QJEEM23Ma7cff: JEE Main 2023 Mathematics - Class 12 JEEM23Ma7cff Solution: Let be the maximum integral value of in for which the roots of…

Question

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Let

q

be the maximum integral value of

p

in

[0,10]

for which the roots of the equation

x^2-p x+\frac{5}{4} p=0

are rational. Then the area of the region

\left\{(x, y): 0 \leq y \leq(x-q)^2, 0 \leq x \leq q\right\}

is :

Accepted Answer

Given equation : $4{x^2} - 4px + 5p = 0$

JEE Main 2023 (Online) 30th January Evening Shift Mathematics - Area Under The Curves Question 54 English Explanation

for rational roots, D must be perfect square

$D = 16{p^2} - 4 \times 4 \times 5p = 16p(p - 5)$

So, max. Integral value of $p = 9$ for making D is perfect square

$\therefore$ $q = 9$

Area of shared region

$= \int\limits_0^9 {{{(x - 9)}^2}dx}$

$= \left. {{{{{(x - 9)}^3}} \over 3}} \right|_0^9 = {{{9^3}} \over 3} = 243$ sq. units

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