Welcome to **AMBiPi (**Read as**: एम्बीपाई) (Amans Maths Blogs)**. JEE Mains and JEE Advanced exams are the engineering entrance exams for taking admission in IITs, NITs and other engineering colleges. In this article, you will get * JEE Main Math Previous Year Papers Sets Relations Functions Questions Answer Keys Solutions*.

### JEE Mains Mathematics Sets Relations and Functions Questions Bank

**Enter Questions No: 1**

The domain of sin^{-1}[log_{3 }(x/3)] is

**[2002]**

**Option A**: [1, 9]

**Option B**: [-1, 9]

**Option C**: [-9, 1]

**Option D**: [-9, -1]

**Show/Hide Answer Key**

**Option A: [1, 9]**

**Enter Questions No: 1**

If f(x + y) = f(x) ∙ f(y) for all x and y and f(5) = 2, f'(0) = 3, then find f'(5).

**[2002]**

**Option A**: 0

**Option B**: 1

**Option C**: 6

**Option D**: 2

**Show/Hide Answer Key**

**Option C: 6**

**Enter Questions No: 1**

Which one is NOT periodic?

**[2002]**

**Option A**: |sin 3x| + sin^{2}x

**Option B**: cos √x + cos^{2}x

**Option C**: cos 4x + tan^{2}x

**Option D**: cos 2x + sin x

**Show/Hide Answer Key**

**Option B: cos √x + cos ^{2}x**

**Enter Questions No: 1**

The period of sin^{2}θ is

**[2002]**

**Option A**: π^{2}

**Option B**: π

**Option C**: π^{3}

**Option D**: π/2

**Show/Hide Answer Key**

**Option B: π**

**Enter Questions No: 1**

The function f(x) = log(x + √(x^{2} + 1))

**[2003]**

**Option A**: An Even Function

**Option B**: An Odd Function

**Option C**: A Periodic Function

**Option D**: Neither an Even nor an Odd Function

**Show/Hide Answer Key**

**Option B: An Odd Function**

**Enter Questions No: 1**

Domain of definition of the function f(x) = 3 / (4 – x^{2}) + log_{10}(x^{3} – x) is

**[2003]**

**Option A**: (1, 2)

**Option B**: (-1, 0) ∪ (1, 2)

**Option C**: (1, 2) ∪ (2, ∞)

**Option D**: (-1, 0) ∪ (1, 2) ∪ (2, ∞)

**Show/Hide Answer Key**

**Option D: (-1, 0) ∪ (1, 2) ∪ (2, ∞)**

**Enter Questions No: 1**

If f : R → R satisfies f(x + y) = f(x) + f(y), for all x, y ∊ R and f(1) = 7, then

**[2003]**

**Option A**: 7n / 2

**Option B**: 7(n + 1) / 2

**Option C**: 7n(n + 1)

**Option D**: 7n(n + 1) / 2

**Show/Hide Answer Key**

**Option D: 7n(n + 1) / 2**

**Enter Questions No: 1**

A function f from the set of natural numbers to integers defined by

is

**[2003]**

**Option A**: One-One but Not Onto

**Option B**: Onto but Not One-One

**Option C**: One-One and Onto Both

**Option D**: Neither One-One Nor Onto

**Show/Hide Answer Key**

**Option C: One-One and Onto Both**

**Enter Questions No: 1**

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

**[2004]**

**Option A**: a function

**Option B**: reflexive

**Option C**: not symmetric

**Option D**: transitive

**Show/Hide Answer Key**

**Option C: not symmetric**

**Enter Questions No: 1**

The domain of the function f(x) = sin^{-1}(x – 3) / √(9 – x^{2}) is

**[2004]**

**Option A**: [2, 3]

**Option B**: [2, 3)

**Option C**: [1, 2]

**Option D**: [1, 2)

**Show/Hide Answer Key**

**Option C: One-One and Onto Both**

**Enter Questions No: 1**

The graph of the function y = f(x) is symmetrical about the line x = 2, then

**[2004]**

**Option A**: f(x + 2) = f(x – 2)

**Option B**: f(2 + x) = f(2 – x)

**Option C**: f(x) = f(-x)

**Option D**: f(x) = -f(-x)

**Show/Hide Answer Key**

**Option B: f(2 + x) = f(2 – x)**

**Enter Questions No: 1**

If f : R → S, defined by f(x) = sin x – √3cos x + 1 is onto, then the interval of S is

**[2004]**

**Option A**: [0, 3]

**Option B**: [-1, 1]

**Option C**: [0, 1]

**Option D**: [-1, 3]

**Show/Hide Answer Key**

**Option D: [-1, 3]**

**Enter Questions No: 1**

The range of the function f(x) = ^{7-x}P_{x-3} is

**[2004]**

**Option A**: {1, 2, 3}

**Option B**: {1, 2, 3, 4, 5}

**Option C**: {1, 2, 3, 4}

**Option D**: {1, 2, 3, 4, 5, 6}

**Show/Hide Answer Key**

**Option A: {1, 2, 3}**

**Enter Questions No: 1**

Let R = {(3, 3), (6, 6), (9,9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12} be a relation on the set A = {3, 6, 9, 12}. The relation is

**[2005]**

**Option A**: reflexive and transitive only

**Option B**: reflexive only

**Option C**: an equivalence relation

**Option D**: reflexive and symmetric only

**Show/Hide Answer Key**

**Option A: reflexive and transitive only**

**Enter Questions No: 1**

Let f : (-1, 1) → B, be a function defined by f(x) = tan^{-1}(2x / (1 – x^{2})), then the function f is both one-one and onto when B is the interval

**[2005]**

**Option A**: (0, π/2)

**Option B**: [0, π/2)

**Option C**: [-π/2, π/2]

**Option D**: (-π/2, π/2)

**Show/Hide Answer Key**

**Option D: (-π/2, π/2)**

**Enter Questions No: 1**

A real valued function f(x) satisfies the functional equation f(x – y) = f(x)f(y) – f(a – x)f(a + y) where a is a given constant and f(0) = 0, f(2a – x) is equal to

**[2005]**

**Option A**: -f(x)

**Option B**: f(x)

**Option C**: f(a) + f(a – x)

**Option D**: f(-x)

**Show/Hide Answer Key**

**Option A: -f(x)**

**Enter Questions No: 1**

Let W denote the words in English dictionary. Define the relation R by R = {(x, y) ∈ W × W | the words x and y have at least one letter in common}. Then, R is

**[2006]**

**Option A**: not reflexive, symmetric and transitive

**Option B**: reflexive, symmetric and not transitive

**Option C**: reflexive, symmetric and transitive

**Option D**: reflexive, not symmetric and transitive

**Show/Hide Answer Key**

**Option B: reflexive, symmetric and not transitive**

**Enter Questions No: 1**

The largest interval lying in (-π/2, π/2) for which the function

is defined, is

**[2007]**

**Option A**: [0, π]

**Option B**: (-π/2, π/2)

**Option C**: [-π/4, π/2)

**Option D**: [0, π/2)

**Show/Hide Answer Key**

**Option D: [0, π/2)**

**Enter Questions No: 1**

Let R be the real line. Consider the following subsets of the plane R × R. S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x-y is an integer}. Which one of the following is true?

**[2008]**

**Option A**: neither S nor T is an equivalence relation on R

**Option B**: both S and T are equivalence relations on R

**Option C**: S is an equivalence relation on R but T is not

**Option D**: T is an equivalence relation on R but S is not

**Show/Hide Answer Key**

**Option D: T is an equivalence relation on R but S is not**

**Enter Questions No: 1**

Let f : N → Y be a function defined as f(x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}. If f is invertible function, then its inverse function is

**[2008]**

**Option A**: g(y) = (3y + 4) / 3

**Option B**: g(y) = 4 + (y + 3) / 4

**Option C**: g(y) = (y + 3) / 4

**Option D**: g(y) = log | sin (x – π / 4) | + c

**Show/Hide Answer Key**

**Option D: g(y) = log | sin (x – π / 4) | + c**

**Enter Questions No: 1**

If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then

**[2009]**

**Option A**: A = B

**Option B**: A = C

**Option C**: B = C

**Option D**: A ∩ B = φ

**Show/Hide Answer Key**

**Option C: B = C**

**Enter Questions No: 1**

For real x, let f(x) = x^{3} + 5x + 1, then

**[2009]**

**Option A**: f is one-one but not onto R

**Option B**: f is onto R but not one-one

**Option C**: f is one-one and onto R

**Option D**: f is neither one-one nor onto R

**Show/Hide Answer Key**

**Option C: f is one-one and onto R**

**Enter Questions No: 1**

Consider the following relations:

R = {(x, y) | x, y are real numbers and x = wy for some rational number w}

S = {(m/n, p/q) | m,n, p and q are integers such that n, q ≠ 0 and qm = pn}. Then,

**[2010]**

**Option A**: neither R nor S is an equivalence relation

**Option B**: S is an equivalence relation but R is not an equivalence relation

**Option C**: R and S both are equivalence relations

**Option D**: R is an equivalence relation but S is not an equivalence relation

**Show/Hide Answer Key**

**Option B: S is an equivalence relation but R is not an equivalence relation**

**Enter Questions No: 1**

The domain of the function f(x) = 1 / √(|x| – x) is

**[2011]**

**Option A**: (0, ∞)

**Option B**: (-∞, 0)

**Option C**: (-∞, ∞) – {0}

**Option D**: (-∞, ∞)

**Show/Hide Answer Key**

**Option B: (-∞, 0)**

**Enter Questions No: 1**

The range of the function f(x) = x / (1 + |x|), x ∈ R, is

**[2012]**

**Option A**: R

**Option B**: (-1, 1)

**Option C**: R – {0}

**Option D**: [-1, 1]

**Show/Hide Answer Key**

**Option B: (-1, 1)**

**Enter Questions No: 1**

Let A and B be two sets containing two elements and four elements respectively. The number of subsets of A × B having three or more elements is

**[2013]**

**Option A**: 220

**Option B**: 219

**Option C**: 211

**Option D**: 256

**Show/Hide Answer Key**

**Option B: 219**

**Enter Questions No: 1**

If X = {4n – 3n – 1 : n ∈ N} and Y = {9(n – 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to

**[2014]**

**Option A**: N

**Option B**: Y – X

**Option C**: X

**Option D**: Y

**Show/Hide Answer Key**

**Option D: Y**

**Enter Questions No: 1**

If g is the inverse of a function f and f ‘(x) = 1 / (1 + x^{5}), then g ‘(x) is equal to

**[2014]**

**Option A**: 1 + x^{5}

**Option B**: 5x^{4}

**Option C**: 1 / [1 + {g(x)}^{5}]

**Option D**: 1 + {g(x)}^{5}

**Show/Hide Answer Key**

**Option D: 1 + {g(x)} ^{5}**

**Enter Questions No: 1**

If a ∈ R and the equation -3(x – [x])^{2} + 2(x – [x]) + a^{2} = 0 (where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval

**[2014]**

**Option A**: (-1, 0) ∪ (0, 1)

**Option B**: (1, 2)

**Option C**: (-2, -1)

**Option D**: (-∞, -2) ∪ (2, ∞)

**Show/Hide Answer Key**

**Option A: (-1, 0) ∪ (0, 1)**

**Enter Questions No: 1**

Let f(n) = [1/3 + 3n/100]n, where [n] denotes the greatest integer less than or equal to n. Then

is equal to

**[2014]**

**Option A**: 56

**Option B**: 689

**Option C**: 1287

**Option D**: 1399

**Show/Hide Answer Key**

**Option D: 1399**

**Enter Questions No: 1**

Let f be an odd function defined on the set of real numbers such that for x ≥ 0, f(x) = 3 sin x + 4 cos x. Then, f(x) at x = -11π/6 is equal to

**[2014]**

**Option A**: 3/2 + 2√3

**Option B**: –3/2 + 2√3

**Option C**: 3/2 – 2√3

**Option D**: –3/2 – 2√3

**Show/Hide Answer Key**

**Option C: 3/2 – 2√3**

**Enter Questions No: 1**

Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set A × B, each having at least three elements os

**[2015]**

**Option A**: 256

**Option B**: 275

**Option C**: 510

**Option D**: 219

**Show/Hide Answer Key**

**Option D: 219**

**Enter Questions No: 1**

If f(x) + 2f(1 / x) = 3x, x ≠ 0, and S = {x ∈ R : f(x) = f(-x)} ; then S

**[2016]**

**Option A**: contains exactly one element

**Option B**: contains exactly two elements

**Option C**: contains more than two elements

**Option D**: is an empty set

**Show/Hide Answer Key**

**Option B: contains exactly two elements**

**Enter Questions No: 1**

Let a, b, c ∈ R. If f(x) = ax^{2} + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, for all x, y ∈ R, then

is equal to

**[2017]**

**Option A**: 330

**Option B**: 165

**Option C**: 190

**Option D**: 255

**Show/Hide Answer Key**

**Option A: 330**

**Enter Questions No: 1**

The function f : R → [-1/2, 1/2] defined as f(x) = x / (1 + x^{2}) is

**[2017]**

**Option A**: invertible

**Option B**: injective but not surjective

**Option C**: surjective but not injective

**Option D**: neither injective nor surjective

**Show/Hide Answer Key**

**Option C: surjective but not injective**

**Enter Questions No: 1**

Let f(x) = a^{x} (a > 0) be written as f(x) = f_{1}(x) + f_{2}(x), where f_{1}(x) is an even function and f_{2}(x) is an odd function. Then, f_{1}(x + y) + f_{2}(x – y) equals

**[2019]**

**Option A**: 2f_{1}(x)f_{1}(y)

**Option B**: 2f_{1}(x + y)f_{1}(x – y)

**Option C**: 2f_{1}(x)f_{2}(y)

**Option D**: 2f_{1}(x + y)f_{2}(x – y)

**Show/Hide Answer Key**

**Option A: 2f _{1}(x)f_{1}(y)**

**Enter Questions No: 1**

If f(x) = log_{e}((1 – x) / (1 + x)), |x| < 1, then f(2x / (1 + x_{2})) is equal to

**[2019]**

**Option A**: 2f(x)

**Option B**: 2f(x^{2})

**Option C**: f(f(x))^{2}

**Option D**: –2f(x)

**Show/Hide Answer Key**

**Option A: 2f(x)**

**Enter Questions No: 1**

Let f : R → R be defined by f(x) = x / (1 + x^{2}), Then the range of f is

**[2019]**

**Option A**: [-1/2, 1/2]

**Option B**: R – [-1, 1]

**Option C**: R – [-1/2, 1/2]

**Option D**: (-1, 1) – {0}

**Show/Hide Answer Key**

**Option A: [-1/2, 1/2]**

**Enter Questions No: 1**

The domain of the definition of the function f(x) = 1 / (4 – x^{2}) + log_{10}(x^{3} – x) is

**[2019]**

**Option A**: (-1, 0) ∪ (1, 2) ∪ (3, ∞)

**Option B**: (-2, -1) ∪ (-1, 0) ∪ (2, ∞)

**Option C**: (-1, 0) ∪ (1, 2) ∪ (2, ∞)

**Option D**: (1, 2) ∪ (2, ∞)

**Show/Hide Answer Key**

**Option C: (-1, 0) ∪ (1, 2) ∪ (2, ∞)**

**Enter Questions No: 1**

Let f(1, 3) → R be a function defined by f(x) = x[x] / (1 + x^{2}), where [x] denotes the greatest integer ≤ x. Then, the range of f is

**[2020]**

**Option A**: (2/5, 3/5) ∪ (3/4, 4/5)

**Option B**: (2/5, 1/2) ∪ (3/5, 4/5)

**Option C**: (2/5, 4/5)

**Option D**: (3/5, 4/5)

**Show/Hide Answer Key**

**Option B: (2/5, 1/2) ∪ (3/5, 4/5)**

**Enter Questions No: 1**

Let f(x) be a quadratic polynomial such that f(-1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root is lies in

**[2020]**

**Option A**: (-1, 0)

**Option B**: (1, 3

**Option C**: (-3, -1)

**Option D**: (0, 1)

**Show/Hide Answer Key**

**Option A: (-1, 0)**

**Enter Questions No: 1**

Let [t] denote the greatest integer ≤ t. Then, the equation in x, [x]^{2} + [x + 2] – 7 = 0 has

**[2020]**

**Option A**: exactly two solutions

**Option B**: exactly four integral solutions

**Option C**: no integral solutions

**Option D**: infinitely many solutions

**Show/Hide Answer Key**

**Option D: infinitely many solutions**

**Enter Questions No: 1**

If R = {(x, y) : x, y ∈ Z, x^{2} + 3y^{2} ≤ 8} is a relation on the set of integers Z, then the domain of R^{-1} is

**[2020]**

**Option A**: {-2, -1, 1, 2}

**Option B**: {0, 1}

**Option C**: {-2, -1, 0, 1, 2}

**Option D**: {-1, 0, 1}

**Show/Hide Answer Key**

**Option D: {-1, 0, 1}**

**Enter Questions No: 1**

The domain of the function f(x) = sin^{-1}(|x| + 5)/(x^{2} + 1) is (-∞, -a] ∪ [a, ∞), Then a is equal to

**[2020]**

**Option A**: √17 / 2

**Option B**: (√17 – 1) / 2

**Option C**: (√17 + 1) / 2

**Option D**: √17 / 2 + 1

**Show/Hide Answer Key**

**Option C: (√17 + 1) / 2**