JEE Mathematics 2023 Question with Solution

Given: The quadratic equation is $x^2 - px + \frac{5p}{4} = 0$ and its roots are rational. We must first find the maximum integral value $q$ of $p$ in $[0,10]$, then evaluate the area of the region $0 \leq y \leq (x-q)^2,\; 0 \leq x \leq q$.

Find: The required area and the correct option.

For the roots to be rational, the discriminant must be a perfect square.

From the extracted solution, the maximum integral value in $[0,10]$ is

Now the region is bounded by

Hence the area is

Evaluating,

Therefore, the area is $243$.

The solution concludes with option B, but the listed options show $243$ as option A. Therefore, there is a source discrepancy. Based on the computed value, the defensible correct option among the given options is A.

Given: After obtaining $q = 9$ from the rational-root condition, the curve is $y = (x-9)^2$ on $0 \leq x \leq 9$.

Find: Area under the curve above the $x$-axis from $x=0$ to $x=9$.

Since the shaded part lies under $y=(x-9)^2$ and above the $x$-axis from $x=0$ to $x=9$, its area is

Now,

So,

Therefore, the required area is $243$, so among the displayed options the correct choice is A.