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Let S be the set of all real roots of the equation, 3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :

JEE · Math · previous-year question

  1. A.contains exactly two elements.
  2. B.is an empty set.
  3. C.is a singleton.correct
  4. D.contains at least four elements.

Answer

C. is a singleton.

Explanation

Let 3x = t ; t $$>$$ 0 t(t – 1) + 2 = |t – 1| + |t – 2| t2 – t + 2 = |t – 1| + |t – 2| Case-I : t $$<$$ 1 t2 – t + 2 = 1 – t + 2 – t $$ \Rightarrow $$ t2 + 2 = 3 – t $$ \Rightarrow $$ t2 + t – 1 = 0 $$ \Rightarrow $$ t = $${{ - 1 \pm \sqrt 5 } \over 2}$$ $$ \Rightarrow $$ t = $${{\sqrt 5 - 1} \over 2}$$ [ As t $$>$$ 0] Case-II : 1 $$ \le $$ t $$<$$ 2 $$ \Rightarrow $$ t2 – t + 2 = t – 1 + 2 – t $$ \Rightarrow $$ t2 – t + 1 = 0 D $$<$$ 0 so no real solution. Case-III : t $$ \ge $$ 2 $$ \Rightarrow $$ t2 – t + 2 = t – 1 + t – 2 $$ \Rightarrow $$ t2 – 3t - 5 = 0 $$ \Rightarrow $$ D $$<$$ 0 so no real solution.

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