Sets, Relations & Functions · JEE Advanced previous year questions

2024 8 questions

1. Q172024 · 26 May 2024 · Paper 1

   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)
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   Answer: C

2. Q12024 · 26 May 2024 · Paper 2

   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.$\frac{5}{24}$
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   Answer: B

3. Q82024 · 26 May 2024 · Paper 2· Numerical answer

   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 ____.

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   Answer: 51

4. Q102024 · 26 May 2024 · Paper 2· Numerical answer

   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 ____.

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   Answer: 1

5. Q142024 · 26 May 2024 · Paper 2· Numerical answer

   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 ____.

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   Answer: 19.99 to 20.01

6. Q152024 · 26 May 2024 · Paper 2· Numerical answer

   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 ____.

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   Answer: 35.99 to 36.01

7. Q162024 · 26 May 2024 · Paper 2· Numerical answer

   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 ____.

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   Answer: 0 to 0

8. Q172024 · 26 May 2024 · Paper 2· Numerical answer

   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 ____.

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   Answer: 0.24 to 0.26