First slide

Theory of equations

Question

$Let conditions C_{1} and C_{2} be defined as follows: C_{1} : b^{2} - 4 a c \geq 0, C_{2} : a, - b, c are of same sign. The roots of a x^{2} + b x + c = 0$
$are real and positive, if$

Moderate

A

$both C_{1} and C_{2} are satisfied$
B

$only C_{2} is satisfied$
C

$only C_{1} is satisfied$
D

None of these

Solution

$\begin{array}{l} C_{1} : b^{2} - 4 a c \geq 0 \\ C_{2} : a, - b, c are of same sign \end{array}$
$a x^{2} + b x + c = 0 has real roots.$
$Then D \geq 0 i.e., C_{1} must be satisfied.$
$\begin{array}{l} (i) Let a, - b, c > 0 \\ Then - \frac{b}{2 a} > 0 \end{array}$
$\begin{array}{l} (ii) Let a, - b, c < 0 \\ Then - \frac{b}{2 a} > 0 \end{array}$
$Hence, for roots to be + ve, C_{2} must be satisfied.$
$Thus, both C_{1}, C_{2} are satisfied.$

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