Sets, Relations & Functions · JEE Main previous year questions

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2023 67 questions

Q132023 · 13 Apr 2023 · Shift 1
For the differentiable function $f : R - {0} \to R$ , let $3 f (x) + 2 f (\frac{1}{x}) = \frac{1}{x} - 10$ , then $∣ f (3) + f^{'} (\frac{1}{4}) ∣$ is equal to
1. A. $13$
2. B. $\frac{29}{5}$
3. C. $\frac{33}{5}$
4. D. $7$
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Answer: A
Q132023 · 13 Apr 2023 · Shift 2
The range of $f (x) = 4 sin^{- 1} (\frac{x ^{2}}{x ^{2} + 1})$ is
1. A. $[0, 2 π]$
2. B. $(0, 2 π)$
3. C. $[0, π)$
4. D. $[0, π]$
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Answer: B
Q222023 · 13 Apr 2023 · Shift 2· Numerical answer
Let $A = {- 4, - 3, - 2, 0, 1, 3, 4}$ and $R = {(a, b) \in A \times A : b = ∣ a ∣ or b^{2} = a + 1}$ be a relation on $A$ . Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is _________
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Answer: 7
Q102023 · 12 Apr 2023 · Shift 1
If the local maximum value of the function $f (x) = (\frac{3 e}{2 s i n x})^{s i n^{2} x}$ , $x \in (0, \frac{π}{2})$ , is $\frac{k}{e}$ , then $(\frac{k}{e})^{8} + \frac{k ^{8}}{e ^{5}} + k^{8}$ is equal to
1. A. $e^{3} + e^{6} + e^{10}$
2. B. $e^{3} + e^{5} + e^{11}$
3. C. $e^{3} + e^{6} + e^{11}$
4. D. $e^{5} + e^{6} + e^{11}$
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Answer: C
Q122023 · 12 Apr 2023 · Shift 1
Let $D$ be the domain of the function $f (x) = sin^{- 1} (lo g_{3 x} (\frac{6 + 2 l o g _{3} x}{- 5 x}))$ . If the range of the function $g : D \to R$ defined by $g (x) = x - [x]$ , ( $[x]$ is the greatest integer function), is $(α, β)$ , then $α^{2} + \frac{5}{β}$ is equal to
1. A. $45$
2. B. $136$
3. C. $46$
4. D. $135$
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Answer: D
Q262023 · 12 Apr 2023 · Shift 1· Numerical answer
The number of relations, on the set ${1, 2, 3}$ containing $(1, 2)$ and $(2, 3)$ , which are reflexive and transitive but not symmetric, is _________.
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Answer: 3
Q62023 · 11 Apr 2023 · Shift 2
Let $A = {1, 3, 4, 6, 9}$ and $B = {2, 4, 5, 8, 10}$ . Let $R$ be a relation defined on $A \times B$ such that $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2} and b_{1} \leq a_{2}}$ . Then the number of elements in the set $R$ is :
1. A. $180$
2. B. $26$
3. C. $52$
4. D. $160$
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Answer: D
Q112023 · 11 Apr 2023 · Shift 2
Let $f$ and $g$ be two functions defined by $f (x) = {x + 1, ∣ x - 1∣, x < 0 x \geq 0 and g (x) = {x + 1, 1, x < 0 x \geq 0$ Then $(g \circ f) (x)$ is :
1. A.continuous everywhere but not differentiable at $x = 1$
2. B.differentiable everywhere
3. C.not continuous at $x = - 1$
4. D.continuous everywhere but not differentiable exactly at one point
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Answer: D
Q142023 · 11 Apr 2023 · Shift 2
The domain of the function $f (x) = \frac{1}{[ x ] ^{2} - 3 [ x ] - 10}$ is : (where $[x]$ denotes the greatest integer less than or equal to $x$ )
1. A. $(- \infty, - 2) \cup [6, \infty)$
2. B. $(- \infty, - 3] \cup [6, \infty)$
3. C. $(- \infty, - 2) \cup (5, \infty)$
4. D. $(- \infty, - 3] \cup (5, \infty)$
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Answer: A
Q222023 · 11 Apr 2023 · Shift 2· Numerical answer
Let $A = {1, 2, 3, 4, 5}$ and $B = {1, 2, 3, 4, 5, 6}$ . Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to _________.
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Answer: 360
Q232023 · 10 Apr 2023 · Shift 1· Numerical answer
The number of elements in the set ${n \in Z : ∣ n^{2} - 10 n + 19∣ < 6}$ is _________.
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Answer: 6
Q122023 · 10 Apr 2023 · Shift 2
Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$ . Then the number of elements in the relation $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1} divides b_{2} and a_{2} divides b_{1}}$ is :
1. A.18
2. B.24
3. C.36
4. D.12
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Answer: C
Q242023 · 10 Apr 2023 · Shift 2· Numerical answer
Let S be the set of values of $λ$ , for which the system of equations $6 λ x - 3 y + 3 z = 4 λ^{2},$ $2 x + 6 λ y + 4 z = 1,$ $3 x + 2 y + 3 λ z = λ$ has no solution. Then $12 \sum_{λ \in S} ∣ λ ∣$ is equal to _________.
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Answer: 24
Q252023 · 10 Apr 2023 · Shift 2· Numerical answer
If the domain of the function $f (x) = sec^{- 1} (\frac{2 x}{5 x + 3})$ is $[α, β) \cup (γ, δ]$ , then $∣3 α + 10 (β + γ) + 21 δ ∣$ is equal to _________.
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Answer: 24
Q132023 · 8 Apr 2023 · Shift 1
Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is :
1. A. $782$
2. B. $772$
3. C. $752$
4. D. $792$
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Answer: D
Q222023 · 8 Apr 2023 · Shift 1· Numerical answer
Let $A = {0, 3, 4, 6, 7, 8, 9, 10}$ and $R$ be the relation defined on $A$ such that $R = {(x, y) \in A \times A : x - y is odd positive integer or x - y = 2}$ . The minimum number of elements that must be added to the relation $R$ , so that it is a symmetric relation, is equal to ___________.
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Answer: 19
Q52023 · 8 Apr 2023 · Shift 2
Let $A = {1, 2, 3, 4, 5, 6, 7}$ . Then the relation $R = {(x, y) \in A \times A : x + y = 7}$ is :
1. A.reflexive but neither symmetric nor transitive
2. B.transitive but neither symmetric nor reflexive
3. C.symmetric but neither reflexive nor transitive
4. D.an equivalence relation
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Answer: C
Q132023 · 8 Apr 2023 · Shift 2
Let $S$ be the set of all values of $θ \in [- π, π]$ for which the system of linear equations $x + y + 3 z = 0$ $- x + (tan θ) y + 7 z = 0$ $x + y + (tan θ) z = 0$ has non-trivial solution. Then $\frac{120}{π} \sum_{θ \in S} θ$ is equal to :
1. A. $40$
2. B. $30$
3. C. $10$
4. D. $20$
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Answer: D
Q212023 · 8 Apr 2023 · Shift 2· Numerical answer
Let $[t]$ denote the greatest integer function. If $\int_{0}^{2.4} [x^{2}] d x = α + β 2 + γ 3 + δ 5$ , then $α + β + γ + δ$ is equal to ___________.
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Answer: 6
Q242023 · 8 Apr 2023 · Shift 2· Numerical answer
Let $R = {a, b, c, d, e}$ and $S = {1, 2, 3, 4}$ . Total number of onto functions $f : R \to S$ such that $f (a) \neq = 1$ , is equal to ___________.
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Answer: 180
Q252023 · 8 Apr 2023 · Shift 2· Numerical answer
Let $k$ and $m$ be positive real numbers such that the function $f (x) = {3 x^{2} + k x + 1, m x^{2} + k^{2}, 0 < x < 1 x \geq 1$ is differentiable for all $x > 0$ . Then $\frac{8 f ^{'} ( 8 )}{f ^{'} ( \frac{1}{8} )}$ is equal to ___________.
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Answer: 309
Q272023 · 8 Apr 2023 · Shift 2· Numerical answer
If domain of the function $lo g_{e} (\frac{6 x ^{2} + 5 x + 1}{2 x - 1}) + cos^{- 1} (\frac{2 x ^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ]$ , then $18 (α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to ___________.
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Answer: 20
Q12023 · 6 Apr 2023 · Shift 1
Let $A = {x \in R : [x + 3] + [x + 4] \leq 3}$ , $B = ⎩ ⎨ ⎧ x \in R : 3^{x} (r = 1 \sum \infty \frac{3}{1 0 ^{r}})^{x - 3} < 3^{- 3 x} ⎭ ⎬ ⎫,$ where [t] denotes greatest integer function. Then,
1. A. $B \subset C, A \neq = B$
2. B. $A \subset B, A \neq = B$
3. C. $A = B$
4. D. $A \cap B = ϕ$
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Answer: C
Q232023 · 6 Apr 2023 · Shift 1· Numerical answer
Let the point $(p, p + 1)$ lie inside the region $E = {(x, y) : 3 - x \leq y \leq 9 - x^{2}, 0 \leq x \leq 3}$ . If the set of all values of $p$ is the interval $(a, b)$ , then $b^{2} + b - a^{2}$ is equal to ________.
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Answer: 3
Q242023 · 6 Apr 2023 · Shift 1· Numerical answer
Let $a \in Z$ and $[t]$ be the greatest integer $\leq t$ . Then the number of points, where the function $f (x) = [a + 13 sin x], x \in (0, π)$ is not differentiable, is ________.
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Answer: 25