Other Maths · JEE Main previous year questions

2021 44 questions

1. Q252021 · 1 Sept 2021 · Shift 2· Numerical answer

   Let f(x) be a polynomial of degree 3 such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10f(10)$ is equal to :

   Show answer

   Answer: 26

2. Q22021 · 26 Aug 2021 · Shift 2

   Let $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right)$. Then $A^{2025}-A^{2020}$ is equal to :

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

   Answer: A

3. Q142021 · 26 Aug 2021 · Shift 2

   The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\frac{1+{\sin }^{2}x}{1+{\pi }^{\sin x}}\right)dx$ is

   1. A.$\frac{\pi }{2}$
   2. B.$\frac{5\pi }{4}$
   3. C.$\frac{3\pi }{4}$
   4. D.$\frac{3\pi }{2}$
   Show answer

   Answer: C

4. Q272021 · 26 Aug 2021 · Shift 2· Numerical answer

   The least positive integer $n$ such that $\frac{(2i{)}^n}{(1-i{)}^{n-2}}$, $i=\sqrt{-1}$ is a positive integer, is ________.

   Show answer

   Answer: 6

5. Q292021 · 26 Aug 2021 · Shift 2· Numerical answer

   Let $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ denotes ${}^n{C}_k$ and $\left[\begin{array}{c}n\\ k\end{array}\right]=\{\begin{array}{ll}\left(\genfrac{}{}{0ex}{}{n}{k}\right),& \text{if }0\le k\le n\\ 0,& \text{otherwise}\end{array}$ If $A_k={\sum }_{i=0}^9\left(\genfrac{}{}{0ex}{}{9}{i}\right)\left[\begin{array}{c}12\\ 12-k+i\end{array}\right]+{\sum }_{i=0}^8\left(\genfrac{}{}{0ex}{}{8}{i}\right)\left[\begin{array}{c}13\\ 13-k+i\end{array}\right]$ and $A_4-A_3=190p$, then $p$ is equal to :

   Show answer

   Answer: 49

6. Q12021 · 27 Jul 2021 · Shift 2

   The point P (a, b) undergoes the following three transformations successively : (a) reflection about the line y = x. (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle $\frac{\pi }{4}$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point P are $\left(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$, then the value of 2a + b is equal to :

   1. A.13
   2. B.9
   3. C.5
   4. D.7
   Show answer

   Answer: B

7. Q102021 · 27 Jul 2021 · Shift 2

   Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :

   1. A.(1, 2)
   2. B.(2, 2)
   3. C.(2, 1)
   4. D.(1, 3)
   Show answer

   Answer: B

8. Q252021 · 27 Jul 2021 · Shift 2· Numerical answer

   Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number $(10{)}^{10}.(11{)}^{11}.(13{)}^{13}$ is equal to ____.

   Show answer

   Answer: 924

9. Q12021 · 25 Jul 2021 · Shift 1

   Let $f(x)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3,x\in \left[-\frac{\pi }{6},\frac{\pi }{2}\right]$. Then, f is :

   1. A.increasing in $\left(-\frac{\pi }{6},\frac{\pi }{2}\right)$
   2. B.decreasing in $\left(0,\frac{\pi }{2}\right)$
   3. C.increasing in $\left(-\frac{\pi }{6},0\right)$
   4. D.decreasing in $\left(-\frac{\pi }{6},0\right)$
   Show answer

   Answer: D

10. Q62021 · 25 Jul 2021 · Shift 1

    If b is very small as compared to the value of a, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\dots +\frac{1}{a-nb}=\alpha n+\beta n^2+\gamma n^3$, then the value of $\gamma$ is :

    1. A.$\frac{a^2+b}{3a^3}$
    2. B.$\frac{a+b}{3a^2}$
    3. C.$\frac{b^2}{3a^3}$
    4. D.$\frac{a+b^2}{3a^3}$
    Show answer

    Answer: C

11. Q102021 · 25 Jul 2021 · Shift 1

    Let $f:[0,\infty )\to [0,\infty )$ be defined as $f(x)={\int }_0^x[y]dy$ where [x] is the greatest integer less than or equal to x. Which of the following is true?

    1. A.f is continuous at every point in $[0,\infty )$ and differentiable except at the integer points.
    2. B.f is both continuous and differentiable except at the integer points in $[0,\infty )$.
    3. C.f is continuous everywhere except at the integer points in $[0,\infty )$.
    4. D.f is differentiable at every point in $[0,\infty )$.
    Show answer

    Answer: A

12. Q292021 · 25 Jul 2021 · Shift 1· Numerical answer

    If the value of $${\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\dots \text{upto }\infty \right)}^{{\log }_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots \text{upto }\infty \right)}$$ is $l$, then $l^2$ is equal to ____.

    Show answer

    Answer: 3

13. Q62021 · 25 Jul 2021 · Shift 2

    The lowest integer which is greater than ${\left(1+\frac{1}{10^{100}}\right)}^{10^{100}}$ is ____.

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

    Answer: A

14. Q92021 · 25 Jul 2021 · Shift 2

    If [x] be the greatest integer less than or equal to x, then $\sum _{n=8}^{100}\left[\frac{(-1{)}^{n}n}{2}\right]$ is equal to :

    1. A.0
    2. B.4
    3. C.$-2$
    4. D.2
    Show answer

    Answer: B

15. Q112021 · 25 Jul 2021 · Shift 2

    If $\left|\vec{a}\right|=2,\left|\vec{b}\right|=5$ and $\left|\vec{a}\times \vec{b}\right|=8$, then $\left|\vec{a}\cdot \vec{b}\right|$ is equal to :

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

    Answer: A

16. Q142021 · 25 Jul 2021 · Shift 2

    If $P=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]$, then $P^{50}$ is :

    1. A.$\left[\begin{array}{cc}1& 0\\ 25& 1\end{array}\right]$
    2. B.$\left[\begin{array}{cc}1& 50\\ 0& 1\end{array}\right]$
    3. C.$\left[\begin{array}{cc}1& 25\\ 0& 1\end{array}\right]$
    4. D.$\left[\begin{array}{cc}1& 0\\ 50& 1\end{array}\right]$
    Show answer

    Answer: A

17. Q232021 · 25 Jul 2021 · Shift 2· Numerical answer

    If $\left(\vec{a}+3\vec{b}\right)$ is perpendicular to $\left(7\vec{a}-5\vec{b}\right)$ and $\left(\vec{a}-4\vec{b}\right)$ is perpendicular to $\left(7\vec{a}-2\vec{b}\right)$, then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is ____.

    Show answer

    Answer: 60

18. Q272021 · 25 Jul 2021 · Shift 2· Numerical answer

    Let $n\in N$ and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms ${}^n{C}_0,3.{}^n{C}_1,5.{}^n{C}_2,7.{}^n{C}_3,\dots$ is equal to $2^{100}$ . 101, then $2\left[\frac{n-1}{2}\right]$ is equal to ____.

    Show answer

    Answer: 98

19. Q282021 · 25 Jul 2021 · Shift 2· Numerical answer
    
    Show answer

    Answer: 3

20. Q302021 · 25 Jul 2021 · Shift 2· Numerical answer

    If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of k is ____.

    Show answer

    Answer: 1

21. Q92021 · 22 Jul 2021 · Shift 2

    Let $[x]$ denote the greatest integer less than or equal to x. Then, the values of $x\in \mathbb{R}$ satisfying the equation $[e^x{]}^2+[e^x+1]-3=0$ lie in the interval :

    1. A.$[0,\frac{1}{e})$
    2. B.$[{\log }_{e}2,{\log }_{e}3)$
    3. C.$[1,e)$
    4. D.$[0,{\log }_{e}2)$
    Show answer

    Answer: D

22. Q152021 · 18 Mar 2021 · Shift 1

    The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\dots \infty }}}}$ is equal to

    1. A.$1.5+\sqrt{3}$
    2. B.$2+\sqrt{3}$
    3. C.$3+2\sqrt{3}$
    4. D.$4+\sqrt{3}$
    Show answer

    Answer: A

23. Q202021 · 18 Mar 2021 · Shift 1

    $\frac{1}{3^2-1}+\frac{1}{5^2-1}+\frac{1}{7^2-1}+\cdots +\frac{1}{(201{)}^2-1}$ is equal to

    1. A.$\frac{101}{404}$
    2. B.$\frac{25}{101}$
    3. C.$\frac{101}{408}$
    4. D.$\frac{99}{400}$
    Show answer

    Answer: B

24. Q232021 · 18 Mar 2021 · Shift 1· Numerical answer
    
    Show answer

    Answer: 4

25. Q262021 · 18 Mar 2021 · Shift 1· Numerical answer

    If $f(x)=\int \frac{5x^8+7x^6}{(x^2+1+2x^7{)}^2}dx$, $(x\ge 0)$, $f(0)=0$ and $f(1)=\frac{1}{K}$, then the value of K is

    Show answer

    Answer: 4