Sets, Relations & Functions · JEE Main previous year questions

2022 44 questions

1. Q292022 · 28 Jun 2022 · Shift 2· Numerical answer

   Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f:S\times S\to S:f\text{ is onto and }f(a,b)=f(b,a)\ge a\forall (a,b)\in S\times S\}$ is ________.

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   Answer: 37

2. Q122022 · 27 Jun 2022 · Shift 1

   If two straight lines whose direction cosines are given by the relations $l+m-n=0$, $3l^2+m^2+cnl=0$ are parallel, then the positive value of c is :

   1. A.6
   2. B.4
   3. C.3
   4. D.2
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   Answer: A

3. Q212022 · 27 Jun 2022 · Shift 1· Numerical answer

   Let $f:\mathrm{R}\to \mathrm{R}$ be a function defined by $f(x)=\frac{2e^{2x}}{e^{2x}+e}$. Then $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+.....+f\left(\frac{99}{100}\right)$ is equal to ____.

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   Answer: 99

4. Q32022 · 27 Jun 2022 · Shift 2

   If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $$f(x)={\int }_0^{x^2}\frac{t^2-5t+4}{2+e^t}dt$$ then the ordered pair $(m,n)$ is equal to

   1. A.$(3,2)$
   2. B.$(2,3)$
   3. C.$(2,2)$
   4. D.$(3,4)$
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   Answer: B

5. Q42022 · 27 Jun 2022 · Shift 2

   Let $f$ be a differentiable function in $\left(0,\frac{\pi }{2}\right)$. If ${\int }_{\cos x}^1{t}^{2}f(t)dt={\sin }^{3}x+\cos x$, then $\frac{1}{\sqrt{3}}{f}^{\prime }\left(\frac{1}{\sqrt{3}}\right)$ is equal to

   1. A.$6-9\sqrt{2}$
   2. B.$6-\frac{9}{\sqrt{2}}$
   3. C.$\frac{9}{2}-6\sqrt{2}$
   4. D.$\frac{9}{\sqrt{2}}-6$
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   Answer: B

6. Q212022 · 27 Jun 2022 · Shift 2· Numerical answer

   Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define $f:S\to S$ as $$f(n)=\{\begin{array}{ll}2n& ,\text{ if }n=1,2,3,4,5\\ 2n-11& ,\text{ if }n=6,7,8,9,10\end{array}$$ Let $g:S\to S$ be a function such that $fog(n)=\{\begin{array}{ll}n+1& ,\text{ if }n\text{ is odd}\\ n-1& ,\text{ if }n\text{ is even}\end{array}$. Then $g(10)(g(1)+g(2)+g(3)+g(4)+g(5))$ is equal to ________.

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   Answer: 190

7. Q282022 · 27 Jun 2022 · Shift 2· Numerical answer

   Let $S=\{E_1,E_2,\dots ,E_8\}$ be a sample space of a random experiment such that $P(E_n)=\frac{n}{36}$ for every $n=1,2,\dots ,8$. Then the number of elements in the set $\{A\subseteq S:P(A)\ge \frac{4}{5}\}$ is ________.

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   Answer: 19

8. Q62022 · 26 Jun 2022 · Shift 1

   Let $f,g:R\to R$ be two real valued functions defined as $f(x)=\{\begin{array}{ll}-|x+3|& ,x<0\\ e^x& ,x\ge 0\end{array}$ and $g(x)=\{\begin{array}{ll}{x}^2+k_{1}x& ,x<0\\ 4x+k_2& ,x\ge 0\end{array}$, where $k_1$ and $k_2$ are real constants. If (gof) is differentiable at $x=0$, then $(\mathrm{gof})(-4)+(\mathrm{gof})(4)$ is equal to :

   1. A.$4(e^4+1)$
   2. B.$2(2e^4+1)$
   3. C.$4e^4$
   4. D.$2(2e^4-1)$
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   Answer: D

9. Q72022 · 26 Jun 2022 · Shift 1

   The sum of the absolute minimum and the absolute maximum values of the function $f(x)=|3x-x^2+2|-x$ in the interval $[-1,2]$ is :

   1. A.$\frac{\sqrt{17}+3}{2}$
   2. B.$\frac{\sqrt{17}+5}{2}$
   3. C.5
   4. D.$\frac{9-\sqrt{17}}{2}$
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   Answer: A

10. Q122022 · 26 Jun 2022 · Shift 1

    Let $f(x)=2{\cos }^{-1}x+4{\cot }^{-1}x-3x^2-2x+10,x\in [-1,1]$. If $[a,b]$ is the range of the function f, then $4a-b$ is equal to :

    1. A.11
    2. B.$11-\pi$
    3. C.$11+\pi$
    4. D.$15-\pi$
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    Answer: B

11. Q142022 · 26 Jun 2022 · Shift 1

    Let S be the set of all the natural numbers, for which the line $\frac{x}{a}+\frac{y}{b}=2$ is a tangent to the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2$ at the point (a, b), $ab\ne 0$. Then :

    1. A.$S=\varphi$
    2. B.$n(S)=1$
    3. C.$S=\{2k:k\in \mathrm{N}\}$
    4. D.$S=\mathrm{N}$
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    Answer: D

12. Q12022 · 26 Jun 2022 · Shift 2

    Let $f:\mathrm{R}\to \mathrm{R}$ be defined as $f(x)=x-1$ and $g:\mathrm{R}-\{1,-1\}\to \mathrm{R}$ be defined as $g(x)=\frac{x^2}{x^2-1}$. Then the function fog is :

    1. A.one-one but not onto
    2. B.onto but not one-one
    3. C.both one-one and onto
    4. D.neither one-one nor onto
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    Answer: D

13. Q22022 · 25 Jun 2022 · Shift 1

    Let $f:\mathbb{N}\to \mathbb{R}$ be a function such that $f(x+y)=2f(x)f(y)$ for natural numbers x and y. If $f(1)=2$, then the value of $\alpha$ for which $\sum _{k=1}^{10}f(\alpha +k)=\frac{512}{3}(2^{20}-1)$ holds, is :

    1. A.$2$
    2. B.$3$
    3. C.$4$
    4. D.$6$
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    Answer: C

14. Q42022 · 25 Jun 2022 · Shift 1

    Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=x^3+x-5$. If $g(x)$ is a function such that $f(g(x))=x$, $\forall x\in \mathbb{R}$, then $g^{\prime }(63)$ is equal to $\underline{}$.

    1. A.$\frac{1}{49}$
    2. B.$\frac{3}{49}$
    3. C.$\frac{43}{49}$
    4. D.$\frac{91}{49}$
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    Answer: A

15. Q102022 · 25 Jun 2022 · Shift 1

    Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be two functions defined by $f(x)={\log }_e(x^2+1)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^x}$. Then, for which of the following range of $\alpha$, the inequality $f\left(g\left(\frac{(\alpha -1{)}^2}{3}\right)\right)>f\left(g\left(\alpha -\frac{5}{3}\right)\right)$ holds ?

    1. A.$(2,3)$
    2. B.$(-2,-1)$
    3. C.$(1,2)$
    4. D.$(-1,1)$
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    Answer: A

16. Q242022 · 25 Jun 2022 · Shift 1· Numerical answer

    Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)={\left(2\left(1-\frac{x^{25}}{2}\right)(2+x^{25})\right)}^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, then the greatest integer less than or equal to $g(1)$ is $\underline{}$.

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    Answer: 2

17. Q212022 · 25 Jun 2022 · Shift 2· Numerical answer

    Let $A=\left(\begin{array}{cc}2& -2\\ 1& -1\end{array}\right)$ and $B=\left(\begin{array}{cc}-1& 2\\ -1& 2\end{array}\right)$. Then the number of elements in the set $\{(n,m):n,m\in \{1,2,\dots ,10\}\text{ and }n{A}^n+m{B}^m=I\}$ is $\underline{}$.

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    Answer: 1

18. Q152022 · 24 Jun 2022 · Shift 2

    Let ${\lambda }^{\ast }$ be the largest value of $\lambda$ for which the function $f_{\lambda }(x)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in \mathrm{R}$. Then $f_{{\lambda }^{\ast }}(1)+f_{{\lambda }^{\ast }}(-1)$ is equal to :

    1. A.36
    2. B.48
    3. C.64
    4. D.72
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    Answer: D

19. Q242022 · 24 Jun 2022 · Shift 2· Numerical answer

    The sum of all the elements of the set $\{\alpha \in \{1,2,\dots ,100\}:HCF(\alpha ,24)=1\}$ is ____.

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    Answer: 1633