Sets, Relations & Functions · JEE Main previous year questions

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2022 44 questions

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
Q292022 · 28 Jun 2022 · Shift 2· Numerical answer
Let $S = {1, 2, 3, 4}$ . Then the number of elements in the set ${f : S \times S \to S : f is onto and f (a, b) = f (b, a) \geq a \forall (a, b) \in S \times S}$ is ________.
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Answer: 37
Q122022 · 27 Jun 2022 · Shift 1
If two straight lines whose direction cosines are given by the relations $l + m - n = 0$ , $3 l^{2} + m^{2} + c n l = 0$ are parallel, then the positive value of c is :
1. A.6
2. B.4
3. C.3
4. D.2
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Answer: A
Q212022 · 27 Jun 2022 · Shift 1· Numerical answer
Let $f : R \to R$ be a function defined by $f (x) = \frac{2 e ^{2 x}}{e ^{2 x} + e}$ . Then $f (\frac{1}{100}) + f (\frac{2}{100}) + f (\frac{3}{100}) + ..... + f (\frac{99}{100})$ is equal to ____.
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Answer: 99
Q32022 · 27 Jun 2022 · Shift 2
If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f (x) = \int_{0}^{x^{2}} \frac{t ^{2} - 5 t + 4}{2 + e ^{t}} d t$ then the ordered pair $(m, n)$ is equal to
1. A. $(3, 2)$
2. B. $(2, 3)$
3. C. $(2, 2)$
4. D. $(3, 4)$
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Answer: B
Q42022 · 27 Jun 2022 · Shift 2
Let $f$ be a differentiable function in $(0, \frac{π}{2})$ . If $\int_{c o s x}^{1} t^{2} f (t) d t = sin^{3} x + cos x$ , then $\frac{1}{3} f^{'} (\frac{1}{3})$ is equal to
1. A. $6 - 92$
2. B. $6 - \frac{9}{2}$
3. C. $\frac{9}{2} - 62$
4. D. $\frac{9}{2} - 6$
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Answer: B
Q212022 · 27 Jun 2022 · Shift 2· Numerical answer
Let $S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ . Define $f : S \to S$ as $f (n) = {2 n 2 n - 11, if n = 1, 2, 3, 4, 5, if n = 6, 7, 8, 9, 10$ Let $g : S \to S$ be a function such that $f o g (n) = {n + 1 n - 1, if n is odd, if n is even$ . Then $g (10) (g (1) + g (2) + g (3) + g (4) + g (5))$ is equal to ________.
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Answer: 190
Q282022 · 27 Jun 2022 · Shift 2· Numerical answer
Let $S = {E_{1}, E_{2}, \dots, E_{8}}$ be a sample space of a random experiment such that $P (E_{n}) = \frac{n}{36}$ for every $n = 1, 2, \dots, 8$ . Then the number of elements in the set ${A \subseteq S : P (A) \geq \frac{4}{5}}$ is ________.
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Answer: 19
Q62022 · 26 Jun 2022 · Shift 1
Let $f, g : R \to R$ be two real valued functions defined as $f (x) = {- ∣ x + 3∣ e^{x}, x < 0, x \geq 0$ and $g (x) = {x^{2} + k_{1} x 4 x + k_{2}, x < 0, x \geq 0$ , where $k_{1}$ and $k_{2}$ are real constants. If (gof) is differentiable at $x = 0$ , then $(gof) (- 4) + (gof) (4)$ is equal to :
1. A. $4 (e^{4} + 1)$
2. B. $2 (2 e^{4} + 1)$
3. C. $4 e^{4}$
4. D. $2 (2 e^{4} - 1)$
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Answer: D
Q72022 · 26 Jun 2022 · Shift 1
The sum of the absolute minimum and the absolute maximum values of the function $f (x) = ∣3 x - x^{2} + 2∣ - x$ in the interval $[- 1, 2]$ is :
1. A. $\frac{17 + 3}{2}$
2. B. $\frac{17 + 5}{2}$
3. C.5
4. D. $\frac{9 - 17}{2}$
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Answer: A
Q122022 · 26 Jun 2022 · Shift 1
Let $f (x) = 2 cos^{- 1} x + 4 cot^{- 1} x - 3 x^{2} - 2 x + 10, x \in [- 1, 1]$ . If $[a, b]$ is the range of the function f, then $4 a - b$ is equal to :
1. A.11
2. B. $11 - π$
3. C. $11 + π$
4. D. $15 - π$
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Answer: B