Mathematical Reasoning & Induction · JEE Advanced previous year questions

2023 1 question

1. Q32023 · 4 Jun 2023 · Paper 1· One or more correct

   Let $f:[0,1]\to [0,1]$ be the function defined by $f(x)=\frac{x^3}{3}-x^2+\frac{5}{9}x+\frac{17}{36}$. Consider the square region $S=[0,1]\times [0,1]$. Let $G=\{(x,y)\in S:y>f(x)\}$ be called the green region and $R=\{(x,y)\in S:y<f(x)\}$ be called the red region. Let $L_h=\{(x,h)\in S:x\in [0,1]\}$ be the horizontal line drawn at a height $h\in [0,1]$. Then which of the following statements is(are) true?

   1. A.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the green region below the line $L_h$
   2. B.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the red region below the line $L_h$
   3. C.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the red region below the line $L_h$
   4. D.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the green region below the line $L_h$
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   Answer: B, C, D