, iii – p, iv – r

i – r, ii – q, iii – p, iv – s

i – r, ii – p, iii – q, iv – s

answer is C.

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Detailed Solution

ii) $a^{n} + b^{n} = {(a + b)}^{n} \Rightarrow n = 1$
iii) consider the given equation as a quadratic in x
$\Rightarrow a = x^{2} + x, a = x^{2} - x + 1 \Rightarrow x^{2} + x - a = 0, x^{2} - x + 1 - a = 0$
For 4 real roots $Δ_{1} \geq 0, Δ_{2} \geq 0$
At $a \in [\frac{3}{4}, α)$
iv) (i) $x^{2} + x - 57 = 0 \to 2 s o l$

ii) $x^{2} + x - 57 = 1 \to 2 s o l$
iii) $x^{2} + x - 57 = - 1, 3 x^{2} + 3. e v e n, 10 x e v e n \to x = 7 o n l y s o l$
iv) $3 x^{2} + 3 = 10 x \to 2 s o l$

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	Column – I		Column – II
i)	If $\| \| x^{2} \| - 2 \| x \| - 3 \| = \| 2 k + 1 \|$ has exactly five solutions then the number of possible values of k is	p)	0
ii)	The sum of the possible values of x satisfying the equation ${(31 + 8 \sqrt{15})}^{x^{2} - 3} + 1 = {(32 + 8 \sqrt{15})}^{x^{2} - 3}$ is	q)	8
iii)	The set of values of real parameter a for which the equation $x^{4} - 2 a x^{2} + x + a^{2} - a = 0$ has all real solutions is given by $[\frac{m}{n}, \infty), m, n$ are relatively prime integers then $1 + m + n$ is	r)	2
iv)	The total number of solutions of the equation ${(x^{2} + x - 57)}^{3 x^{2} + 3} = {(x^{2} + x - 57)}^{10 x}$ is	s)	7
		t)	9