Condition for real and distinct roots

CAT 2019 Slot 2 · QA · Medium · Linear & Quadratic equations

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Let A be a real number. Then the roots of the equation x2−4x−log⁡2A=0x^2 - 4x - \log_2 A = 0 are real and distinct if and only if

  1. A.

    A<1/16A < 1/16

  2. B.

    A>1/8A > 1/8

  3. C.

    A>1/16A > 1/16

  4. D.

    A<1/8A < 1/8

Answer

C

Explanation

For the quadratic equation to have real and distinct roots, its discriminant must be strictly positive: D=(−4)2−4(1)(−log⁡2A)>0D = (-4)^2 - 4(1)(-\log_2 A) > 0 16+4log⁡2A>016 + 4\log_2 A > 0 log⁡2A>−4  ⟹  A>2−4=116\log_2 A > -4 \implies A > 2^{-4} = \frac{1}{16}

Also A>0A > 0 for log⁡2A\log_2 A to be defined, which is inherently satisfied when A>1/16A > 1/16.

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