First slide

Theory of equations

Question

$If m is selected at random from set {1, 2 \dots \dots .10} and the probability that the quadratic$ $equation 2 x^{2} + 2 m x + m + 1 = 0 has real roots, is$

Easy

A

$\frac{1}{8}$
B

$\frac{1}{9}$
C

$\frac{4}{5}$
D

$\frac{1}{2}$

Solution

$\begin{array}{l} 2 x^{2} + 2 m x + m + 1 = 0 \\ For real roots, D \geq 0 \\ \Rightarrow 4 m^{2} - 8 (m + 1) \geq 0 \\ m^{2} - 2 m - 2 \geq 0 \\ ∴ (m - 1)^{2} \geq 3 \end{array}$
$\begin{array}{l} \Rightarrow m \geq 1 + \sqrt{3} = 2.7 \\ \Rightarrow m = 3, 4, 5, 6, 7, 8, 9, 10 are favourable cases \end{array}$
$P (A) = \frac{8}{10}$

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