Sets, Relations & Functions · JEE Advanced previous year questions

2025 8 questions

1. Q12025 · 18 May 2025 · Paper 1

   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.$-6$
   Show answer

   Answer: C

2. Q32025 · 18 May 2025 · Paper 1

   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 $f$
   Show answer

   Answer: C

3. Q62025 · 18 May 2025 · Paper 1· One or more correct

   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
   Show answer

   Answer: A, D

4. Q82025 · 18 May 2025 · Paper 1· Numerical answer

   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 ____.

   Show answer

   Answer: 105

5. Q122025 · 18 May 2025 · Paper 1· Numerical answer

   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 ____.

   Show answer

   Answer: 96

6. Q22025 · 18 May 2025 · Paper 2

   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.$\frac{17}{2}-{\log }_{e}4$
   Show answer

   Answer: B

7. Q142025 · 18 May 2025 · Paper 2· Numerical answer

   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 ____.

   Show answer

   Answer: 0.2 to 0.3

8. Q162025 · 18 May 2025 · Paper 2· Numerical answer

   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 $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.)

   Show answer

   Answer: 21