Let for , , where , . Then, is equal to ________.

18 Explanation: Given that: Putting and : Putting : Diye gaye calculation ke mutabiq Step 2: ki value Image ke formula ke hisaab se: Step 3: nikalna Step 4: Final Value Value Value Value Answer: 18

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let for $x \in \mathbb{R}$ , $S_0(x) = x$ , $S_k(x) = C_k x + k \int_{0}^{\pi} S_{k-1}(t)dt$ where $C_0 = 1$ , $C_k = 1 - \int_{0}^{1} S_{k-1}(x)dx, k = 1, 2, 3, \dots$ . Then, $S_2(3) + 6C_3$ is equal to ________.

Choose the correct answer:

Explanation

Given that:

$S_k(x) = C_k x + k \int_0^x S_{k-1}(t) \, dt$

Putting $k = 2$ and $x = 3$ :

S_2(3) = C_2(3) + 2 \int_0^3 S_1(t) \, dt \quad \dots(1)

Putting $k = 1$ :

S_1(x) = C_1 x + \int_0^x S_0(t) \, dt = C_1 x + \frac{x^2}{2}

S_1(x) = C_1 x + \int_0^x S_0(t) \, dt = C_1 x + \frac{x^2}{2}

Choose the correct answer:

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