Sets, Relations & Functions · JEE Advanced previous year questions

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i,b_i\in \mathbb{R}$ for $i\in \{1,2,3\}$. Define the functions $f:\mathbb{R}\to \mathbb{R}$, $g:\mathbb{R}\to \mathbb{R}$, and $h:\mathbb{R}\to \mathbb{R}$ by $$f(x)=a_1+10x+a_2{x}^2+a_3{x}^3+x^4,$$ $$g(x)=b_1+3x+b_2{x}^2+b_3{x}^3+x^4,$$ $$h(x)=f(x+1)-g(x+2).$$ If $f(x)\ne g(x)$ for every $x\in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is

1. A.8
2. B.2
3. C.$-4$
4. D.$-6$

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f:\mathbb{R}\to \mathbb{R}$ by $$f(x)=\{\begin{array}{ll}2-2x^2-x^2\sin \frac{1}{x}& \text{if }x\ne 0,\\ 2& \text{if }x=0.\end{array}$$ Then which one of the following statements is TRUE?

1. A.The function $f$ is $\text{NOT}$ differentiable at $x=0$
2. B.There is a positive real number $\delta$, such that $f$ is a decreasing function on the interval $(0,\delta )$
3. C.For any positive real number $\delta$, the function $f$ is $\text{NOT}$ an increasing function on the interval $(-\delta ,0)$
4. D.$x=0$ is a point of local minima of $f$

Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f:\mathbb{N}\to \mathbb{Z}$ and $g:\mathbb{Z}\to \mathbb{N}$ defined by $$f(n)=\{\begin{array}{ll}(n+1)/2& \text{if }n\text{ is odd},\\ (4-n)/2& \text{if }n\text{ is even},\end{array}$$ and $$g(n)=\{\begin{array}{ll}3+2n& \text{if }n\ge 0,\\ -2n& \text{if }n<0.\end{array}$$ Define $(g\circ f)(n)=g(f(n))$ for all $n\in \mathbb{N}$, and $(f\circ g)(n)=f(g(n))$ for all $n\in \mathbb{Z}$. Then which of the following statements is (are) TRUE?

1. A.$g\circ f$ is $\text{NOT}$ one-one and $g\circ f$ is $\text{NOT}$ onto
2. B.$f\circ g$ is $\text{NOT}$ one-one but $f\circ g$ is onto
3. C.$g$ is one-one and $g$ is onto
4. D.$f$ is $\text{NOT}$ one-one but $f$ is onto

Let the set of all relations $R$ on the set $\{a,b,c,d,e,f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal{S}$. Then the number of elements in $\mathcal{S}$ is ____.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x)>0$ for all $x\in \mathbb{R}$, and $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$. Let the real numbers $a_1,a_2,\dots ,a_{50}$ be in an arithmetic progression. If $f(a_{31})=64f(a_{25})$, and $$\sum _{i=1}^{50}f(a_i)=3(2^{25}+1),$$ then the value of $$\sum _{i=6}^{30}f(a_i)$$ is ____.

Let $\mathbb{R}$ denote the set of all real numbers. Then the area of the region $$\left\{(x,y)\in \mathbb{R}\times \mathbb{R}:x>0,y>\frac{1}{x},5x-4y-1>0,4x+4y-17<0\right\}$$ is

1. A.$\frac{17}{16}-{\log }_{e}4$
2. B.$\frac{33}{8}-{\log }_{e}4$
3. C.$\frac{57}{8}-{\log }_{e}4$
4. D.$\frac{17}{2}-{\log }_{e}4$

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to (0,4)$ be functions defined by $$f(x)={\log }_e(x^2+2x+4),\text{and}g(x)=\frac{4}{1+e^{-2x}}.$$ Define the composite function $f\circ g^{-1}$ by $(f\circ g^{-1})(x)=f(g^{-1}(x))$, where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $f\circ g^{-1}$ at $x=2$ is ____.

If $$\alpha ={\int }_{\frac{1}{2}}^2\frac{{\tan }^{-1}x}{2x^2-3x+2}dx,$$ then the value of $\sqrt{7}\tan \left(\frac{2\alpha \sqrt{7}}{\pi }\right)$ is ____. (Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.)

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be functions defined by $$f(x)=\{\begin{array}{ll}x|x|\sin \left(\frac{1}{x}\right),& x\ne 0,\\ 0,& x=0,\end{array}\text{and}g(x)=\{\begin{array}{ll}1-2x,& 0\le x\le \frac{1}{2},\\ 0,& \text{otherwise}.\end{array}$$ Let $a,b,c,d\in \mathbb{R}$. Define the function $h:\mathbb{R}\to \mathbb{R}$ by $$h(x)=af(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+dg(x),x\in \mathbb{R}.$$ Match each entry in List-I to the correct entry in List-II. List-I: (P) If $a=0$, $b=1$, $c=0$, and $d=0$, then (Q) If $a=1$, $b=0$, $c=0$, and $d=0$, then (R) If $a=0$, $b=0$, $c=1$, and $d=0$, then (S) If $a=0$, $b=0$, $c=0$, and $d=1$, then List-II: (1) $h$ is one-one. (2) $h$ is onto. (3) $h$ is differentiable on $\mathbb{R}$. (4) the range of $h$ is $[0,1]$. (5) the range of $h$ is $\{0,1\}$. The correct option is

1. A.(P) $\to$ (4) \quad (Q) $\to$ (3) \quad (R) $\to$ (1) \quad (S) $\to$ (2)
2. B.(P) $\to$ (5) \quad (Q) $\to$ (2) \quad (R) $\to$ (4) \quad (S) $\to$ (3)
3. C.(P) $\to$ (5) \quad (Q) $\to$ (3) \quad (R) $\to$ (2) \quad (S) $\to$ (4)
4. D.(P) $\to$ (4) \quad (Q) $\to$ (2) \quad (R) $\to$ (1) \quad (S) $\to$ (3)

Considering only the principal values of the inverse trigonometric functions, the value of $$\tan \left({\sin }^{-1}\left(\frac{3}{5}\right)-2{\cos }^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$$ is

1. A.$\frac{7}{24}$
2. B.$\frac{-7}{24}$
3. C.$\frac{-5}{24}$
4. D.$\frac{5}{24}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{R}$, and $g:\mathbb{R}\to (0,\infty )$ be a function such that $g(x+y)=g(x)g(y)$ for all $x,y\in \mathbb{R}$. If $f\left(\frac{-3}{5}\right)=12$ and $g\left(\frac{-1}{3}\right)=2$, then the value of $\left(f\left(\frac{1}{4}\right)+g(-2)-8\right)g(0)$ is ____.

Let the function $f:\mathbb{R}\to \mathbb{R}$ be defined by $$f(x)=\frac{\sin x}{e^{\pi x}}\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}+\frac{2}{e^{\pi x}}\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}.$$ Then the number of solutions of $f(x)=0$ in $\mathbb{R}$ is ____.

PARAGRAPH “I” Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $(a,b)\in R$, we have $|a-b|\ge 2$. Let $Y=\{R\in X:\text{The range of }R\text{ has exactly one element}\}$ and $Z=\{R\in X:R\text{ is a function from }S\text{ to }S\}$. Let $n(A)$ denote the number of elements in a set $A$. (There are two questions based on PARAGRAPH “I”, the question given below is one of them) If $n(X)={}^m{C}_6$, then the value of $m$ is ____.

PARAGRAPH “I” Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $(a,b)\in R$, we have $|a-b|\ge 2$. Let $Y=\{R\in X:\text{The range of }R\text{ has exactly one element}\}$ and $Z=\{R\in X:R\text{ is a function from }S\text{ to }S\}$. Let $n(A)$ denote the number of elements in a set $A$. (There are two questions based on PARAGRAPH “I”, the question given below is one of them) If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is ____.

PARAGRAPH “II” Let $f:\left[0,\frac{\pi }{2}\right]\to [0,1]$ be the function defined by $f(x)={\sin }^{2}x$ and let $g:\left[0,\frac{\pi }{2}\right]\to [0,\infty )$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$. (There are two questions based on PARAGRAPH “II”, the question given below is one of them) The value of $2{\int }_0^{\frac{\pi }{2}}f(x)g(x)dx-{\int }_0^{\frac{\pi }{2}}g(x)dx$ is ____.

PARAGRAPH “II” Let $f:\left[0,\frac{\pi }{2}\right]\to [0,1]$ be the function defined by $f(x)={\sin }^{2}x$ and let $g:\left[0,\frac{\pi }{2}\right]\to [0,\infty )$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$. (There are two questions based on PARAGRAPH “II”, the question given below is one of them) The value of $\frac{16}{{\pi }^3}{\int }_0^{\frac{\pi }{2}}f(x)g(x)dx$ is ____.

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 ____.

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 $X_f=1$