How many real solutions exist for the equation ?

Answer and explanation

Correct answer: C.

  1. Step 1: Notice that both functions and are even functions, symmetric across the -axis.
  2. Step 2: For , the equation is . Since and , by the Intermediate Value Theorem there is at least one root in . Since and for , , there are no roots for . On , is strictly decreasing, giving exactly one positive solution .
  3. Step 3: By symmetry, is also a solution because . At , . Thus, there are exactly real solutions.
Answer
.
Why not the others: finds only the positive solution and neglects negative symmetry; assumes the graphs do not intersect; Infinitely many confuses this with periodic intersections where would be unbounded.

Practise more Trigonometric functions and inverse trigonometric functions.

Free questions with an explanation for every one, and full timed JEE Main mock tests.

All Trigonometric functions and inverse trigonometric functions questionsTake a mock test

More Trigonometric functions and inverse trigonometric functions questions