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Probability

Section: Mathematics

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Question : 33 of 92

Marks: +1, -0

If

l, m

represent any two elements (identical or different) of the set

\{1,2,3,4,5

6,7\}

, then the probability that

l x^2 + m x + 1 > 0 \forall x \in R

[AP EAPCET 22 May 2025 S1]

$\; \frac{12}{7 C_2}$
$\; \frac{22}{7^2}$
$\; \frac{10}{7 C_2}$
$\; \frac{36}{7^2}$

Solution:

Given set

\{1,2,3,4,5,6,7\}

Two number can be taken from set

=49

ways.

Now, for

l x^2 + m x + 1 > 0

m^2 - 4 l < 0 \Rightarrow m^2 < 4 l

If

l=1 \Rightarrow m^2 < 4 \Rightarrow m=1

\; l=2 \Rightarrow m^2 < 8 \Rightarrow m=1,2

\; l=3 \Rightarrow m^2 < 12 \Rightarrow m=1,2,3

\; l=4 \Rightarrow m^2 < 16 \Rightarrow m=1,2,3

\; l=5 \Rightarrow m^2 < 20 \Rightarrow m=1,2,3,4

\; l=6 \Rightarrow m^2 < 24 \Rightarrow m=1,2,3,4

\; l=7 \Rightarrow m^2 < 28 \Rightarrow m=1,2,3,4,5

\; \therefore \; \text{ Required probability } \; = \; \frac{22}{49} = \; \frac{22}{7^2}

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