Question

The number of $x$ satisfying $| x - 2 |^{10x^2 - 1} = |x - 2|^{3x}$ is

⭕ 5
⭕ 6
⭕ 2
⭕ 4​

Answer 1

$\large\underline{\sf{Solution-}}$

Given equation is

${\bf :\longmapsto\: { |x - 2| }^{ {10x}^{2} - 1} = { |x - 2| }^{3x}}$

Using hit and trial method,

If we take x = 2, we get

$\red{\rm :\longmapsto\: { |2 - 2| }^{ {10x}^{2} - 1} = { |2 - 2| }^{3x}}$

$\red{\rm :\longmapsto\: {0 }^{ {10(2)}^{2} - 1} = { 0 }^{3 \times 2}}$

$\red{\rm :\longmapsto\: 0 = 0}$

$\red{\bf\implies \:x = 2 \: is \: its \: solution}$

Consider again,

$\purple{\bf :\longmapsto\: { |x - 2| }^{ {10x}^{2} - 1} = { |x - 2| }^{3x}}$

Again, If we take x = 3, we get

$\purple{\bf :\longmapsto\: { |3 - 2| }^{ {10(3)}^{2} - 1} = { |3 - 2| }^{3 \times 3}}$

$\purple{\bf :\longmapsto\: { |1| }^{ 90 - 1} = { |1| }^{9}}$

$\purple{\bf :\longmapsto\: { |1| }^{ 89} = { |1| }^{9}}$

$\purple{\bf :\longmapsto\: 1 = 1}$

$\purple{\bf\implies \:x = 3 \: is \: its \: solution}$

Again, Consider

$\green{\bf :\longmapsto\: { |x - 2| }^{ {10x}^{2} - 1} = { |x - 2| }^{3x}}$

If we take x = 1, we get

$\green{\bf :\longmapsto\: { |1 - 2| }^{ {10(1)}^{2} - 1} = { |1 - 2| }^{3 \times 1}}$

$\green{\bf :\longmapsto\: { | - 1| }^{ 10 - 1} = { | - 1| }^{3}}$

$\green{\bf :\longmapsto\: { | 1| }^{9} = { | 1| }^{3}}$

$\green{\bf :\longmapsto\:1 = 1 }$

$\green{\bf\implies \:x = 1 \: is \: its \: solution}$

Again Consider

${\bf :\longmapsto\: { |x - 2| }^{ {10x}^{2} - 1} = { |x - 2| }^{3x}}$

$\rm \implies\: {10x}^{2} - 1 = 3x$

$\rm \implies\: {10x}^{2} - 3x - 1 = 0$

$\rm \implies\: {10x}^{2} - 5x + 2x- 1 = 0$

$\rm :\longmapsto\:5x(2x - 1) + 1(2x - 1) = 0$

$\rm :\longmapsto\:(2x - 1)(5x + 1) = 0$

$\bf\implies \:x = \dfrac{1}{2} \: \: or \: \: - \: \dfrac{1}{5}$

Hence, Solution is

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{x = 2} \\ \\ &\sf{x = 3}\\ \\ &\sf{x = 1}\\ \\ &\sf{x = - \dfrac{1}{5} }\\ \\ &\sf{x = \dfrac{1}{2} } \end{cases}\end{gathered}\end{gathered}$

So, it implies number of solution of this equation is 5.

So, option (1) is correct.