Sets, Relations & Functions · JEE Main previous year questions

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2024 59 questions

Q12024 · 9 Apr 2024 · Shift 1
If the domain of the function $f (x) = sin^{- 1} (\frac{x - 1}{2 x + 3})$ is $R - (α, β)$ , then $12 α β$ is equal to :
1. A.40
2. B.36
3. C.24
4. D.32
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Answer: D
Q212024 · 9 Apr 2024 · Shift 1· Numerical answer
Let $A = {2, 3, 6, 7}$ and $B = {4, 5, 6, 8}$ . Let $R$ be a relation defined on $A \times B$ by $(a_{1}, b_{1}) R (a_{2}, b_{2})$ if and only if $a_{1} + a_{2} = b_{1} + b_{2}$ . Then the number of elements in $R$ is ______.
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Answer: 25
Q252024 · 9 Apr 2024 · Shift 1· Numerical answer
If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m, n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq (2022)^{2}$ is equal to ______.
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Answer: 1010
Q282024 · 9 Apr 2024 · Shift 1· Numerical answer
Let the set of all positive values of $λ$ , for which the point of local minimum of the function $(1 + x (λ^{2} - x^{2}))$ satisfies $\frac{x ^{2} + x + 2}{x ^{2} + 5 x + 6} < 0$ , be $(α, β)$ . Then $α^{2} + β^{2}$ is equal to ______.
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Answer: 39
Q32024 · 9 Apr 2024 · Shift 2
Let the range of the function $f (x) = \frac{1}{2 + s i n 3 x + c o s 3 x}$ , $x \in R$ be $[a, b]$ . If $α$ and $β$ are respectively the A.M. and the G.M. of $a$ and $b$ , then $\frac{α}{β}$ is equal to
1. A. $π$
2. B. $π$
3. C. $2$
4. D. $2$
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Answer: C
Q212024 · 9 Apr 2024 · Shift 2· Numerical answer
Let $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$ and $B = {x : (x, y) \in A}$ . Then the number of one-one functions from $A$ to $B$ is equal to ____.
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Answer: 24
Q82024 · 8 Apr 2024 · Shift 1
The number of critical points of the function $f (x) = (x - 2)^{2/3} (2 x + 1)$ is
1. A. $2$
2. B. $1$
3. C. $0$
4. D. $3$
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Answer: A
Q172024 · 8 Apr 2024 · Shift 1
Let $[t]$ be the greatest integer less than or equal to $t$ . Let $A$ be the set of all prime factors of 2310 and $f : A \to Z$ be the function $f (x) = [lo g_{2} (x^{2} + [\frac{x ^{3}}{5}])]$ . The number of one-to-one functions from $A$ to the range of $f$ is
1. A. $20$
2. B. $120$
3. C. $25$
4. D. $24$
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Answer: B
Q182024 · 8 Apr 2024 · Shift 1
If the set $R = {(a, b) : a + 5 b = 42, a, b \in N}$ has $m$ elements and $\sum_{n = 1}^{m} (1 - i^{n!}) = x + i y$ , where $i = - 1$ , then the value of $m + x + y$ is
1. A. $12$
2. B. $4$
3. C. $8$
4. D. $5$
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Answer: A
Q102024 · 8 Apr 2024 · Shift 2
Let $f (x) = {- a x + a if - a \leq x \leq 0 if 0 < x \leq a$ where $a > 0$ and $g (x) = (f (∣ x ∣) - ∣ f (x) ∣) /2$ . Then the function $g : [- a, a] \to [- a, a]$ is
1. A.neither one-one nor onto.
2. B.both one-one and onto.
3. C.one-one.
4. D.onto
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Answer: A
Q132024 · 8 Apr 2024 · Shift 2
Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is
1. A.reflexive but not symmetric.
2. B.an equivalence relation.
3. C.reflexive and symmetric but not transitive.
4. D.transitive but not symmetric.
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Answer: C
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. $x^{2} + 6 x + 8 = 0$
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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. $e^{π} > π^{e}$
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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
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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. $[\frac{1}{8}, \frac{1}{6}]$
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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
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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
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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 ____.
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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.
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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
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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
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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
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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
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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
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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.
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Answer: D