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CAT 2019 Slot 2 — QA Question 13

Discriminant and Roots of Quadratic EquationEasy

Let A be a real number. Then the roots of the equation x2 - 4x - log2A = 0 are real and distinct if and only if

Answer & solution

  • A

    A < 1/16

  • B

    A < 1/8

  • C

    A > 1/8

  • A > 1/16

Solution

Easy

A quadratic has real, distinct roots exactly when its discriminant is strictly positive. Set up D>0D>0 for the given equation and solve the resulting logarithmic inequality for AA.

1

Write the discriminant. For x2−4x−log⁡2A=0x^{2}-4x-\log_{2}A=0 we have a=1, b=−4, c=−log⁡2Aa=1,\ b=-4,\ c=-\log_{2}A.

D=b2−4ac⇒ D=(−4)2−4(1)(−log⁡2A)(substitute coefficients)⇒ D=16+4log⁡2A\begin{aligned} &D=b^{2}-4ac\\ &\Rightarrow\ D=(-4)^{2}-4(1)(-\log_{2}A) \quad\text{(substitute coefficients)}\\ &\Rightarrow\ D=16+4\log_{2}A \end{aligned}
2

Impose D>0D>0. Real and distinct roots require a strictly positive discriminant.

16+4log⁡2A>0⇒ log⁡2A>−4(divide by 4, subtract)⇒ A>2−4(rewrite in exponential form)⇒ A>116\begin{aligned} &16+4\log_{2}A>0\\ &\Rightarrow\ \log_{2}A>-4 \quad\text{(divide by }4\text{, subtract)}\\ &\Rightarrow\ A>2^{-4} \quad\text{(rewrite in exponential form)}\\ &\Rightarrow\ A>\tfrac{1}{16} \end{aligned}
A>116A>\dfrac{1}{16}
CAT 2019 Slot 2 QA Q13: Let A be a real number. Then the roots of the equation x 2 - 4x - log 2 A = 0 are real and distinct if and onl — Solution | TheCATExam