Number of Non-Negative Real Roots

CAT 2003 Slot 1 · Quantitative Ability · Medium · Algebra

This is a medium Quantitative Ability question from the CAT 2003 Slot 1 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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The number of non-negative real roots of $2^x - x - 1 = 0$ equals

A.
0
B.
1
C.
2
D.
3

Answer

Explanation

Equation: $2^x - 1 = x$ . For $x = 0$ , $2^0 - 1 = 0 = x$ . For $x = 1$ , $2^1 - 1 = 1 = x$ . For $x = 2$ , $2^2 - 1 = 3 \neq 2$ . By testing or taking derivatives, $f(x) = 2^x - x - 1$ is strictly convex for $x > 0$ . Thus, it can have at most 2 real roots. The non-negative roots are $x = 0$ and $x = 1$ . Hence, there are 2 roots.

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Number of Non-Negative Real Roots

Explanation

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