JEE Mathematics 2023 Question with Solution

Given: The probability of head on each toss is $\frac{1}{4}$, so $N$ follows a geometric distribution with

Find: The value of $q-p$ when the probability of the quadratic $64x^2+5Nx+1=0$ having no real root is $\frac{p}{q}$.

For the quadratic equation to have no real root, its discriminant must be negative:

So,

Hence,

Since $N$ is a positive integer, the possible values are $N=1,2,3$.

Now compute

Using the geometric distribution,

Therefore,

Thus $\frac{p}{q}=\frac{37}{64}$. Since the fraction must be written as $\frac{p}{q}$ with $p$ and $q$ coprime, we have $p=37$ and $q=64$. Therefore,

So the required answer is $27$.

The solution shows an intermediate mismatch in labeling $p$ and $q$, but the final computed value of $q-p$ is correctly $27$.

Given: No real root occurs when $N=1,2,3$. Find: The probability of this event quickly.

Instead of adding three terms separately, use

For a geometric random variable, $P(N\ge 4)$ means the first three tosses are all tails:

Hence,

So $p=37, q=64$ and

Therefore, the required answer is $27$.