GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 48

Question

Consider the quadratic equation $x^2+\dfrac{x}{2}+c=0$, where $c$ is chosen uniformly randomly from the interval $[0,1]$. What is the probability that the given quadratic equation has a real solution?
 

The solutions of $a x^2+b x+c=0$ are given by $x=\dfrac{-b \pm \sqrt{b^2-4 a c}}{2a}$.

• A. $1 / 2$
• B. $1 / 4$
• C. $1 / 8$
• D. $1 / 16$

Answer 1

$\frac{1}{16}$ since a real solution occurs precisely when $b^2-4 c=\frac{1}{4}-4 c \geq 0$, i.e., $0 \leq c \leq \frac{1}{16}$, which is $\frac{1}{16}^{\text {th }}$ fraction of the interval $[0,1]$ over which $c$ ranges.