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JEE Main 20208 JanuaryEvening Shift
Exam
JEE Main
Year
2020
Date
8 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2020 · 8 January · Evening Shift · Mathematics · Quadratic Equation

Let S , be the set of all real roots of the equation, 3 ^ x ( 3 ^ x...

Let SS, be the set of all real roots of the equation, 3x(3x−1)+2=∣3x−1∣+∣3x−2∣{3}^{x}\left({3}^{x} - 1\right) + 2 = \left|{3}^{x} - 1\right| + \left|{3}^{x} - 2\right|, thenLet S , be the set of all real roots of the equation, 3 ^ x ( 3 ^ x - 1) + 2 = | 3 ^ x - 1| + | 3 ^ x - 2| , then

Options

  1. Acontains exactly two elements.
  2. Bis a singleton.
  3. Cis an empty set.
  4. Dcontains at least four elements.

Correct answer

B. is a singleton.

Step-by-step solution

Let 3x=t{3}^{x} = t
t(t−1)+2=∣t−1∣+∣t−2∣t\left(t - 1\right) + 2 = \left|t - 1\right| + \left|t - 2\right|
t2−t+2=∣t−1∣+∣t−2∣{t}^{2} - t + 2 = \left|t - 1\right| + \left|t - 2\right|

Let, f(t)=t2−t+2 and g(t)=∣t−1∣+∣t−2∣f\left(t\right) = {t}^{2} - t + 2 \text{ and } g\left(t\right) = \left|t - 1\right| + \left|t - 2\right|
Let S , be the set of all real roots of the equation, 3 ^ x ( 3 ^ x - 1) + 2 = | 3 ^ x - 1| + | 3 ^ x - 2| , then
f(t) and g(t)f\left(t\right) \text{ and } g\left(t\right) has a solution at t=at = a.
3x=a⇒x=log⁡3a{3}^{x} = a ⇒ x = \log_{3} a.
So, it is a singleton set.

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