Sets, Relations & Functions · JEE Main previous year questions

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2025 42 questions

Q42025 · 29 Jan 2025 · Shift 2
If the domain of the function $lo g_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $lo g_{(x - 1)} (\frac{2 x ^{2} + 3 x - 2}{x ^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is equal to:
1. A. $186$
2. B. $179$
3. C. $195$
4. D. $174$
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Answer: A
Q72025 · 29 Jan 2025 · Shift 2
If the set of all $a \in R$ , for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z; α < x < β}$ , then $\sum_{x \in X} x^{2}$ is equal to:
1. A. $2139$
2. B. $2119$
3. C. $2109$
4. D. $2129$
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Answer: A
Q212025 · 29 Jan 2025 · Shift 2· Numerical answer
If $24 0 \int \frac{π}{4} (sin 4 x - \frac{π}{12} + [2 sin x]) dx = 2 π + α$ , where $[\cdot]$ denotes the greatest integer function, then $α$ is equal to ____.
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Answer: 12
Q42025 · 28 Jan 2025 · Shift 1
The relation $R = {(x, y) : x, y \in Z and x + y is even}$ is:
1. A.reflexive and transitive but not symmetric
2. B.reflexive and symmetric but not transitive
3. C.an equivalence relation
4. D.symmetric and transitive but not reflexive
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Answer: C
Q152025 · 28 Jan 2025 · Shift 1
The sum of all local minimum values of the function $f (x) = ⎩ ⎨ ⎧ 1 - 2 x, \frac{1}{3} (7 + 2∣ x ∣), \frac{11}{18} (x - 4) (x - 5), x < - 1 - 1 \leq x \leq 2 x > 2$ is
1. A. $\frac{167}{72}$
2. B. $\frac{157}{72}$
3. C. $\frac{171}{72}$
4. D. $\frac{131}{72}$
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Answer: B
Q182025 · 28 Jan 2025 · Shift 1
Let $f : R \to R$ be a function defined by $f (x) = (2 + 3 a) x^{2} + (\frac{a + 2}{a - 1}) x + b, a \neq = 1$ . If $f (x + y) = f (x) + f (y) + 1 - \frac{2}{7} x y$ , then the value of $28 \sum_{i = 1}^{5} ∣ f (i) ∣$ is
1. A.735
2. B.675
3. C.715
4. D.545
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Answer: B
Q232025 · 24 Jan 2025 · Shift 1· Numerical answer
Let $S = {p_{1}, p_{2} \dots, p_{10}}$ be the set of first ten prime numbers. Let $A = S \cup P$ , where $P$ is the set of all possible products of distinct elements of $S$ . Then the number of all ordered pairs $(x, y), x \in S$ , $y \in A$ , such that $x$ divides $y$ , is ____ .
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Answer: 5120
Q32025 · 24 Jan 2025 · Shift 2
Let $(2, 3)$ be the largest open interval in which the function $f (x) = 2 lo g_{e} (x - 2) - x^{2} + a x + 1$ is strictly increasing and $(b, c)$ be the largest open interval, in which the function $g (x) = (x - 1)^{3} (x + 2 - a)^{2}$ is strictly decreasing. Then $100 (a + b - c)$ is equal to :
1. A.360
2. B.420
3. C.160
4. D.280
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Answer: A
Q192025 · 24 Jan 2025 · Shift 2
Let $f : (0, \infty) \to R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f^{'} (x) = 2 x f (x) + 3$ , with $f (1) = 4$ . Then $2 f (2)$ is equal to :
1. A.19
2. B.23
3. C.29
4. D.39
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Answer: D
Q202025 · 24 Jan 2025 · Shift 2
The function $f : (- \infty, \infty) \to (- \infty, 1)$ , defined by $f (x) = \frac{2 ^{x} - 2 ^{- x}}{2 ^{x} + 2 ^{- x}}$ 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: A
Q242025 · 24 Jan 2025 · Shift 2· Numerical answer
Number of functions $f : {1, 2, \dots, 100} \to {0, 1}$ , that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________.
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Answer: 392
Q72025 · 23 Jan 2025 · Shift 2
Let $X = R \times R$ . Define a relation $R$ on $X$ as : $(a_{1}, b_{1}) R (a_{2}, b_{2}) \Leftrightarrow b_{1} = b_{2}$ Statement I: $R$ is an equivalence relation. Statement II : For some $(a, b) \in X$ , the set $S = {(x, y) \in X : (x, y) R (a, b)}$ represents a line parallel to $y = x$ . In the light of the above statements, choose the correct answer from the options given below :
1. A.Both Statement I and Statement II are true
2. B.Statement I is true but Statement II is false
3. C.Both Statement I and Statement II are false
4. D.Statement I is false but Statement II is true
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Answer: B
Q92025 · 23 Jan 2025 · Shift 2
Let the range of the function $f (x) = 6 + 16 cos x \cdot cos (\frac{π}{3} - x) \cdot cos (\frac{π}{3} + x) \cdot sin 3 x \cdot cos 6 x, x \in R$ be $[α, β]$ . Then the distance of the point $(α, β)$ from the line $3 x + 4 y + 12 = 0$ is :
1. A.11
2. B.10
3. C.8
4. D.9
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Answer: A
Q122025 · 22 Jan 2025 · Shift 1
Let $f (x)$ be a real differentiable function such that $f (0) = 1$ and $f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)$ for all $x, y \in R$ . Then $\sum_{n = 1}^{100} lo g_{e} f (n)$ is equal to :
1. A.2406
2. B.5220
3. C.2525
4. D.2384
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Answer: C
Q162025 · 22 Jan 2025 · Shift 1
The number of non-empty equivalence relations on the set ${1, 2, 3}$ is :
1. A.7
2. B.4
3. C.5
4. D.6
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Answer: C
Q192025 · 22 Jan 2025 · Shift 1
Let $f : R \to R$ be a twice differentiable function such that $f (x + y) = f (x) f (y)$ for all $x, y \in R$ . If $f^{'} (0) = 4 a$ and $f$ satisfies $f^{''} (x) - 3 a f^{'} (x) - f (x) = 0, a > 0$ , then the area of the region $R = {(x, y) ∣ 0 \leq y \leq f (a x), 0 \leq x \leq 2}$ is :
1. A. $e^{2} - 1$
2. B. $e^{4} + 1$
3. C. $e^{2} + 1$
4. D. $e^{4} - 1$
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Answer: A
Q212025 · 22 Jan 2025 · Shift 1· Numerical answer
Let the function, $f (x) = {- 3 a x^{2} - 2, a^{2} + b x, x < 1 x ⩾ 1$ be differentiable for all $x \in R$ , where $a > 1$ , $b \in R$ . If the area of the region enclosed by $y = f (x)$ and the line $y = - 20$ is $α + β 3$ , $α, β \in Z$ , then the value of $α + β$ is ____ .
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Answer: 34