Sets, Relations & Functions · JEE Main previous year questions

2020 26 questions

1. Q102020 · 6 Sept 2020 · Shift 1

   If $f(x+y)=f(x)f(y)$ and ${\sum }_{x=1}^{\infty }f(x)=2$, $x,y\in \mathbb{N}$, where $\mathbb{N}$ is the set of all natural number, then the value of $\frac{f(4)}{f(2)}$ is :

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

2. Q232020 · 6 Sept 2020 · Shift 1· Numerical answer

   Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m.n is ______.

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   Answer: 28

3. Q62020 · 6 Sept 2020 · Shift 2

   Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=\max \{x,x^2\}$. Let S denote the set of all points in $\mathbb{R}$, where f is not differentiable. Then S is equal to :

   1. A.$\{0,1\}$
   2. B.$\{0\}$
   3. C.$\varphi$(an empty set)
   4. D.$\{1\}$
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   Answer: A

4. Q72020 · 6 Sept 2020 · Shift 2

   The set of all real values of $\lambda$ for which the function $$f(x)=(1-{\cos }^{2}x)(\lambda +\sin x),x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$$ has exactly one maxima and exactly one minima, is :

   1. A.$\left(-\frac{3}{2},\frac{3}{2}\right)-\{0\}$
   2. B.$\left(-\frac{3}{2},\frac{3}{2}\right)$
   3. C.$\left(-\frac{1}{2},\frac{1}{2}\right)-\{0\}$
   4. D.$\left(-\frac{1}{2},\frac{1}{2}\right)$
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   Answer: A

5. Q82020 · 6 Sept 2020 · Shift 2

   For all twice differentiable functions $f:\mathbb{R}\to \mathbb{R}$, with $f(0)=f(1)=f^{\prime }(0)=0$

   1. A.$f^{\prime \prime }(x)\ne 0$, at every point $x\in (0,1)$
   2. B.$f^{\prime \prime }(x)=0$, for some $x\in (0,1)$
   3. C.$f^{\prime \prime }(0)=0$
   4. D.$f^{\prime \prime }(x)=0$, at every point $x\in (0,1)$
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   Answer: B

6. Q102020 · 6 Sept 2020 · Shift 2

   For a suitably chosen real constant a, let a function, $f:\mathbb{R}-\{-a\}\to \mathbb{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further suppose that for any real number $x\ne -a$ and $f(x)\ne -a$, $(f\circ f)(x)=x$. Then $f\left(-\frac{1}{2}\right)$ is equal to :

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

   Answer: D

7. Q222020 · 6 Sept 2020 · Shift 2· Numerical answer

   Suppose that a function $f:\mathbb{R}\to \mathbb{R}$ satisfies $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$ and $f(1)=3$. If ${\sum }_{i=1}^{n}f(i)=363$ then n is equal to ______ .

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   Answer: 5

8. Q92020 · 3 Sept 2020 · Shift 1

   Consider the two sets : $A=\{m\in \mathbb{R}:\text{both the roots of }{x}^2-(m+1)x+m+4=0\text{ are real}\}$ and $B=[-3,5)$. Which of the following is not true?

   1. A.$A\cap B=\{-3\}$
   2. B.$B-A=(-3,5)$
   3. C.$A\cup B=\mathbb{R}$
   4. D.$A-B=(-\infty ,-3)\cup (5,\infty )$
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   Answer: D

9. Q112020 · 3 Sept 2020 · Shift 1

   The function, $f(x)=(3x-7)x^{2/3},x\in \mathbb{R}$, is increasing for all $x$ lying in :

   1. A.$(-\infty ,0)\cup \left(\frac{3}{7},\infty \right)$
   2. B.$(-\infty ,0)\cup \left(\frac{14}{15},\infty \right)$
   3. C.$\left(-\infty ,\frac{14}{15}\right)$
   4. D.$\left(-\infty ,\frac{14}{15}\right)\cup (0,\infty )$
   Show answer

   Answer: B

10. Q22020 · 3 Sept 2020 · Shift 2

    Let ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ be two relation defined as follows : ${\mathrm{R}}_1=\{(a,b)\in {\mathbb{R}}^2:a^2+b^2\in \mathbb{Q}\}$ and ${\mathrm{R}}_2=\{(a,b)\in {\mathbb{R}}^2:a^2+b^2\notin \mathbb{Q}\}$, where $\mathbb{Q}$ is the set of all rational numbers. Then :

    1. A.Neither ${\mathrm{R}}_1$ nor ${\mathrm{R}}_2$ is transitive.
    2. B.${\mathrm{R}}_2$ is transitive but ${\mathrm{R}}_1$ is not transitive.
    3. C.${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are both transitive.
    4. D.${\mathrm{R}}_1$ is transitive but ${\mathrm{R}}_2$ is not transitive.
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    Answer: A

11. Q52020 · 2 Sept 2020 · Shift 1

    Let S be the set of all $\lambda \in \mathbb{R}$ for which the system of linear equations $2x-y+2z=2$ $x-2y+\lambda z=-4$ $x+\lambda y+z=4$ has no solution. Then the set S :

    1. A.contains more than two elements.
    2. B.contains exactly two elements.
    3. C.is a singleton.
    4. D.is an empty set.
    Show answer

    Answer: B

12. Q102020 · 2 Sept 2020 · Shift 1

    If $\mathrm{R}=\{(x,y):x,y\in \mathbb{Z},x^2+3y^2\le 8\}$ is a relation on the set of integers $\mathbb{Z}$, then the domain of ${\mathrm{R}}^{-1}$ is :

    1. A.$\{0,1\}$
    2. B.$\{-2,-1,1,2\}$
    3. C.$\{-1,0,1\}$
    4. D.$\{-2,-1,0,1,2\}$
    Show answer

    Answer: C

13. Q122020 · 2 Sept 2020 · Shift 1

    The domain of the function $f(x)={\sin }^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty ,-a]\cup [a,\infty )$. Then a is equal to :

    1. A.$\frac{\sqrt{17}-1}{2}$
    2. B.$\frac{1+\sqrt{17}}{2}$
    3. C.$\frac{\sqrt{17}}{2}+1$
    4. D.$\frac{\sqrt{17}}{2}$
    Show answer

    Answer: B

14. Q52020 · 2 Sept 2020 · Shift 2

    Let $f:\mathbb{R}\to \mathbb{R}$ be a function which satisfies $f(x+y)=f(x)+f(y)\forall x,y\in \mathbb{R}$. If $f(1)=2$ and $g(n)={\sum }_{k=1}^{(n-1)}f(k),n\in \mathbb{N}$ then the value of $n$, for which $g(n)=20$, is :

    1. A.20
    2. B.9
    3. C.5
    4. D.4
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    Answer: C

15. Q72020 · 2 Sept 2020 · Shift 2

    Let $f:(-1,\infty )\to \mathbb{R}$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x}{\log }_e(1+x),x\ne 0$. Then the function $f$ :

    1. A.decreases in $(-1,\infty )$
    2. B.decreases in $(-1,0)$ and increases in $(0,\infty )$
    3. C.increases in $(-1,\infty )$
    4. D.increases in $(-1,0)$ and decreases in $(0,\infty )$
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    Answer: A

16. Q122020 · 2 Sept 2020 · Shift 2

    Let $A=\{X=(x,y,z{)}^T:PX=0\text{ and }{x}^2+y^2+z^2=1\}$ where $$P=\left[\begin{array}{ccc}1& 2& 1\\ -2& 3& -4\\ 1& 9& -1\end{array}\right],$$ then the set $A$ :

    1. A.is an empty set.
    2. B.contains more than two elements.
    3. C.contains exactly two elements.
    4. D.is a singleton.
    Show answer

    Answer: C

17. Q162020 · 9 Jan 2020 · Shift 2

    Let f and g be differentiable functions on $\mathbb{R}$ such that $f\circ g$ is the identity function. If for some $a,b\in \mathbb{R}$, $g^{\prime }(a)=5$ and $g(a)=b$, then $f^{\prime }(b)$ is equal to :

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

    Answer: D

18. Q132020 · 8 Jan 2020 · Shift 1

    The inverse function of $$f(x)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x\in (-1,1),\text{ is :}$$

    1. A.$\frac{1}{4}{\log }_e\left(\frac{1-x}{1+x}\right)$
    2. B.$\frac{1}{4}({\log }_{8}e){\log }_e\left(\frac{1-x}{1+x}\right)$
    3. C.$\frac{1}{4}({\log }_{8}e){\log }_e\left(\frac{1+x}{1-x}\right)$
    4. D.$\frac{1}{4}{\log }_e\left(\frac{1+x}{1-x}\right)$
    Show answer

    Answer: C

19. Q202020 · 8 Jan 2020 · Shift 1

    If c is a point at which Rolle's theorem holds for the function, $f(x)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, where $a\in \mathrm{R}$, then $f^{\prime \prime }(c)$ is equal to

    1. A.$\frac{1}{12}$
    2. B.$\frac{\sqrt{3}}{7}$
    3. C.$-\frac{1}{12}$
    4. D.$-\frac{1}{24}$
    Show answer

    Answer: A

20. Q62020 · 8 Jan 2020 · Shift 2

    Let $f:(1,3)\to \mathrm{R}$ be a function defined by $f(x)=\frac{x[x]}{1+x^2}$, where $[x]$ denotes the greatest integer $\le x$. Then the range of f is

    1. A.$\left(\frac{2}{5},\frac{1}{2}\right)\cup \left(\frac{3}{5},\frac{4}{5}\right]$
    2. B.$\left(\frac{3}{5},\frac{4}{5}\right)$
    3. C.$\left(\frac{2}{5},\frac{4}{5}\right]$
    4. D.$\left(\frac{2}{5},\frac{3}{5}\right]\cup \left(\frac{3}{4},\frac{4}{5}\right)$
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    Answer: A

21. Q132020 · 8 Jan 2020 · Shift 2

    Let S be the set of all real roots of the equation, $3^x(3^x-1)+2=|3^x-1|+|3^x-2|$. Then S :

    1. A.contains exactly two elements.
    2. B.is an empty set.
    3. C.is a singleton.
    4. D.contains at least four elements.
    Show answer

    Answer: C

22. Q112020 · 7 Jan 2020 · Shift 1

    If $g(x)=x^2+x-1$ and $(g\circ f)(x)=4x^2-10x+5$, then $f\left(\frac{5}{4}\right)$ is equal to:

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

    Answer: C

23. Q212020 · 7 Jan 2020 · Shift 1· Numerical answer

    Let S be the set of points where the function, $f(x)=|2-|x-3||$, $x\in \mathbb{R}$ is not differentiable. Then ${\sum }_{x\in S}f(f(x))$ is equal to _____.

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    Answer: 3

24. Q102020 · 7 Jan 2020 · Shift 2

    Let $y=y(x)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$, is equal to :

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

    Answer: B

25. Q192020 · 7 Jan 2020 · Shift 2

    The value of c in the Lagrange's mean value theorem for the function $f(x)=x^3-4x^2+8x+11$, when $x\in [0,1]$ is:

    1. A.$\frac{2}{3}$
    2. B.$\frac{\sqrt{7}-2}{3}$
    3. C.$\frac{4-\sqrt{5}}{3}$
    4. D.$\frac{4-\sqrt{7}}{3}$
    Show answer

    Answer: D