Sets, Relations & Functions · JEE Main previous year questions

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2024 59 questions

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 _________.
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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 _________.
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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
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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 ____.
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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 ____.
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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
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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
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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 ________ .
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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 ________ .
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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 ____.
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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
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Answer: C
Q132024 · 31 Jan 2024 · Shift 2
If the function $f : (- \infty, - 1] \to (a, b]$ defined by $f (x) = e^{x^{3} - 3 x + 1}$ is one - one and onto, then the distance of the point $P (2 b + 4, a + 2)$ from the line $x + e^{- 3} y = 4$ is :
1. A. $2 1 + e^{6}$
2. B. $1 + e^{6}$
3. C. $3 1 + e^{6}$
4. D. $4 1 + e^{6}$
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Answer: A
Q252024 · 31 Jan 2024 · Shift 2· Numerical answer
Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $(a^{2} + b^{2}) x^{2} - 2 b (a + c) x + (b^{2} + c^{2}) = 0$ . If the set of all possible values of $x$ is the interval $(α, β)$ , then $12 (α^{2} + β^{2})$ is equal to ____.
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Answer: 36
Q272024 · 31 Jan 2024 · Shift 2· Numerical answer
Let $A = {1, 2, 3, \dots\dots\dots\dots, 100}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $2 x = 3 y$ . Let $R_{1}$ be a symmetric relation on $A$ such that $R \subset R_{1}$ and the number of elements in $R_{1}$ is $n$ . Then, the minimum value of $n$ is ____.
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Answer: 66
Q12024 · 30 Jan 2024 · Shift 1
Let $g : R \to R$ be a non constant twice differentiable function such that $g^{'} (\frac{1}{2}) = g^{'} (\frac{3}{2})$ . If a real valued function $f$ is defined as $f (x) = \frac{1}{2} [g (x) + g (2 - x)]$ , then
1. A. $f^{''} (x) = 0$ for at least two $x$ in $(0, 2)$
2. B. $f^{'} (\frac{3}{2}) + f^{'} (\frac{1}{2}) = 1$
3. C. $f^{''} (x) = 0$ for no $x$ in $(0, 1)$
4. D. $f^{''} (x) = 0$ for exactly one $x$ in $(0, 1)$
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Answer: A
Q182024 · 30 Jan 2024 · Shift 1
If the domain of the function $f (x) = cos^{- 1} (\frac{2 - ∣ x ∣}{4}) + {lo g_{e} (3 - x)}^{- 1}$ is $[- α, β) - {γ}$ , then $α + β + γ$ is equal to :
1. A.11
2. B.12
3. C.9
4. D.8
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Answer: A
Q292024 · 30 Jan 2024 · Shift 1· Numerical answer
Let $A = {1, 2, 3, \dots, 7}$ and let $P (A) denote the power set of A. If the number of functions f : A \to P (A)$ such that $a \in f (a), \forall a \in A$ is $m^{n}, m$ and $n \in N$ and $m$ is least, then $m + n$ is equal to _________.
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Answer: 44
Q302024 · 30 Jan 2024 · Shift 1· Numerical answer
If the function $f (x) = {\frac{1}{∣ x ∣}, a x^{2} + 2 b, ∣ x ∣ \geq 2 ∣ x ∣ < 2$ is differentiable on $R$ , then $48 (a + b)$ is equal to _________.
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Answer: 15
Q92024 · 29 Jan 2024 · Shift 1
Let $R$ be a relation on $Z \times Z$ defined by $(a, b) R (c, d)$ if and only if $a d - b c$ is divisible by 5. Then $R$ is
1. A.Reflexive and transitive but not symmetric
2. B.Reflexive and symmetric but not transitive
3. C.Reflexive but neither symmetric nor transitive
4. D.Reflexive, symmetric and transitive
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Answer: B
Q102024 · 29 Jan 2024 · Shift 1
If $f (x) = {2 + 2 x, 1 - \frac{x}{3}, - 1 \leq x < 0 0 \leq x \leq 3$ ; $g (x) = {- x, x, - 3 \leq x \leq 0 0 < x \leq 1$ , then range of $(f \circ g) (x)$ is
1. A. $[0, 1)$
2. B. $[0, 3)$
3. C. $(0, 1]$
4. D. $[0, 1]$
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Answer: D
Q192024 · 29 Jan 2024 · Shift 1
A function $y = f (x)$ satisfies $f (x) sin 2 x + sin x - (1 + cos^{2} x) f^{'} (x) = 0$ with condition $f (0) = 0$ . Then, $f (\frac{π}{2})$ is equal to
1. A. $2$
2. B. $1$
3. C. $- 1$
4. D. $0$
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Answer: B
Q62024 · 29 Jan 2024 · Shift 2
The function $f (x) = \frac{x}{x ^{2} - 6 x - 16}, x \in R - {- 2, 8}$
1. A.decreases in $(- \infty, - 2) \cup (- 2, 8) \cup (8, \infty)$
2. B.increases in $(- \infty, - 2) \cup (- 2, 8) \cup (8, \infty)$
3. C.decreases in $(- 2, 8)$ and increases in $(- \infty, - 2) \cup (8, \infty)$
4. D.decreases in $(- \infty, - 2)$ and increases in $(8, \infty)$
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Answer: A
Q72024 · 29 Jan 2024 · Shift 2
If R is the smallest equivalence relation on the set ${1, 2, 3, 4}$ such that ${(1, 2), (1, 3)} \subset R$ , then the number of elements in R is _________.
1. A.15
2. B.10
3. C.12
4. D.8
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Answer: B
Q152024 · 29 Jan 2024 · Shift 2
The function $f (x) = 2 x + 3 (x)^{\frac{2}{3}}, x \in R$ , has
1. A.exactly one point of local minima and no point of local maxima
2. B.exactly one point of local maxima and exactly one point of local minima
3. C.exactly two points of local maxima and exactly one point of local minima
4. D.exactly one point of local maxima and no point of local minima
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Answer: B
Q232024 · 29 Jan 2024 · Shift 2· Numerical answer
Let the set $C = {(x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N}$ . Then $\sum_{(x, y) \in C} (x + y)$ is equal to _________.
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Answer: 46