Sets, Relations & Functions · JEE Main previous year questions

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2026 47 questions

Q12026 · 8 Apr 2026 · Shift 2
Consider the relation $R$ on the set ${- 2, - 1, 0, 1, 2}$ defined by $(a, b) \in R$ if and only if $1 + ab > 0$ . Then, among the statements : I. The number of elements in $R$ is $17$ II. $R$ is an equivalence relation
1. A.Only I is true
2. B.Only II is true
3. C.Both I and II are true
4. D.Neither I nor II is true
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Answer: A
Q182026 · 8 Apr 2026 · Shift 2
Let $f : (1, \infty) \to R$ be a function defined as $f (x) = \frac{x - 1}{x + 1}$ . Let $f^{i + 1} (x) = f (f^{i} (x))$ , $i = 1, 2, \dots, 25$ , where $f^{1} (x) = f (x)$ . If $g (x) + f^{26} (x) = 0$ , $x \in (1, \infty)$ , then the area of the region bounded by the curves $y = g (x)$ , $2 y = 2 x - 3$ , $y = 0$ and $x = 4$ is :
1. A. $\frac{1}{8} + lo g_{e} 2$
2. B. $\frac{1}{4} + lo g_{e} 2$
3. C. $\frac{5}{6} + 3 lo g_{e} 2$
4. D. $\frac{5}{6} + lo g_{e} 2$
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Answer: A
Q252026 · 8 Apr 2026 · Shift 2· Numerical answer
Let $f$ be a polynomial function such that $lo g_{2} (f (x)) = (lo g_{2} (2 + \frac{2}{3} + \frac{2}{9} + \dots \infty)) \cdot lo g_{3} (1 + \frac{f ( x )}{f ( 1/ x )}), x > 0 and f (6) = 37.$ Then $\sum_{n = 1}^{10} f (n)$ is equal to ____ .
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Answer: 395
Q12026 · 6 Apr 2026 · Shift 1
Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f (x) = sin^{- 1} (\frac{x + [ x ]}{3})$ is $[α, β)$ , then $α^{2} + β^{2}$ is equal to:
1. A.2
2. B.5
3. C.10
4. D.13
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Answer: B
Q42026 · 6 Apr 2026 · Shift 1
Let the set of all values of $k \in R$ such that the equation $z (\overline{z} + 2 + i) + k (2 + 3 i) = 0$ , $z \in C$ , has at least one solution, be the interval $[α, β]$ . Then $9 (α + β)$ is equal to:
1. A. $- 10$
2. B. $- 8$
3. C. $1013$
4. D. $813$
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Answer: A
Q202026 · 6 Apr 2026 · Shift 1
Let $e$ be the base of natural logarithm and let $f : {1, 2, 3, 4} \to {1, e, e^{2}, e^{3}}$ and $g : {1, e, e^{2}, e^{3}} \to {1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $ϕ (x) = [f^{- 1} {g^{- 1} (\frac{1}{2})}]^{x}$ , then the area of the region $R = {(x, y) : x^{2} \leq y \leq ϕ (x), 0 \leq x \leq 1}$ is:
1. A. $\frac{3 - l o g _{e} ( 2 )}{3 l o g _{e} ( 2 )}$
2. B. $\frac{1}{3 l o g _{e} ( 2 )}$
3. C. $3 + lo g_{e} (2)$
4. D. $\frac{3 + l o g _{e} ( 2 )}{2 + l o g _{e} ( 3 )}$
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Answer: A
Q212026 · 6 Apr 2026 · Shift 2· Numerical answer
Let $R = {(x, y) \in N \times N : lo g_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is ____ .
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Answer: 15
Q252026 · 6 Apr 2026 · Shift 2· Numerical answer
Let $f (x) = {x^{3} + 8; x^{2} - 4; x < 0, x \geq 0,$ and $g (x) = {(x - 8)^{1/3}; (x + 4)^{1/2}; x < 0, x \geq 0 .$ Then the number of points, where the function $g o f$ is discontinuous, is ____ .
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Answer: 3
Q42026 · 5 Apr 2026 · Shift 1
Consider the system of linear equations in $x, y, z$ : $x + 2 y + t z = 0$ , $6 x + y + 5 t z = 0$ , $3 x + t^{2} y + f (t) z = 0$ , where $f : R \to R$ is a differentiable function. If this system has infinitely many solutions for all $t \in R$ , then $f$
1. A.is a constant function
2. B.is strictly increasing on $R$
3. C.is strictly decreasing on $R$
4. D.has two critical points
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Answer: B
Q192026 · 5 Apr 2026 · Shift 1
Let $f : R \to R$ be a differentiable function such that $f (\frac{x + y}{3}) = \frac{f ( x ) + f ( y )}{3}$ for all $x, y \in R$ , and $f^{'} (0) = 3$ . Then the minimum value of the function $g (x) = 3 + e^{x} f (x)$ , is:
1. A. $3 (\frac{e + 1}{e})$
2. B. $3 (\frac{e - 1}{e})$
3. C. $\frac{3 - e}{e}$
4. D.3e
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Answer: B
Q212026 · 5 Apr 2026 · Shift 1· Numerical answer
Let $A = {1, 2, 3, 4, 5, 6}$ . The number of one-one functions $f : A \to A$ such that $f (1) \geq 3$ , $f (3) \leq 4$ and $f (2) + f (3) = 5$ , is ____.
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Answer: 72
Q202026 · 5 Apr 2026 · Shift 2
Let $f (x)$ and $g (x)$ be twice differentiable functions satisfying $f^{''} (x) = g^{''} (x)$ for all $x \in R$ , $f^{'} (1) = 2 g^{'} (1) = 4$ and $g (2) = 3 f (2) = 9$ . Then $f (25) - g (25)$ is equal to :
1. A.20
2. B.40
3. C. $- 20$
4. D. $- 40$
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Answer: B
Q212026 · 5 Apr 2026 · Shift 2· Numerical answer
Let $A = {1, 4, 7}$ and $B = {2, 3, 8}$ . Then the number of elements, in the relation $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in ((A \times B) \times (A \times B)) : a_{1} + b_{2} divides a_{2} + b_{1}}$ is ____.
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Answer: 18
Q242026 · 5 Apr 2026 · Shift 2· Numerical answer
Let $f : R \to R$ be a function such that $f (x) + 3 f (\frac{π}{2} - x) = sin x$ , $x \in R$ . Let the maximum value of $f$ on $R$ be $α$ . If the area of the region bounded by the curves $g (x) = x^{2}$ and $h (x) = β x^{3}$ , $β > 0$ , is $α^{2}$ , then $30 β^{3}$ is equal to ____.
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Answer: 16
Q12026 · 4 Apr 2026 · Shift 1
Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to:
1. A.6
2. B.8
3. C.9
4. D.4
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Answer: A
Q22026 · 4 Apr 2026 · Shift 1
What is the ratio of wave number of first line (lowest energy line) of Balmer series of H atomic spectrum to first line of its Brackett series?
1. A.5 : 1
2. B.5 : 0.81
3. C.5 : 1.75
4. D.5 : 27
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Answer: B
Q42026 · 4 Apr 2026 · Shift 1
Given below are two statements : Statement I : For an ideal gas, heat capacity at constant volume is always greater than the heat capacity at constant pressure. Statement II : In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas. 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.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true
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Answer: B
Q192026 · 4 Apr 2026 · Shift 1
Let $\int_{- 2}^{2} ([sin x] + [x sin x]) d x = 2 (3 - cos 2) + β$ , where $[\cdot]$ is the greatest integer function. Then $β sin (\frac{β}{2})$ equals:
1. A.1
2. B.2
3. C.4
4. D.8
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Answer: B
Q252026 · 4 Apr 2026 · Shift 1· Numerical answer
The number of points, at which the function $f (x) = max {6 x, 2 + 3 x^{2}} + ∣ x - 1∣ cos x^{2} - \frac{1}{4}$ , $x \in (- π, π)$ , is not differentiable, is _________.
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Answer: 3
Q12026 · 4 Apr 2026 · Shift 2
For the function $f : [1, \infty) \to [1, \infty)$ defined by $f (x) = (x - 1)^{4} + 1$ , among the two statements: (I) The set $S = {x \in [1, \infty) : f (x) = f^{- 1} (x)}$ contains exactly two elements, and (II) The set $S = {x \in [1, \infty) : f (x) = f^{- 1} (x + 1)}$ is an empty set,
1. A.only (I) is TRUE
2. B.only (II) is TRUE
3. C.both (I) and (II) are TRUE
4. D.neither (I) nor (II) is TRUE
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Answer: A
Q162026 · 4 Apr 2026 · Shift 2
Let for some $α \in R$ , $f : R \to R$ be a function satisfying $f (x + y) = f (x) + 2 y^{2} + y + α x y$ for all $x, y \in R$ . If $f (0) = - 1$ and $f (1) = 2$ , then the value of $\sum_{n = 1}^{5} (α + f (n))$ is:
1. A.110
2. B.140
3. C.150
4. D.170
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Answer: B
Q252026 · 4 Apr 2026 · Shift 2· Numerical answer
Let $f$ be a twice differentiable function such that $f (x) = \int_{0}^{x} tan (t - x) d t - \int_{0}^{x} f (t) tan t d t, x \in (- \frac{π}{2}, \frac{π}{2}) .$ Then $f^{''} (\frac{π}{6}) + 12 f^{'} (- \frac{π}{6}) + f (\frac{π}{6})$ is equal to ____.
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Answer: 5
Q72026 · 2 Apr 2026 · Shift 1
The number of elements in the set $S = {(r, k) : k \in Z and^{36} C_{r + 1} = \frac{6 ( ^{35} C _{r} )}{( k ^{2} - 3 )}}$ , is :
1. A.2
2. B.4
3. C.8
4. D.16
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Answer: B
Q182026 · 2 Apr 2026 · Shift 1
The number of critical points of the function $f (x) = {\frac{s i n x}{x} 1, x \neq = 0, x = 0$ in the interval $(- 2 π, 2 π)$ is equal to :
1. A.1
2. B.3
3. C.5
4. D.7
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Answer: C
Q212026 · 2 Apr 2026 · Shift 1· Numerical answer
If the domain of the function $f (x) = lo g_{(0.6)} (\frac{2 x - 5}{x ^{2} - 4})$ is $(- \infty, a] \cup {b} \cup [c, d) \cup (e, \infty)$ , then the value of $a + b + c + d + e$ is ____.
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Answer: 4