Sets, Relations & Functions · JEE Main previous year questions

2019 25 questions

1. Q292019 · 12 Apr 2019 · Shift 2

   Let S be the set of all $\alpha \in \mathbb{R}$ such that the equation, $\cos 2x+\alpha \sin x=2\alpha -7$ has a solution. Then S is equal to :

   1. A.$[2,6]$
   2. B.$[3,7]$
   3. C.$[1,4]$
   4. D.$\mathbb{R}$
   Show answer

   Answer: A

2. Q212019 · 10 Apr 2019 · Shift 1

   Let $f(x)=e^x-x$ and $g(x)=x^2-x$, $\forall x\in \mathrm{R}$. Then the set of all $x\in \mathrm{R}$, where the function $h(x)=(f\circ g)(x)$ is increasing, is :

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

   Answer: D

3. Q42019 · 10 Apr 2019 · Shift 2

   Let $f(x)={\log }_e(\sin x)$, $(0<x<\pi )$ and $g(x)={\sin }^{-1}(e^{-x})$, $(x\ge 0)$. If $\alpha$ is a positive real number such that a = (fog)'($\alpha$) and b = (fog)($\alpha$), then :

   1. A.$a{\alpha }^2+b\alpha -a=-2{\alpha }^2$
   2. B.$a{\alpha }^2+b\alpha +a=0$
   3. C.$a{\alpha }^2-b\alpha -a=0$
   4. D.$a{\alpha }^2-b\alpha -a=1$
   Show answer

   Answer: D

4. Q32019 · 9 Apr 2019 · Shift 1

   If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set $S=\{x\in R:f(x)=f(0)\}$ Contains exactly :

   1. A.four rational numbers.
   2. B.four irrational numbers.
   3. C.two irrational and one rational number.
   4. D.two irrational and two rational numbes.
   Show answer

   Answer: C

5. Q52019 · 9 Apr 2019 · Shift 1

   All the points in the set $S=\left\{\frac{\alpha +i}{\alpha -i}:\alpha \in R\right\}(i=\sqrt{-1})$ lie on a :

   1. A.straight line whose slope is –1
   2. B.straight line whose slope is 1.
   3. C.circle whose radius is 1.
   4. D.circle whose radius is $\sqrt{2}$.
   Show answer

   Answer: C

6. Q62019 · 9 Apr 2019 · Shift 1

   Let $f(x)=15-|x-10|$; $x\in R$. Then the set of all values of x, at which the function, $g(x)=f(f(x))$ is not differentiable, is :

   1. A.{10,15}
   2. B.{5,10,15,20}
   3. C.{10}
   4. D.{5,10,15}
   Show answer

   Answer: D

7. Q152019 · 9 Apr 2019 · Shift 1

   Let $\sum _{k=1}^{10}f(a+k)=16(2^{10}-1)$ where the function f satisfies $f(x+y)=f(x)f(y)$ for all natural numbers x, y and f(1) = 2. then the natural number 'a' is

   1. A.2
   2. B.16
   3. C.4
   4. D.3
   Show answer

   Answer: D

8. Q222019 · 9 Apr 2019 · Shift 1

   If the function $f:R-\{1,-1\}\to A$ defined by $f(x)=\frac{x^2}{1-x^2}$, is surjective, then A is equal to

   1. A.R − (−1, 0)
   2. B.R − {−1}
   3. C.R − [−1, 0)
   4. D.$[0,\infty )$
   Show answer

   Answer: C

9. Q242019 · 9 Apr 2019 · Shift 1

   Let S be the set of all values of x for which the tangent to the curve $y=f(x)=x^3-x^2-2x$ at (x, y) is parallel to the line segment joining the points (1, f(1)) and (−1, f(−1)), then S is equal to :

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

   Answer: B

10. Q132019 · 9 Apr 2019 · Shift 2

    If $f(x)=[x]-\left[\frac{x}{4}\right],x\in 4$, where $[x]$ denotes the greatest integer function, then

    1. A.Both ${\lim }_{x\to 4-}f(x)$ and ${\lim }_{x\to 4+}f(x)$ exist but are not equal
    2. B.f is continuous at x = 4
    3. C.${\lim }_{x\to 4+}f(x)$ exists but ${\lim }_{x\to 4-}f(x)$ does not exist
    4. D.${\lim }_{x\to 4-}f(x)$ exists but ${\lim }_{x\to 4+}f(x)$ does not exist
    Show answer

    Answer: B

11. Q172019 · 9 Apr 2019 · Shift 2

    The domain of the definition of the function $f(x)=\frac{1}{4-x^2}+{\log }_{10}(x^3-x)$ is

    1. A.$(-1,0)\cup (1,2)\cup (2,\infty )$
    2. B.$(-2,-1)\cup (-1,0)\cup (2,\infty )$
    3. C.$(1,2)\cup (2,\infty )$
    4. D.$(-1,0)\cup (1,2)\cup (3,\infty )$
    Show answer

    Answer: A

12. Q52019 · 8 Apr 2019 · Shift 2

    Let $f(x)=a^x(a>0)$ be written as $f(x)=f_1(x)+f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x+y)+f_1(x-y)$ equals

    1. A.$2f_1(x)f_1(y)$
    2. B.$2f_1(x+y)f_1(x-y)$
    3. C.$2f_1(x)f_2(y)$
    4. D.$2f_1(x+y)f_2(x-y)$
    Show answer

    Answer: A

13. Q142019 · 8 Apr 2019 · Shift 2

    Let $f(x)={\int }_0^{x}g(t)dt$ where g is a non-zero even function. If $f(x+5)=g(x)$, then ${\int }_0^{x}f(t)dt$ equals-

    1. A.$5{\int }_{x+5}^{5}g(t)dt$
    2. B.${\int }_{x+5}^{5}g(t)dt$
    3. C.${\int }_5^{x+5}g(t)dt$
    4. D.$2{\int }_5^{x+5}g(t)dt$
    Show answer

    Answer: B

14. Q32019 · 12 Jan 2019 · Shift 1

    Let $S=\{1,2,3,\dots ,100\}$. The number of non-empty subsets A of S such that the product of elements in A is even is :

    1. A.$2^{50}-1$
    2. B.$2^{50}(2^{50}-1)$
    3. C.$2^{100}-1$
    4. D.$2^{50}+1$
    Show answer

    Answer: B

15. Q102019 · 12 Jan 2019 · Shift 1

    Considering only the principal values of inverse functions, the set $A=\{x\ge 0:{\tan }^{-1}(2x)+{\tan }^{-1}(3x)=\frac{\pi }{4}\}$

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

    Answer: D

16. Q162019 · 12 Jan 2019 · Shift 1

    Let S be the set of all points in $(-\pi ,\pi )$ at which the function, $f(x)=\min \{\sin x,\cos x\}$ is not differentiable. Then S is a subset of which of the following ?

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

    Answer: D

17. Q102019 · 12 Jan 2019 · Shift 2

    Let Z be the set of integers. If $A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is

    1. A.$2^{12}$
    2. B.$2^{18}$
    3. C.$2^{10}$
    4. D.$2^{15}$
    Show answer

    Answer: D

18. Q112019 · 12 Jan 2019 · Shift 2

    If the function f given by $f(x)=x^3-3(a-2)x^2+3ax+7$, for some $a\in \mathbb{R}$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation, $\frac{f(x)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

    1. A.$-7$
    2. B.5
    3. C.7
    4. D.6
    Show answer

    Answer: C

19. Q162019 · 12 Jan 2019 · Shift 2

    Let f be a differentiable function such that $f(1)=2$ and $f^{\prime }(x)=f(x)$ for all $x\in \mathbb{R}$. If $h(x)=f(f(x))$, then $h^{\prime }(1)$ is equal to :

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

    Answer: A

20. Q192019 · 12 Jan 2019 · Shift 2

    The set of all values of $\lambda$ for which the system of linear equations $x-2y-2z=\lambda x$ $x+2y+z=\lambda y$ $-x-y=\lambda z$ has a non-trivial solution :

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

    Answer: C

21. Q22019 · 11 Jan 2019 · Shift 2

    Let K be the set of all real values of x where the function $f(x)=\sin |x|-|x|+2(x-\pi )\cos |x|$ is not differentiable. Then the set K is equal to :

    1. A.$\{0,\pi \}$
    2. B.$\varphi$ (an empty set)
    3. C.$\{r\}$
    4. D.$\{0\}$
    Show answer

    Answer: B

22. Q32019 · 11 Jan 2019 · Shift 2

    The number of functions f from $\{1,2,3,\dots ,20\}$ onto $\{1,2,3,\dots ,20\}$ such that $f(k)$ is a multiple of 3, whenever k is a multiple of 4, is :

    1. A.$6^5\times (15)!$
    2. B.$5^6\times 15$
    3. C.$(15)!\times 6!$
    4. D.$5!\times 6!$
    Show answer

    Answer: C

23. Q112019 · 11 Jan 2019 · Shift 2

    Let a function $f:(0,\infty )\to (0,\infty )$ be defined by $f(x)=\left|1-\frac{1}{x}\right|$. Then f is :

    1. A.not injective but it is surjective
    2. B.neither injective nor surjective
    3. C.injective only
    4. D.both injective as well as surjective
    Show answer

    Answer: B

24. Q52019 · 9 Jan 2019 · Shift 1

    Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined as $$f(x)=\{\begin{array}{ll}5& ;x\le 1\\ a+bx& ;1<x<3\\ b+5x& ;3\le x<5\\ 30& ;x\ge 5\end{array}$$ Then, f is

    1. A.continuous if $a=0$ and $b=5$
    2. B.continuous if $a=-5$ and $b=10$
    3. C.continuous if $a=5$ and $b=5$
    4. D.not continuous for any values of a and b
    Show answer

    Answer: D

25. Q192019 · 9 Jan 2019 · Shift 1

    For $x\in \mathbb{R}-\{0,1\}$, Let $f_1(x)=\frac{1}{x}$, $f_2(x)=1-x$ and $f_3(x)=\frac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $(f_2\circ J\circ f_1)(x)=f_3(x)$ then $J(x)$ is equal to :

    1. A.$f_1(x)$
    2. B.$\frac{1}{x}{f}_3(x)$
    3. C.$f_2(x)$
    4. D.$f_3(x)$
    Show answer

    Answer: D