Question

Let $\left\{\frac{6^p-1}{5}\right\} x^2+(p-5)\left(p^2-4 p+3\right) x+\left[\frac{p-2}{3}\right]=0$ where $\{y\}$ and $[y]$ denote fractional part function and greatest integer function of y respectively. On throwing a normal dice, its outcome is considered as the value of $p$. Identify which of the following statement(s) is(are) correct?

• A. The probability that the equation has exactly one root is infinite is $\frac{1}{2}$.
• B. The probability that the equation has both roots are infinite is $\frac{1}{3}$.
• C. The probability that the equation becomes an identity is $\frac{1}{6}$.
• D. The probability that the equation has two distinct real roots is 1 .

Answer 1

Answer: (A), (B), (C)

$\left\{\frac{6^{ p }-1}{5}\right\}=0 \Rightarrow p=1,2,3,4,5,6$
$( p -5)\left( p ^2-4 p +3\right)=0 \Rightarrow p=1,3,5$
${\left[\frac{ p -2}{3}\right]=0 \Rightarrow 2 \leq p<5 \Rightarrow p=2,3,4 .}$