Other Maths · JEE Main previous year questions

2022 53 questions

1. Q12022 · 30 Jun 2022 · Shift 1

   Let $S_1=\left\{x\in R-\{1,2\}:\frac{(x+2)(x^2+3x+5)}{-2+3x-x^2}\ge 0\right\}$ and $S_2=\{x\in R:3^{2x}-3^{x+1}-3^{x+2}+27\le 0\}$. Then, $S_1\cup S_2$ is equal to :

   1. A.$(-\infty ,-2]\cup (1,2)$
   2. B.$(-\infty ,-2]\cup [1,2]$
   3. C.$(-2,1]\cup [2,\infty )$
   4. D.$(-\infty ,2]$
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   Answer: B

2. Q72022 · 30 Jun 2022 · Shift 1

   Let $f(x)=4x^3-11x^2+8x-5$, $x\in R$. Then f :

   1. A.has a local minima at $x=\frac{1}{2}$
   2. B.has a local minima at $x=\frac{3}{4}$
   3. C.is increasing in $\left(\frac{1}{2},\frac{3}{4}\right)$
   4. D.is decreasing in $\left(\frac{1}{2},\frac{4}{3}\right)$
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   Answer: D

3. Q132022 · 30 Jun 2022 · Shift 1

   The value of $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots +\frac{1}{1+2+3+\cdots +11}$ is equal to:

   1. A.$\frac{20}{11}$
   2. B.$\frac{11}{6}$
   3. C.$\frac{241}{132}$
   4. D.$\frac{21}{11}$
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   Answer: B

4. Q252022 · 30 Jun 2022 · Shift 1· Numerical answer

   A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on $4^{\mathrm{th}}$ day is 30, then number of infected students on $8^{\mathrm{th}}$ day will be ____ .

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   Answer: 90

5. Q92022 · 27 Jun 2022 · Shift 1

   If $\frac{dy}{dx}+\frac{2^{x-y}(2^y-1)}{2^x-1}=0$, $x,y>0$, $y(1)=1$, then $y(2)$ is equal to :

   1. A.$2+{\log }_{2}3$
   2. B.$2+{\log }_{3}2$
   3. C.$2-{\log }_{3}2$
   4. D.$2-{\log }_{2}3$
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   Answer: D

6. Q202022 · 27 Jun 2022 · Shift 1

   The boolean expression $(\sim (p\wedge q))\vee q$ is equivalent to :

   1. A.$q\to (p\wedge q)$
   2. B.$p\to q$
   3. C.$p\to (p\to q)$
   4. D.$p\to (p\vee q)$
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   Answer: D

7. Q182022 · 27 Jun 2022 · Shift 2

   Let $S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+\dots$. Then $4S$ is equal to

   1. A.${\left(\frac{7}{3}\right)}^2$
   2. B.$\frac{7^3}{3^2}$
   3. C.${\left(\frac{7}{3}\right)}^3$
   4. D.$\frac{7^2}{3^3}$
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   Answer: C

8. Q252022 · 27 Jun 2022 · Shift 2· Numerical answer

   If $y(x)={\left(x^x\right)}^x$, $x>0$, then $\frac{d^{2}x}{d{y}^2}+20$ at $x=1$ is equal to ________.

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   Answer: 16

9. Q12022 · 26 Jun 2022 · Shift 1

   Let $f(x)=\frac{x-1}{x+1}$, $x\in R-\{0,-1,1\}$. If $f^{n+1}(x)=f(f^n(x))$ for all $n\in \mathrm{N}$, then $f^6(6)+f^7(7)$ is equal to :

   1. A.$\frac{7}{6}$
   2. B.$-\frac{3}{2}$
   3. C.$\frac{7}{12}$
   4. D.$-\frac{11}{12}$
   Show answer

   Answer: B

10. Q22022 · 26 Jun 2022 · Shift 1

    Let $A=\left\{z\in C:\left|\frac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in C:\arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}\right\}$. Then $A\cap B$ is :

    1. A.a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second and third quadrants only
    2. B.a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second quadrant only
    3. C.an empty set
    4. D.a portion of a circle of radius $\frac{2}{\sqrt{3}}$ that lies in the third quadrant only
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    Answer: B

11. Q42022 · 26 Jun 2022 · Shift 1

    The remainder when $(2021{)}^{2023}$ is divided by 7 is :

    1. A.1
    2. B.2
    3. C.5
    4. D.6
    Show answer

    Answer: C

12. Q182022 · 26 Jun 2022 · Shift 1

    If $\vec{a}\cdot \vec{b}=1$, $\vec{b}\cdot \vec{c}=2$ and $\vec{c}\cdot \vec{a}=3$, then the value of $\left[\vec{a}\times (\vec{b}\times \vec{c}),\vec{b}\times (\vec{c}\times \vec{a}),\vec{c}\times (\vec{b}\times \vec{a})\right]$ is :

    1. A.0
    2. B.$-6\vec{a}\cdot (\vec{b}\times \vec{c})$
    3. C.$-12\vec{c}\cdot (\vec{a}\times \vec{b})$
    4. D.$-12\vec{b}\cdot (\vec{c}\times \vec{a})$
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    Answer: A

13. Q232022 · 26 Jun 2022 · Shift 1· Numerical answer

    Let $f(x)=\max \{|x+1|,|x+2|,.....,|x+5|\}$. Then ${\int }_{-6}^{0}f(x)dx$ is equal to ____.

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    Answer: 21

14. Q262022 · 26 Jun 2022 · Shift 1· Numerical answer

    Let $A=\{n\in \mathrm{N}:H.C.F.(n,45)=1\}$ and Let $B=\{2k:k\in \{1,2,.......,100\}\}$. Then the sum of all the elements of $A\cap B$ is ____.

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    Answer: 5264

15. Q302022 · 26 Jun 2022 · Shift 1· Numerical answer

    Let $A={\sum }_{i=1}^{10}{\sum }_{j=1}^{10}\min \{i,j\}$ and $B={\sum }_{i=1}^{10}{\sum }_{j=1}^{10}\max \{i,j\}$. Then $A+B$ is equal to ____.

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    Answer: 1100

16. Q152022 · 26 Jun 2022 · Shift 2

    If $A={\sum }_{n=1}^{\infty }\frac{1}{(3+(-1{)}^n{)}^n}$ and $B={\sum }_{n=1}^{\infty }\frac{(-1{)}^n}{(3+(-1{)}^n{)}^n}$, then $\frac{A}{B}$ is equal to :

    1. A.$\frac{11}{9}$
    2. B.1
    3. C.$-\frac{11}{9}$
    4. D.$-\frac{11}{3}$
    Show answer

    Answer: C

17. Q212022 · 26 Jun 2022 · Shift 2· Numerical answer

    Let $f:\mathrm{R}\to \mathrm{R}$ satisfy $f(x+y)=2^{x}f(y)+4^{y}f(x),\forall x,y\in \mathrm{R}$. If $f(2)=3$, then $14.\frac{f^{\prime }(4)}{f^{\prime }(2)}$ is equal to ________.

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    Answer: 248

18. Q222022 · 26 Jun 2022 · Shift 2· Numerical answer

    Let p and q be two real numbers such that $p+q=3$ and $p^4+q^4=369$. Then ${\left(\frac{1}{p}+\frac{1}{q}\right)}^{-2}$ is equal to ________.

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    Answer: 4

19. Q232022 · 26 Jun 2022 · Shift 2· Numerical answer

    If $z^2+z+1=0,z\in C$, then $\left|{\sum }_{n=1}^{15}{\left(z^n+(-1{)}^n\frac{1}{z^n}\right)}^2\right|$ is equal to ________.

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    Answer: 2

20. Q292022 · 26 Jun 2022 · Shift 2· Numerical answer

    If ${(}^{40}{C}_0)+{(}^{41}{C}_1)+{(}^{42}{C}_2)+\cdots +{(}^{60}{C}_{20})=\frac{m}{n}{}^{60}{C}_{20}$ m and n are coprime, then m + n is equal to ________.

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    Answer: 102

21. Q52022 · 25 Jun 2022 · Shift 1

    If $\frac{1}{2\cdot 3^{10}}+\frac{1}{2^2\cdot 3^9}+\cdots +\frac{1}{2^{10}\cdot 3}=\frac{K}{2^{10}\cdot 3^{10}}$, then the remainder when $K$ is divided by $6$ is :

    1. A.$1$
    2. B.$2$
    3. C.$3$
    4. D.$5$
    Show answer

    Answer: D

22. Q282022 · 25 Jun 2022 · Shift 1· Numerical answer

    For a natural number n, let ${\alpha }_n=19^n-12^n$. Then, the value of $\frac{31{\alpha }_9-{\alpha }_{10}}{57{\alpha }_8}$ is $\underline{}$.

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    Answer: 4

23. Q152022 · 25 Jun 2022 · Shift 2

    The sum $1+2\cdot 3+3\cdot 3^2+\cdots +10\cdot 3^9$ is equal to :

    1. A.$\frac{2\cdot 3^{12}+10}{4}$
    2. B.$\frac{19\cdot 3^{10}+1}{4}$
    3. C.$5\cdot 3^{10}-2$
    4. D.$\frac{9\cdot 3^{10}+1}{2}$
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    Answer: B

24. Q232022 · 25 Jun 2022 · Shift 2· Numerical answer

    The value of $b>3$ for which $12\underset{3}{\overset{b}{\int }}\frac{1}{(x^2-1)(x^2-4)}dx={\log }_e\left(\frac{49}{40}\right)$, is equal to $\underline{}$.

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    Answer: 6

25. Q42022 · 24 Jun 2022 · Shift 2

    Let $x,y>0$. If $x^3{y}^2=2^{15}$, then the least value of $3x+2y$ is

    1. A.30
    2. B.32
    3. C.36
    4. D.40
    Show answer

    Answer: D