Sets, Relations & Functions · JEE Main previous year questions

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2025 42 questions

Q252025 · 8 Apr 2025 · Shift 2· Numerical answer
Let the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 5}{3 x - 7})$ be $[α, β]$ and the domain of $g (x) = lo g_{2} (2 - 6 lo g_{27} (2 x + 5))$ be $(γ, δ)$ . Then $∣7 (α + β) + 4 (γ + δ) ∣$ is equal to .
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Answer: 96
Q62025 · 4 Apr 2025 · Shift 1
Consider the sets $A = {(x, y) \in R \times R : x^{2} + y^{2} = 25}$ , $B = {(x, y) \in R \times R : x^{2} + 9 y^{2} = 144}$ , $C = {(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4}$ and $D = A \cap B$ . The total number of one-one functions from the set $D$ to the set $C$ is:
1. A. $15120$
2. B. $18290$
3. C. $17160$
4. D. $19320$
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Answer: C
Q162025 · 4 Apr 2025 · Shift 1
Let $f, g : (1, \infty) \to R$ be defined as $f (x) = \frac{2 x + 3}{5 x + 2}$ and $g (x) = \frac{2 - 3 x}{1 - x}$ . If the range of the function $f \circ g : [2, 4] \to R$ is $[α, β]$ , then $\frac{1}{β - α}$ is equal to
1. A. $56$
2. B. $2$
3. C. $29$
4. D. $68$
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Answer: A
Q12025 · 4 Apr 2025 · Shift 2
Let $A = {- 3, - 2, - 1, 0, 1, 2, 3}$ and R be a relation on A defined by $x R y$ if and only if $2 x - y \in {0, 1}$ . Let $l$ be the number of elements in $R$ . Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to:
1. A.17
2. B.18
3. C.15
4. D.16
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Answer: A
Q22025 · 4 Apr 2025 · Shift 2
Let the domains of the functions $f (x) = lo g_{4} lo g_{3} lo g_{7} (8 - lo g_{2} (x^{2} + 4 x + 5))$ and $g (x) = sin^{- 1} (\frac{7 x + 10}{x - 2})$ be $(α, β)$ and $[γ, δ]$ , respectively. Then $α^{2} + β^{2} + γ^{2} + δ^{2}$ is equal to :
1. A.15
2. B.13
3. C.16
4. D.14
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Answer: A
Q52025 · 4 Apr 2025 · Shift 2
Let $a > 0$ . If the function $f (x) = 6 x^{3} - 45 a x^{2} + 108 a^{2} x + 1$ attains its local maximum and minimum values at the points $x_{1}$ and $x_{2}$ respectively such that $x_{1} x_{2} = 54$ , then $a + x_{1} + x_{2}$ is equal to :
1. A.15
2. B.13
3. C.24
4. D.18
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Answer: D
Q12025 · 3 Apr 2025 · Shift 1
If the domain of the function $f (x) = lo g_{e} (\frac{2 x - 3}{5 + 4 x}) + sin^{- 1} (\frac{4 + 3 x}{2 - x})$ is $[α, β)$ , then $α^{2} + 4 β$ is equal to
1. A.4
2. B.3
3. C.7
4. D.5
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Answer: A
Q52025 · 3 Apr 2025 · Shift 1
Let $A = {- 3, - 2, - 1, 0, 1, 2, 3}$ . Let $R$ be a relation on A defined by $x R y$ if and only if $0 \leq x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added in $R$ to make it a reflexive relation. Then $l + m$ is equal to
1. A.18
2. B.20
3. C.17
4. D.19
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Answer: A
Q82025 · 3 Apr 2025 · Shift 1
Let the domain of the function $f (x) = lo g_{2} lo g_{4} lo g_{6} (3 + 4 x - x^{2})$ be $(a, b)$ . If $\int_{0}^{b - a} [x^{2}] d x = p - q - r$ , $p, q, r \in N$ , $g cd (p, q, r) = 1$ , where $[\cdot]$ is the greatest integer function, then $p + q + r$ is equal to
1. A.10
2. B.11
3. C.9
4. D.8
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Answer: A
Q92025 · 3 Apr 2025 · Shift 2
Let $f$ be a function such that $f (x) + 3 f (\frac{24}{x}) = 4 x, x \neq = 0$ . Then $f (3) + f (8)$ is equal to
1. A. $13$
2. B. $11$
3. C. $10$
4. D. $12$
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Answer: B
Q112025 · 3 Apr 2025 · Shift 2
Let $A = {- 2, - 1, 0, 1, 2, 3}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $y = max {x, 1}$ . Let $l$ be the number of elements in $R$ . Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to
1. A. $11$
2. B. $12$
3. C. $14$
4. D. $13$
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Answer: B
Q172025 · 3 Apr 2025 · Shift 2
If the domain of the function $f (x) = lo g_{7} (1 - lo g_{4} (x^{2} - 9 x + 18))$ is $(α, β) \cup (γ, δ)$ , then $α + β + γ + δ$ is equal to
1. A. $17$
2. B. $15$
3. C. $16$
4. D. $18$
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Answer: D
Q182025 · 3 Apr 2025 · Shift 2
Let $f : R \to R$ be a function defined by $f (x) = ∣∣ x + 2∣ - 2∣ x ∣∣$ . If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$ , then $m + n$ is
1. A. $3$
2. B. $4$
3. C. $2$
4. D. $5$
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Answer: A
Q72025 · 2 Apr 2025 · Shift 1
If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ , where $a > 0$ , attains its local maximum and local minimum values at $p$ and $q$ , respectively, such that $p^{2} = q$ , then $f (3)$ is equal to :
1. A.55
2. B.37
3. C.10
4. D.23
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Answer: B
Q152025 · 2 Apr 2025 · Shift 1
Let $f : R \to R$ be a twice differentiable function such that $(sin x cos y) (f (2 x + 2 y) - f (2 x - 2 y)) = (cos x sin y) (f (2 x + 2 y) + f (2 x - 2 y))$ , for all $x, y \in R$ . If $f^{'} (0) = \frac{1}{2}$ , then the value of $24 f^{''} (\frac{5 π}{3})$ is :
1. A.2
2. B.3
3. C.-3
4. D.-2
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Answer: C
Q162025 · 2 Apr 2025 · Shift 1
Let $A$ be the set of all functions $f : Z \to Z$ and $R$ be a relation on $A$ such that $R = {(f, g) : f (0) = g (1) and f (1) = g (0)}$ . Then $R$ is :
1. A.Symmetric and transitive but not reflexive
2. B.Symmetric but neither reflexive nor transitive
3. C.Transitive but neither reflexive nor symmetric
4. D.Reflexive but neither symmetric nor transitive
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Answer: B
Q222025 · 2 Apr 2025 · Shift 1· Numerical answer
Let $f : R \to R$ be a thrice differentiable odd function satisfying $f^{'} (x) \geq 0, f^{''} (x) = f (x), f (0) = 0, f^{'} (0) = 3$ . Then $9 f (lo g_{e} 3)$ is equal to _________ .
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Answer: 36
Q252025 · 2 Apr 2025 · Shift 1· Numerical answer
Let $[.]$ denote the greatest integer function. If $\int_{0}^{e^{3}} [\frac{1}{e ^{x} - 1}] d x = α - lo g_{e} 2$ , then $α^{3}$ is equal to _________ .
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Answer: 8
Q102025 · 2 Apr 2025 · Shift 2
Let $A = {1, 2, 3, \dots, 100}$ and $R$ be a relation on $A$ such that $R = {(a, b) : a = 2 b + 1}$ . Let $(a_{1}, a_{2}), (a_{2}, a_{3}), (a_{3}, a_{4}), \dots, (a_{k}, a_{k + 1})$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$ , for which such a sequence exists, is equal to :
1. A.6
2. B.8
3. C.7
4. D.5
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Answer: D
Q112025 · 2 Apr 2025 · Shift 2
If the domain of the function $f (x) = \frac{1}{10 + 3 x - x ^{2}} + \frac{1}{x + ∣ x ∣}$ is $(a, b)$ , then $(1 + a)^{2} + b^{2}$ is equal to :
1. A.29
2. B.30
3. C.25
4. D.26
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Answer: D
Q222025 · 2 Apr 2025 · Shift 2· Numerical answer
If the set of all $a \in R - {1}$ , for which the roots of the equation $(1 - a) x^{2} + 2 (a - 3) x + 9 = 0$ are positive is $(- \infty, - α] \cup [β, γ)$ , then $2 α + β + γ$ is equal to .
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Answer: 7
Q12025 · 29 Jan 2025 · Shift 2
Let $S = N \cup {0}$ . Define a relation R from S to $R$ by : $R = {(x, y) : lo g_{e} y = x lo g_{e} (\frac{2}{5}), x \in S, y \in R}$ . Then, the sum of all the elements in the range of $R$ is equal to :
1. A. $\frac{3}{2}$
2. B. $\frac{10}{9}$
3. C. $\frac{5}{2}$
4. D. $\frac{5}{3}$
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Answer: D
Q42025 · 29 Jan 2025 · Shift 2
If the domain of the function $lo g_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $lo g_{(x - 1)} (\frac{2 x ^{2} + 3 x - 2}{x ^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is equal to:
1. A. $186$
2. B. $179$
3. C. $195$
4. D. $174$
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Answer: A
Q72025 · 29 Jan 2025 · Shift 2
If the set of all $a \in R$ , for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z; α < x < β}$ , then $\sum_{x \in X} x^{2}$ is equal to:
1. A. $2139$
2. B. $2119$
3. C. $2109$
4. D. $2129$
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Answer: A
Q212025 · 29 Jan 2025 · Shift 2· Numerical answer
If $24 0 \int \frac{π}{4} (sin 4 x - \frac{π}{12} + [2 sin x]) dx = 2 π + α$ , where $[\cdot]$ denotes the greatest integer function, then $α$ is equal to ____.
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Answer: 12