Consider a region R = \ (x, y) ∈ R ^ 2 : x ^ 2 ≤ y ≤ 2x\ . If a line y = α divides the area of region R into two equal parts, then which of the following is true ?

Correct answer: B. 3 α ^ 2 - 8 α ^ 3 / 2 + 8 = 0 ∫ _ 0 ^ 4 ( y - (y)/(2))dy = 2 ∫ _ 0 ^ α ( y - (y)/(2))dy ( 2 y ^ 3 / 2 3 - y ^ 2 2 ) _ 0 ^ 4 = 2( 2 y ^ 3 / 2 3 - y ^ 2 4 ) _ 0 ^ α ⇒ (16)/(3) - 4 = 2((2)/(3) α ^ 3 / 2 - α ^ 2 4 ) ⇒ (4)/(3) = 2((2)/(3) α ^ 3 / 2 - α ^ 2 4 ) ⇒ (2)/(3) = 8 α ^ 3 / 2 - 3 α ^ 2 12 ⇒ 8 α ^ 3 / 2 - 3 α ^ 2 = 8 ⇒ 3 α ^ 2 - 8 α ^ 3 / 2 + 8 = 0

JEE Main 20202 SeptemberEvening Shift

Exam: JEE Main
Year: 2020
Date: 2 September
Shift: Evening Shift
Subject: Mathematics
Chapter: Area Under Curves

JEE Main 2020 · 2 September · Evening Shift · Mathematics · Area Under Curves

Consider a region R = \ (x, y) ∈ R ^ 2 : x ^ 2 ≤ y ≤ 2x\ . If a...

Consider a region $R = \left\{\left(x, y\right) \in {R}^{2}: {x}^{2} \le y \le 2x\right\}$ . If a line $y = \alpha$ divides the area of region $R$ into two equal parts, then which of the following is true ?

$Consider a region R = \ (x, y) ∈ R ^ 2 : x ^ 2 ≤ y ≤ 2x\ . If a line y = α divides the area of region R into two equal$

Options

A ${\alpha}^{3} - 6{\alpha}^{2} + 16 = 0$
B $3{\alpha}^{2} - 8{\alpha}^{3 / 2} + 8 = 0$
C $3{\alpha}^{2} - 8\alpha + 8 = 0$
D ${\alpha}^{3} - 6{\alpha}^{3 / 2} - 16 = 0$

Correct answer

B. $3{\alpha}^{2} - 8{\alpha}^{3 / 2} + 8 = 0$

Step-by-step solution

$Consider a region R = \ (x, y) ∈ R ^ 2 : x ^ 2 ≤ y ≤ 2x\ . If a line y = α divides the area of region R into two equal$

${∫}_{0}^{4}\left(\sqrt{y} - \dfrac{y}{2}\right)dy = 2{∫}_{0}^{\alpha}\left(\sqrt{y} - \dfrac{y}{2}\right)dy$

${\left(\dfrac{2{y}^{3 / 2}}{3} - \dfrac{{y}^{2}}{2}\right)}_{0}^{4} = {2\left(\dfrac{2{y}^{3 / 2}}{3} - \dfrac{{y}^{2}}{4}\right)}_{0}^{\alpha}$

$⇒ \dfrac{16}{3} - 4 = 2\left(\dfrac{2}{3}{\alpha}^{3 / 2} - \dfrac{{\alpha}^{2}}{4}\right)$

$⇒ \dfrac{4}{3} = 2\left(\dfrac{2}{3}{\alpha}^{3 / 2} - \dfrac{{\alpha}^{2}}{4}\right)$

$⇒ \dfrac{2}{3} = \dfrac{8{\alpha}^{3 / 2} - 3{\alpha}^{2}}{12}$

$⇒ 8{\alpha}^{3 / 2} - 3{\alpha}^{2} = 8$

$⇒ 3{\alpha}^{2} - 8{\alpha}^{3 / 2} + 8 = 0$

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