Sets, Relations & Functions · JEE Main previous year questions

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2022 44 questions

Q12022 · 29 Jul 2022 · Shift 1
Let $R$ be a relation from the set ${1, 2, 3, \dots, 60}$ to itself such that $R = {(a, b) : b = pq$ , where $p, q ⩾ 3$ are prime numbers $}$ . Then, the number of elements in $R$ is :
1. A.600
2. B.660
3. C.540
4. D.720
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Answer: B
Q132022 · 29 Jul 2022 · Shift 1
The number of points, where the function $f : R \to R$ , $f (x) = ∣ x - 1∣ cos ∣ x - 2∣ sin ∣ x - 1∣ + (x - 3) x^{2} - 5 x + 4$ , is NOT differentiable, is :
1. A.1
2. B.2
3. C.3
4. D.4
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Answer: B
Q262022 · 29 Jul 2022 · Shift 1· Numerical answer
Let $S = {4, 6, 9}$ and $T = {9, 10, 11, \dots, 1000}$ . If $A = {a_{1} + a_{2} + \dots + a_{k} : k \in N, a_{1}, a_{2}, a_{3}, \dots, a_{k} \in S}$ , then the sum of all the elements in the set $T - A$ is equal to ____.
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Answer: 11
Q142022 · 29 Jul 2022 · Shift 2
The domain of the function $f (x) = sin^{- 1} (\frac{x ^{2} - 3 x + 2}{x ^{2} + 2 x + 7})$ is :
1. A. $[1, \infty)$
2. B. $[- 1, 2]$
3. C. $[- 1, \infty)$
4. D. $(- \infty, 2]$
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Answer: C
Q172022 · 29 Jul 2022 · Shift 2
The number of elements in the set $S = {x \in R : 2 cos (\frac{x ^{2} + x}{6}) = 4^{x} + 4^{- x}}$ is :
1. A.1
2. B.3
3. C.0
4. D.infinite
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Answer: A
Q22022 · 28 Jul 2022 · Shift 1
Considering only the principal values of the inverse trigonometric functions, the domain of the function $f (x) = cos^{- 1} (\frac{x ^{2} - 4 x + 2}{x ^{2} + 3})$ is :
1. A. $(- \infty, \frac{1}{4}]$
2. B. $[- \frac{1}{4}, \infty)$
3. C. $(- 1/3, \infty)$
4. D. $(- \infty, \frac{1}{3}]$
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Answer: B
Q72022 · 28 Jul 2022 · Shift 1
For $α \in N$ , consider a relation $R$ on $N$ given by $R = {(x, y) : 3 x + α y is a multiple of 7}$ . The relation $R$ is an equivalence relation if and only if :
1. A. $α = 14$
2. B. $α$ is a multiple of 4
3. C.4 is the remainder when $α$ is divided by 10
4. D.4 is the remainder when $α$ is divided by 7
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Answer: D
Q212022 · 28 Jul 2022 · Shift 1· Numerical answer
Let $S$ be the set of all passwords which are six to eight characters long, where each character is either an alphabet from ${A, B, C, D, E}$ or a number from ${1, 2, 3, 4, 5}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from ${1, 2, 3, 4, 5}$ is $α \times 5^{6}$ , then $α$ is equal to ____.
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Answer: 7073
Q232022 · 28 Jul 2022 · Shift 1· Numerical answer
Let $f : [0, 1] \to R$ be a twice differentiable function in $(0, 1)$ such that $f (0) = 3$ and $f (1) = 5$ . If the line $y = 2 x + 3$ intersects the graph of $f$ at only two distinct points in $(0, 1)$ , then the least number of points $x \in (0, 1)$ , at which $f^{''} (x) = 0$ , is ____.
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Answer: 2
Q62022 · 28 Jul 2022 · Shift 2
The function $f (x) = x e^{x (1 - x)}$ , $x \in R$ , is :
1. A.increasing in $(- \frac{1}{2}, 1)$
2. B.decreasing in $(\frac{1}{2}, 2)$
3. C.increasing in $(- 1, - \frac{1}{2})$
4. D.decreasing in $(- \frac{1}{2}, \frac{1}{2})$
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Answer: A
Q302022 · 28 Jul 2022 · Shift 2· Numerical answer
Let $S = [- π, \frac{π}{2}) - {- \frac{π}{2}, - \frac{π}{4}, - \frac{3 π}{4}, \frac{π}{4}}$ . Then the number of elements in the set $∣ A = {θ \in S : tan θ (1 + 5 tan (2 θ)) = 5 - tan (2 θ)}$ is _________.
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Answer: 5
Q12022 · 27 Jul 2022 · Shift 1
Let $R_{1}$ and $R_{2}$ be two relations defined on $R$ by $a R_{1} b \Leftrightarrow ab \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b$ Then,
1. A. $R_{1}$ is an equivalence relation but not $R_{2}$
2. B. $R_{2}$ is an equivalence relation but not $R_{1}$
3. C.both $R_{1}$ and $R_{2}$ are equivalence relations
4. D.neither $R_{1}$ nor $R_{2}$ is an equivalence relation
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Answer: D
Q22022 · 27 Jul 2022 · Shift 1
Let $f, g : N - {1} \to N$ be functions defined by $f (a) = α$ , where $α$ is the maximum of the powers of those primes $p$ such that $p^{α}$ divides $a$ , and $g (a) = a + 1$ , for all $a \in N - {1}$ . Then, the function $f + g$ is
1. A.one-one but not onto
2. B.onto but not one-one
3. C.both one-one and onto
4. D.neither one-one nor onto
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Answer: D
Q72022 · 27 Jul 2022 · Shift 1
Let $f : R \to R$ be a function defined as $f (x) = a sin (\frac{π [ x ]}{2}) + [2 - x], a \in R$ where $[t]$ is the greatest integer less than or equal to $t$ . If $lim_{x \to - 1} f (x)$ exists, then the value of $\int_{0}^{4} f (x) d x$ is equal to
1. A. $- 1$
2. B. $- 2$
3. C.1
4. D.2
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Answer: B
Q12022 · 27 Jul 2022 · Shift 2
The domain of the function $f (x) = sin^{- 1} [2 x^{2} - 3] + lo g_{2} (lo g_{\frac{1}{2}} (x^{2} - 5 x + 5))$ , where $[t]$ is the greatest integer function, is :
1. A. $(- \frac{5}{2}, \frac{5 - 5}{2})$
2. B. $(\frac{5 - 5}{2}, \frac{5 + 5}{2})$
3. C. $(1, \frac{5 - 5}{2})$
4. D. $[1, \frac{5 + 5}{2})$
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Answer: C
Q82022 · 27 Jul 2022 · Shift 2
$\int_{0}^{2} (∣2 x^{2} - 3 x ∣ + [x - \frac{1}{2}]) d x$ , where $[t]$ is the greatest integer function, is equal to :
1. A. $\frac{7}{6}$
2. B. $\frac{19}{12}$
3. C. $\frac{31}{12}$
4. D. $\frac{3}{2}$
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Answer: B
Q212022 · 27 Jul 2022 · Shift 2· Numerical answer
The number of functions $f$ , from the set $A = {x \in N : x^{2} - 10 x + 9 \leq 0}$ to the set $B = {n^{2} : n \in N}$ such that $f (x) \leq (x - 3)^{2} + 1$ , for every $x \in A$ , is ____.
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Answer: 1440
Q262022 · 27 Jul 2022 · Shift 2· Numerical answer
Let $f$ be a differentiable function satisfying $f (x) = \frac{2}{3} \int_{0}^{3} f (\frac{λ ^{2} x}{3}) d λ$ , $x > 0$ and $f (1) = 3$ . If $y = f (x)$ passes through the point $(α, 6)$ , then $α$ is equal to ____.
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Answer: 12
Q82022 · 26 Jul 2022 · Shift 1
Let $f (x) = {x^{3} - x^{2} + 10 x - 7, - 2 x + lo g_{2} (b^{2} - 4), x \leq 1 x > 1$ . Then the set of all values of b, for which f(x) has maximum value at $x = 1$ , is :
1. A. $(- 6, - 2)$
2. B. $(2, 6)$
3. C. $[- 6, - 2) \cup (2, 6]$
4. D. $[- 6, - 2) \cup (2, 6]$
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Answer: C
Q32022 · 25 Jul 2022 · Shift 2
The number of bijective functions $f : {1, 3, 5, 7, \dots, 99} \to {2, 4, 6, 8, \dots, 100}$ , such that $f (3) \geq f (9) \geq f (15) \geq f (21) \geq \dots \geq f (99)$ , is ___________.
1. A. $^{50} P_{17}$
2. B. $^{50} P_{33}$
3. C. $33! \times 17!$
4. D. $\frac{50 !}{2}$
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Answer: B
Q212022 · 25 Jul 2022 · Shift 2· Numerical answer
Let $A = {1, 2, 3, 4, 5, 6, 7}$ . Define $B = {T \subseteq A : either 1 \in / T or 2 \in T}$ and $C = {T \subseteq A : the sum of all the elements of T is a prime number}$ . Then the number of elements in the set $B \cup C$ is ___________.
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Answer: 107
Q242022 · 25 Jul 2022 · Shift 2· Numerical answer
The sum of the maximum and minimum values of the function $f (x) = ∣5 x - 7∣ + [x^{2} + 2 x]$ in the interval $[\frac{5}{4}, 2]$ , where $[t]$ is the greatest integer $\leq t$ , is ___________.
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Answer: 15
Q62022 · 28 Jun 2022 · Shift 2
Let $f, g : R \to R$ be functions defined by $f (x) = {[x] ∣1 - x ∣, x < 0, x \geq 0 and g (x) = {e^{x} - x (x - 1)^{2} - 1, x < 0, x \geq 0$ where $[x]$ denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :
1. A.one point
2. B.two points
3. C.three points
4. D.four points
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Answer: B
Q162022 · 28 Jun 2022 · Shift 2
The probability that a randomly chosen one-one function from the set ${a, b, c, d}$ to the set ${1, 2, 3, 4, 5}$ satisfies $f (a) + 2 f (b) - f (c) = f (d)$ is :
1. A. $\frac{1}{24}$
2. B. $\frac{1}{40}$
3. C. $\frac{1}{30}$
4. D. $\frac{1}{20}$
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Answer: D
Q272022 · 28 Jun 2022 · Shift 2· Numerical answer
Let $A = (1 + i - i 10)$ where $i = - 1$ . Then, the number of elements in the set ${n \in {1, 2, \dots, 100} : A^{n} = A}$ is ________.
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Answer: 25