Sets, Relations & Functions · JEE Main previous year questions

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2023 67 questions

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
Q252023 · 6 Apr 2023 · Shift 1· Numerical answer
Let $A = {1, 2, 3, 4, \dots, 10}$ and $B = {0, 1, 2, 3, 4}$ . The number of elements in the relation $R = {(a, b) \in A \times A : 2 (a - b)^{2} + 3 (a - b) \in B}$ is ________.
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Answer: 18
Q42023 · 6 Apr 2023 · Shift 2
Let $a \neq = b$ be two non-zero real numbers. Then the number of elements in the set $X = {z \in C : Re (a z^{2} + b z) = a and Re (b z^{2} + a z) = b}$ is equal to :
1. A.0
2. B.2
3. C.1
4. D.Infinite
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Answer: D
Q82023 · 6 Apr 2023 · Shift 2
Let $f (x)$ be a function satisfying $f (x) + f (π - x) = π^{2}, \forall x \in R$ . Then $\int_{0}^{π} f (x) sin x d x$ is equal to :
1. A. $π^{2}$
2. B. $\frac{π ^{2}}{2}$
3. C. $2 π^{2}$
4. D. $\frac{π ^{2}}{4}$
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Answer: A
Q132023 · 6 Apr 2023 · Shift 2
Let the sets A and B denote the domain and range respectively of the function $f (x) = \frac{1}{⌈ x ⌉ - x}$ , where $⌈ x ⌉$ denotes the smallest integer greater than or equal to $x$ . Then among the statements (S1) : $A \cap B = (1, \infty) - N$ and (S2) : $A \cup B = (1, \infty)$
1. A.only (S2) is true
2. B.only (S1) is true
3. C.neither (S1) nor (S2) is true
4. D.both (S1) and (S2) are true
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Answer: B
Q32023 · 1 Feb 2023 · Shift 1
Let $R$ be a relation on $R$ , given by $R = {(a, b) : 3 a - 3 b + 7 is an irrational number}$ . Then $R$ is
1. A.an equivalence relation
2. B.reflexive and symmetric but not transitive
3. C.reflexive and transitive but not symmetric
4. D.reflexive but neither symmetric nor transitive
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Answer: D
Q232023 · 1 Feb 2023 · Shift 1· Numerical answer
Let $f : R \to R$ be a differentiable function such that $f^{'} (x) + f (x) = \int_{0}^{2} f (t) d t$ . If $f (0) = e^{- 2}$ , then $2 f (0) - f (2)$ is equal to ________.
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Answer: 1
Q62023 · 31 Jan 2023 · Shift 1
Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $a d (b - c) = b c (a - d)$ . Then $R$ is
1. A.symmetric and transitive but not reflexive
2. B.reflexive and symmetric but not transitive
3. C.transitive but neither reflexive nor symmetric
4. D.symmetric but neither reflexive nor transitive
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Answer: D
Q72023 · 31 Jan 2023 · Shift 1
If the domain of the function $f (x) = \frac{[ x ]}{1 + x ^{2}}$ , where $[x]$ is greatest integer $\leq x$ , is $[2, 6)$ , then its range is
1. A. $(\frac{5}{37}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$
2. B. $(\frac{5}{37}, \frac{2}{5}]$
3. C. $(\frac{5}{26}, \frac{2}{5}]$
4. D. $(\frac{5}{26}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$
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Answer: B
Q162023 · 31 Jan 2023 · Shift 1
Let a differentiable function $f$ satisfy $f (x) + \int_{3}^{x} \frac{f ( t )}{t} d t = x + 1, x \geq 3$ . Then $12 f (8)$ is equal to :
1. A.19
2. B.34
3. C.17
4. D.1
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Answer: C
Q22023 · 31 Jan 2023 · Shift 2
Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} - 8 x + 12}$ . Then range of $f$ is
1. A. $(- \infty, - \frac{21}{4}] \cup [1, \infty)$
2. B. $(- \infty, - \frac{21}{4}) \cup (0, \infty)$
3. C. $(- \infty, - \frac{21}{4}] \cup [0, \infty)$
4. D. $(- \infty, - \frac{21}{4}] \cup [\frac{21}{4}, \infty)$
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Answer: C
Q92023 · 31 Jan 2023 · Shift 2
Among the relations $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and $T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ ,
1. A. $S$ is transitive but $T$ is not
2. B.both $S$ and $T$ are symmetric
3. C.neither $S$ nor $T$ is transitive
4. D. $T$ is symmetric but $S$ is not
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Answer: D
Q102023 · 31 Jan 2023 · Shift 2
The absolute minimum value, of the function $f (x) = x^{2} - x + 1 + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ , is :
1. A. $\frac{3}{4}$
2. B. $\frac{3}{2}$
3. C. $\frac{1}{4}$
4. D. $\frac{5}{4}$
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Answer: A
Q22023 · 30 Jan 2023 · Shift 1
The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is :
1. A.7
2. B.3
3. C.4
4. D.5
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Answer: A
Q62023 · 30 Jan 2023 · Shift 1
Suppose $f : R \to (0, \infty)$ be a differentiable function such that $5 f (x + y) = f (x) \cdot f (y), \forall x, y \in R$ . If $f (3) = 320$ , then $n = 0 \sum 5 f (n)$ is equal to :
1. A.6875
2. B.6525
3. C.6575
4. D.6825
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Answer: D
Q252023 · 30 Jan 2023 · Shift 1· Numerical answer
Let $S = {1, 2, 3, 4, 5, 6}$ . Then the number of one-one functions $f : S \to P (S)$ , where $P (S)$ denote the power set of $S$ , such that $f (n) \subset f (m)$ where $n < m$ is ____.
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Answer: 3240
Q22023 · 29 Jan 2023 · Shift 1
Let $B$ and $C$ be the two points on the line $y + x = 0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y - 2 x = 2$ such that $△ A B C$ is an equilateral triangle. Then, the area of the $△ A B C$ is :
1. A. $\frac{10}{3}$
2. B. $23$
3. C. $33$
4. D. $\frac{8}{3}$
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Answer: D
Q72023 · 29 Jan 2023 · Shift 1
Let $f (x) = x + \frac{a}{π ^{2} - 4} sin x + \frac{b}{π ^{2} - 4} cos x, x \in R$ be a function which satisfies $f (x) = x + \int_{0}^{π /2} sin (x + y) f (y) d y$ . then $(a + b)$ is equal to
1. A. $- 2 π (π + 2)$
2. B. $- π (π - 2)$
3. C. $- π (π + 2)$
4. D. $- 2 π (π - 2)$
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Answer: A
Q132023 · 29 Jan 2023 · Shift 1
The domain of $f (x) = \frac{l o g _{(x + 1)} ( x - 2 )}{e ^{2 l o g_{e} x} - ( 2 x + 3 )}, x \in R$ is
1. A. $(- 1, \infty) - {3}$
2. B. $R - {- 1, 3}$
3. C. $(2, \infty) - {3}$
4. D. $R - {3}$
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Answer: C
Q142023 · 29 Jan 2023 · Shift 1
Let $f : R \to R$ be a function such that $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} + 1}$ . Then
1. A. $f (x)$ is many-one in $(- \infty, - 1)$
2. B. $f (x)$ is one-one in $(- \infty, \infty)$
3. C. $f (x)$ is one-one in $[1, \infty)$ but not in $(- \infty, \infty)$
4. D. $f (x)$ is many-one in $(1, \infty)$
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Answer: C
Q152023 · 29 Jan 2023 · Shift 1
Let $[x]$ denote the greatest integer $\leq x$ . Consider the function $f (x) = max {x^{2}, 1 + [x]}$ . Then the value of the integral $\int_{0}^{2} f (x) d x$ is
1. A. $\frac{5 + 4 2}{3}$
2. B. $\frac{4 + 5 2}{3}$
3. C. $\frac{8 + 4 2}{3}$
4. D. $\frac{1 + 5 2}{3}$
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Answer: A
Q222023 · 29 Jan 2023 · Shift 1· Numerical answer
Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in N$ and $f (1) = \frac{1}{5}$ . If $n = 1 \sum m \frac{f ( n )}{n ( n + 1 ) ( n + 2 )} = \frac{1}{12}$ , then $m$ is equal to ____.
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Answer: 10
Q262023 · 29 Jan 2023 · Shift 1· Numerical answer
Let $f : R \to R$ be a differentiable function that satisfies the relation $f (x + y) = f (x) + f (y) - 1, \forall x, y \in R$ . If $f^{'} (0) = 2$ , then $∣ f (- 2) ∣$ is equal to ____.
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Answer: 3
Q22023 · 29 Jan 2023 · Shift 2
Let R be a relation defined on $N$ as $a R b$ if $2 a + 3 b$ is a multiple of $5, a, b \in N$ . Then R is
1. A.an equivalence relation
2. B.non reflexive
3. C.symmetric but not transitive
4. D.transitive but not symmetric
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Answer: A