Sets, Relations & Functions · JEE Advanced previous year questions

2023 2 questions

1. Q92023 · 4 Jun 2023 · Paper 1· Numerical answer

   Let $n\ge 2$ be a natural number and $f:[0,1]\to \mathbb{R}$ be the function defined by $$f(x)=\{\begin{array}{ll}n(1-2nx)& \text{if }0\le x\le \frac{1}{2n}\\ 2n(2nx-1)& \text{if }\frac{1}{2n}\le x\le \frac{3}{4n}\\ 4n(1-nx)& \text{if }\frac{3}{4n}\le x\le \frac{1}{n}\\ \frac{n}{n-1}(nx-1)& \text{if }\frac{1}{n}\le x\le 1\end{array}$$ If $n$ is such that the area of the region bounded by the curves $x=0$, $x=1$, $y=0$ and $y=f(x)$ is 4, then the maximum value of the function $f$ is ____.

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   Answer: 8

2. Q72023 · 4 Jun 2023 · Paper 2· One or more correct

   Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^{2}f}{d{x}^2}(x)>0$ for all $x\in (-1,1)$. For $f\in S$, let $X_f$ be the number of points $x\in (-1,1)$ for which $f(x)=x$. Then which of the following statements is(are) true?

   1. A.There exists a function $f\in S$ such that $X_f=0$
   2. B.For every function $f\in S$, we have $X_f\le 2$
   3. C.There exists a function $f\in S$ such that $X_f=2$
   4. D.There does NOT exist any function $f$ in $S$ such that $X_f=1$
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   Answer: A, B, C