Sets, Relations & Functions · JEE Main previous year questions

387 questions asked across 19 exam years, newest first

Year-wise papers Topic-wise questions

Practise this topic

2024 59 questions

Q172024 · 8 Apr 2024 · Shift 2
If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1, a > 0$ has a local maximum at $x = α$ and a local minimum at $x = α^{2}$ , then $α$ and $α^{2}$ are the roots of the equation :
1. A. $x^{2} - 6 x + 8 = 0$
2. B. $8 x^{2} - 6 x + 1 = 0$
3. C. $8 x^{2} + 6 x - 1 = 0$
4. D. $x^{2} + 6 x + 8 = 0$
Show answer
Answer: A
Q12024 · 6 Apr 2024 · Shift 2
If the function $f (x) = (\frac{1}{x})^{2 x}; x > 0$ attains the maximum value at $x = \frac{1}{e}$ then :
1. A. $e^{π} < π^{e}$
2. B. $e^{2 π} < (2 π)^{e}$
3. C. $(2 e)^{π} > π^{(2 e)}$
4. D. $e^{π} > π^{e}$
Show answer
Answer: D
Q32024 · 6 Apr 2024 · Shift 2
Let $A = {1, 2, 3, 4, 5}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $4 x \leq 5 y$ . Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m + n$ is equal to :
1. A.23
2. B.26
3. C.25
4. D.24
Show answer
Answer: C
Q42024 · 6 Apr 2024 · Shift 2
Let $f (x) = \frac{1}{7 - s i n 5 x}$ be a function defined on $R$ . Then the range of the function $f (x)$ is equal to :
1. A. $[\frac{1}{8}, \frac{1}{5}]$
2. B. $[\frac{1}{7}, \frac{1}{6}]$
3. C. $[\frac{1}{7}, \frac{1}{5}]$
4. D. $[\frac{1}{8}, \frac{1}{6}]$
Show answer
Answer: D
Q142024 · 6 Apr 2024 · Shift 2
A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to:
1. A.125
2. B.160
3. C.150
4. D.180
Show answer
Answer: C
Q172024 · 6 Apr 2024 · Shift 2
Suppose for a differentiable function $h, h (0) = 0, h (1) = 1$ and $h^{'} (0) = h^{'} (1) = 2$ . If $g (x) = h (e^{x}) e^{h (x)}$ , then $g^{'} (0)$ is equal to:
1. A.4
2. B.5
3. C.3
4. D.8
Show answer
Answer: A
Q212024 · 6 Apr 2024 · Shift 2· Numerical answer
Let $[t]$ denote the greatest integer less than or equal to $t$ . Let $f : [0, \infty) \to R$ be a function defined by $f (x) = [\frac{x}{2} + 3] - [x]$ . Let $S$ be the set of all points in the interval $[0, 8]$ at which $f$ is not continuous. Then $\sum_{a \in S} a$ is equal to ____.
Show answer
Answer: 17
Q22024 · 5 Apr 2024 · Shift 2
Let $f, g : R \to R$ be defined as : $f (x) = ∣ x - 1∣$ and $g (x) = {e^{x}, x + 1, x \geq 0 x \leq 0.$ Then the function $f (g (x))$ is
1. A.neither one-one nor onto.
2. B.one-one but not onto.
3. C.both one-one and onto.
4. D.onto but not one-one.
Show answer
Answer: A
Q132024 · 5 Apr 2024 · Shift 2
Let the set $S = {2, 4, 8, 16, \dots, 512}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = ϕ$ . The maximum number of such possible partitions of $S$ is equal to:
1. A.1640
2. B.1520
3. C.1710
4. D.1680
Show answer
Answer: D
Q72024 · 4 Apr 2024 · Shift 1
Let $f (x) = x^{5} + 2 e^{x /4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g \circ f) (x) = x$ for all $x \in R$ . Then the value of $8 g^{'} (2)$ is :
1. A.4
2. B.2
3. C.16
4. D.8
Show answer
Answer: C
Q92024 · 4 Apr 2024 · Shift 1
If the domain of the function $sin^{- 1} (\frac{3 x - 22}{2 x - 19}) + lo g_{e} (\frac{3 x ^{2} - 8 x + 5}{x ^{2} - 3 x - 10})$ is $(α, β]$ , then $3 α + 10 β$ is equal to:
1. A.95
2. B.100
3. C.97
4. D.98
Show answer
Answer: C
Q172024 · 4 Apr 2024 · Shift 1
Let the sum of the maximum and the minimum values of the function $f (x) = \frac{2 x ^{2} - 3 x + 8}{2 x ^{2} + 3 x + 8}$ be $\frac{m}{n}$ , where $g cd (m, n) = 1$ . Then $m + n$ is equal to :
1. A.217
2. B.182
3. C.201
4. D.195
Show answer
Answer: C
Q62024 · 4 Apr 2024 · Shift 2
Let $f (x) = 3 x - 2 + 4 - x$ be a real valued function. If $α$ and $β$ are respectively the minimum and the maximum values of $f$ , then $α^{2} + 2 β^{2}$ is equal to
1. A.42
2. B.38
3. C.24
4. D.44
Show answer
Answer: A
Q182024 · 4 Apr 2024 · Shift 2
Let a relation $R$ on $N \times N$ be defined as: $(x_{1}, y_{1}) R (x_{2}, y_{2})$ if and only if $x_{1} \leq x_{2}$ or $y_{1} \leq y_{2}$ . Consider the two statements: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?
1. A.Only (II) is correct.
2. B.Both (I) and (II) are correct.
3. C.Neither (I) nor (II) is correct.
4. D.Only (I) is correct.
Show answer
Answer: D
Q262024 · 4 Apr 2024 · Shift 2· Numerical answer
Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{1 + 9 x ^{2}}$ . If the composition of $f$ , $10 times (f \circ f \circ f \circ \dots \circ f) (x) = \frac{2 ^{10} x}{1 + 9 α x ^{2}}$ , then the value of $3 α + 1$ is equal to _________.
Show answer
Answer: 1024
Q282024 · 4 Apr 2024 · Shift 2· Numerical answer
Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2$ and $f (4) = - 2$ . Then, the minimum number of zeros of $(3 f^{'} f^{''} + f f^{'''}) (x)$ is _________.
Show answer
Answer: 5
Q112024 · 1 Feb 2024 · Shift 1
Let $f : R \to R$ and $g : R \to R$ be defined as $f (x) = {lo g_{e} x, e^{- x}, x > 0 x \leq 0$ and $g (x) = {x, e^{x}, x ⩾ 0 x < 0$ . Then, gof : $R \to R$ is :
1. A.one-one but not onto
2. B.neither one-one nor onto
3. C.onto but not one-one
4. D.both one-one and onto
Show answer
Answer: B
Q222024 · 1 Feb 2024 · Shift 1· Numerical answer
The number of elements in the set $S = {(x, y, z) : x, y, z \in Z, x + 2 y + 3 z = 42, x, y, z ⩾ 0}$ equals ____.
Show answer
Answer: 169
Q302024 · 1 Feb 2024 · Shift 1· Numerical answer
Let $A = {1, 2, 3, \dots, 20}$ . Let $R_{1}$ and $R_{2}$ be two relations on $A$ such that $R_{1} = {(a, b) : b$ is divisible by $a}$ $R_{2} = {(a, b) : a$ is an integral multiple of $b}$ . Then, number of elements in $R_{1} - R_{2}$ is equal to ____.
Show answer
Answer: 46
Q12024 · 1 Feb 2024 · Shift 2
If the domain of the function $f (x) = \frac{x ^{2} - 25}{( 4 - x ^{2} )} + lo g_{10} (x^{2} + 2 x - 15)$ is $(- \infty, α) \cup [β, \infty)$ , then $α^{2} + β^{3}$ is equal to :
1. A.140
2. B.175
3. C.125
4. D.150
Show answer
Answer: D
Q42024 · 1 Feb 2024 · Shift 2
Consider the relations $R_{1}$ and $R_{2}$ defined as $a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1$ for all $a, b \in R$ and $(a, b) R_{2} (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in N \times N$ . Then :
1. A. $R_{1}$ and $R_{2}$ both are equivalence relations
2. B.Only $R_{1}$ is an equivalence relation
3. C.Only $R_{2}$ is an equivalence relation
4. D.Neither $R_{1}$ nor $R_{2}$ is an equivalence relation
Show answer
Answer: C
Q272024 · 1 Feb 2024 · Shift 2· Numerical answer
Let $A = I_{2} - 2 M M^{T}$ , where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^{T} M = I_{1}$ holds. If $λ$ is a real number such that the relation $A X = λ X$ holds for some non-zero real matrix $X$ of order $2 \times 1$ , then the sum of squares of all possible values of $λ$ is equal to ________ .
Show answer
Answer: 2
Q302024 · 1 Feb 2024 · Shift 2· Numerical answer
The lines $L_{1}, L_{2}, \dots, L_{20}$ are distinct. For $n = 1, 2, 3, \dots, 10$ all the lines $L_{2 n - 1}$ are parallel to each other and all the lines $L_{2 n}$ pass through a given point $P$ . The maximum number of points of intersection of pairs of lines from the set ${L_{1}, L_{2}, \dots, L_{20}}$ is equal to ________ .
Show answer
Answer: 101
Q292024 · 31 Jan 2024 · Shift 1· Numerical answer
Let $A = {1, 2, 3, 4}$ and $R = {(1, 2), (2, 3), (1, 4)}$ be a relation on $A$ . Let $S$ be the equivalence relation on $A$ such that $R \subset S$ and the number of elements in $S$ is $n$ . Then, the minimum value of $n$ is ____.
Show answer
Answer: 16
Q102024 · 31 Jan 2024 · Shift 2
Let $f, g : (0, \infty) \to R$ be two functions defined by $f (x) = \int_{- x}^{x} (∣ t ∣ - t^{2}) e^{- t^{2}} d t$ and $g (x) = \int_{0}^{x^{2}} t^{1/2} e^{- t} d t$ . Then, the value of $9 (f (lo g_{e} 9) + g (lo g_{e} 9))$ is equal to :
1. A.10
2. B.9
3. C.8
4. D.6
Show answer
Answer: C