Verified Solution • 2025–26 Session moderate

QJEEM24Ma4866: JEE Main 2024 Mathematics - Class 11 JEEM24Ma4866 Solution: Let a be the sum of all coefficients in the expansion of ( 1 −…

Let $a$ be the sum of all coefficients in the expansion of ${(1 - 2 x + 2 x^{2})}^{2023} {(3 - 4 x^{2} + 2 x^{3})}^{2024}$ and $b = lim_{x \to 0} (\frac{\int_{0}^{x} \frac{\log (1 + t)}{t^{2024} + 1} d t}{x^{2}})$ . If the equation $c x^{2} + d x + e = 0$ and $2 b x^{2} + a x + 4 = 0$ have a common root, where $c, d, e \in R$ , then $d : c :$ e equals

Multiple Choice Options

A 2:01:04
B 1:01:04
C 1:02:04
D 4:01:04

Step-by-Step Verified Solution

Put $x = 1$

$∴ a = 1$

$b = lim_{x \to 0} \frac{\int_{0}^{x} \frac{\ln (1 + t)}{1 + t^{2024}} dt}{x^{2}}$

Using L' HOPITAL Rule

$b = lim_{x \to 0} \frac{\ln (1 + x)}{(1 + x^{2024})} \times \frac{1}{2 x} = \frac{1}{2}$

Now, ${cx}^{2} + dx + e = 0, x^{2} + x + 4 = 0$

$(D < 0)$

$∴ \frac{c}{1} = \frac{d}{1} = \frac{e}{4}$

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