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JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced PDF Download

2024

Q1: Let f : ℝ → ℝ be a function such that f(x + y) = f(x) + f(y) for all x, y ∈ ℝ, and g : ℝ → (0, ∞) be a function such that g(x + y) = g(x)g(y) for all x, y ∈ ℝ. If f(-3/5) = 12 and g(-1/3) = 2, then the value of ( f(1/4) + g(-2) - 8 ) g(0) is ______.     JEE Advanced 2024 Paper 2]
Ans:
51
f(x + y) = f(x) + f(y) … (1)
⇒ f(nx) = n f(x) ∀ n ∈ ℕ … (2)

Now, put y = -x in eq. (1)
f(x) + f(-x) = f(0) { f(0) = 0 }
⇒ f(-x) = -f(x)
⇒ f is an odd function
From eq. (2)
f(-nx) = n f(-x)
⇒ f(-nx) = -n f(x)
⇒ f(mx) = m f(x) ∀ m ∈ ℤ⁻ … (3)
From eq. (2) and eq. (3)
f(nx) = n f(x) ∀ n ∈ ℤ … (4)
Now, put x = pq where p, q ∈ ℤ, q ≠ 0
f (npq) = n f (pq) ∀ n ∈ ℤ
Put n = q
f(p) = q f pq
⇒ p f(1) = q f (pq) {from eq. (4)}
Let f(1) = a
Then, p a = q f (pq)
⇒ f (pq) = apq
⇒ f(x) = a x ∀ x ∈ ℚ
Now, f (-35) = a (-35) = 12 ⇒ a = -20
⇒ f(x) = -20x ∀ x ∈ ℚ … (5)
From the given functional equation, it is not possible to find a unique function for irrational values of ‘x’. There are infinitely many such functions satisfying the given functional equation for irrational values of x, but in this problem, we finally need the function at rational values of ‘x’ only. So, for rational values of x, we get a unique function mentioned in (5).
Now, g(x + y) = g(x) ⋅ g(y)
⇒ ln(g(x + y)) = ln(g(x)) + ln(g(y))
Let ln(g(x)) = h(x)
⇒ h(x + y) = h(x) + h(y)
⇒ h(x) = k x ∀ x ∈ ℚ
⇒ g(x) = ekx ∀ x ∈ ℚ … (6)
And g (-13) = e -k3 = 2 ⇒ K = -3 ln(2)
⇒ K = ln (18)
⇒ g(x) = e(ln(1/8) . x) = 18x = 2-3x∀ x ∈ ℚ
Now, f (14) = -5, g(-2) = 2⁶ = 64
g(0) = 1
So, (f 14 + g(-2) - 8 g(0))
= (-5 + 64 - 8)(1) = 51

Q2: Let the function f : ℝ → ℝ be defined by

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced.
Then the number of solutions of f(x) = 0 in ℝ is ______.  [JEE Advanced 2024 Paper 2]
Ans:
1
f(x) m= x2023 + 2024x + 2025eπx (x² - x + 3) (sin x + 2)
∴ (sin x + 2) is never zero.
∴ For x2023 + 2024x + 20251 = 0
Let ϕ(x) = x2023 + 2024x + 2025.
ϕ'(x) = 2023x2022 + 2024 > 0 ∀ x ∈ ℝ
∴ ϕ(x) is a strictly increasing function.
∴ ϕ(x) = 0 for exactly one value of x.
∴ f(x) = 0 has one solution.

2023


Q1: Let S = (0, 1) ∪ (1, 2) ∪ (3, 4) and T = {0, 1, 2, 3}. Then which of the following statements is(are) true?
(a) There are infinitely many functions from S to T
(b) There are infinitely many strictly increasing functions from S to T
(c) The number of continuous functions from S to T is at most 120
(d) Every continuous function from S to T is differentiable      [JEE Advanced 2023 Paper 1]
Ans: 
(a), (c) & (d)
S = (0, 1) ∪ (1, 2) ∪ (3, 4) and T = {0, 1, 2, 3}

Let domain and co-domain of a function y = f(x) are S and T respectively.
(A) There are infinitely many elements in domain and four elements in co-domain.
 There are infinitely many functions from S to T.
 Option (A) is correct
(B) If number of elements in domain is greater than number of elements in co-domain, then number of strictly increasing function is zero.
 Option (B) is incorrect
(C) Maximum number of continuous functions = 4 × 4 × 4 = 64
(Every subset (0, 1),(1, 2),(3, 4) has four choices)
∵ 64<120⇒ option (C) is correct.
(D) For every point at which f(x) is continuous, f(x) = 0
 Every continuous function from S to T is differentiable.
Option (D) is correct. 

Q2: Let f : [0, 1] → [0, 1] be the function defined by JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced. Consider the square region S = [0, 1] × [0, 1]. Let JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced be called the green region and JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced be called the red region. Let JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced be the horizontal line drawn at a height ℎ ∈ [0, 1]. Then which of the following statements is(are) true?
(a) There exists an ℎ ∈ [1/4, 2/3] such that the area of the green region above the line L equals the area of the green region below the line L
(b) There exists an ℎ ∈ [1/4, 2/3] such that the area of the red region above the line L equals the area of the red region below the line L
(c) There exists an ℎ ∈ [1/4, 2/3] such that the area of the green region above the line L equals the area of the red region below the line L
(d) There exists an ℎ ∈ [1/4, 2/3] such that the area of the red region above the line L equals the area of the green region below the line [JEE Advanced 2023 Paper 1]
Ans: 
(b), (c) & (d)
Given, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & AdvancedJEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced option (A) is incorrect

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced option (B) is correct.

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

⇒ h= 1/2 ⇒ option (C) is correct.
(D) ∵ Option (C) is correct ⇒ option (D) is also correct.

2022

Q1: Let |M| denote the determinant of a square matrix M. Let JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced be the function defined by 

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

where

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

Let p(x) be a quadratic polynomial whose roots are the maximum and minimum values of the function g(θ), and p(2)  =2 − √2. Then, which of the following is/are TRUE ?
(a) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

(b) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced
(c) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced
(d) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced            [JEE Advanced 2022 Paper 1]Ans: (a) & (c)
Given,
JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

Here, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

and JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

and JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

Also, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced  (skew symmetric)  

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

For option (A) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced Correct.
For option (B) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced Incorrect.
For option (C) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced Correct.
For option (D) JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced Incorrect. 

2020


Q1: Let the function f : [0, 1]  R be defined by               [JEE Advanced 2020 Paper 2]

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & AdvancedThen the value of JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced is ..........
Ans: 
19
The given function f : [0, 1]  R be define by
JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

So, f(x) + f(1 − x) = 1 .....(i)

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

= JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

= JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced (19 times) {from Eq. (i)}
= 19. 

Q2: Let the function JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advancedbe defined by JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & AdvancedSuppose the function f has a local minimum at θ precisely when JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced, where JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced. Then the value of JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced is ............. [JEE Advanced 2020 Paper 2]
Ans: 
0.5
The given function f : R → R be defined by

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

The local minimum of function 'f' occurs when

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

but JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

Where, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

So, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced = 0.50

Q3: Let f : [0, 2]  R be the function defined by

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

If JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced are such that JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advancedthen the value of β - α is .......... [JEE Advanced 2020 Paper 1]
Ans:
1
The given function f : [0, 2] → R defined by

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

As, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advancedso,

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

Therefore the value of (β - α) = 1

Q3: For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by
JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & AdvancedFor a polynomial f, let f' and f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (mf' + mf''), where f  S, is .............. [JEE Advanced 2020 Paper 1]
Ans: 
5
Given set S of polynomials with real coefficients

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

and for a polynomial f ∈ S, Let

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advancedit have 1 and 1 as repeated roots twice, so graph of f(x) touches the X-axis at x = 1 and x = 1, so f'(x) having at least three roots x = 1, 1 and α. Where α ∈ (1, 1) and f''(x) having at least two roots in interval (1, 1)
So, mf' = 3 and mf'' = 2
 Minimum possible value of (mf' + mf'') = 5 

Q4: If the function f : R → R is defined by f(x) = |x| (x − sin x), then which of the following statements is TRUE?
(a) f is one-one, but NOT onto
(b) f is onto, but NOT one-one
(c) f is BOTH one-one and onto
(d) f is NEITHER one-one NOR onto             [JEE Advanced 2020 Paper 1]
Ans:
(c)
The given function f : R → R is

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

The function 'f' is a odd and continuous function and as JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced, so range is R, therefore, 'f' is a onto function.

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

 f is strictly increasing function. ∀x ∈(0, ∞).
Similarly, for x < 0, x + sin x > 0 and ( x) (1  cos x) > 0, therefore, f′(x)> 0∀ x ∈(−∞, 0)
 f is strictly increasing function, ∀x ∈ (0, )
Therefore 'f' is a strictly increasing function for x R and it implies that f is one-one function. 

2018

Q1: Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of 1 / 5!(β − α) is .................. [JEE Advanced 2018 Paper 2]
Ans:
119
Given, X has exactly 5 elements and Y has exactly 7 elements.
 n(X) = 5
and n(Y) = 7
Now, number of one-one functions from X to Y is
JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & AdvancedNumber of onto functions from Y to X is  β

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

1, 1, 1, 1, 3 or 1, 1, 1, 2, 2

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced= 4 x 35 - 21= 140 - 21
= 119

Q2: Let JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced and JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced(Here, the inverse trigonometric function sin−1 x assumes values in [−π/2, π/2].).
Let f : E1  R be the function defined by JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced and g : E2  R be the function defined by JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced .      [JEE Advanced 2018 Paper 2]

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

The correct option is :
(a) P → 4; Q → 2; R → 1 ; S → 1
(b) P → 3; Q → 3; R → 6 ; S → 5
(c) P → 4; Q → 2; R → 1 ; S → 6
(d) P → 4; Q → 3; R → 6 ; S → 5                  [JEE Advanced 2018 Paper 2]
Ans:
(a)
We have,
JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced


JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

and

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

So, JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

The domain of f and g are JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced
and Range of JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced

Range of f is R  {0} or (−∞, 0)  (0, )
Range of g is JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced
Now, P  4, Q  2, R  1, S  1
Hence, option (a) is correct answer.

The document JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on JEE Advanced Previous Year Questions (2018 - 2024): Sets, Relations and Functions - Mathematics (Maths) for JEE Main & Advanced

1. What are the key topics covered in Sets, Relations, and Functions for JEE Advanced?
Ans. The key topics include types of sets (finite, infinite, equal, null, etc.), operations on sets (union, intersection, difference), Venn diagrams, relations (types, properties), functions (one-to-one, onto, bijective), and the concept of domain and range. Understanding these concepts is crucial for solving problems related to Sets, Relations, and Functions in JEE Advanced.
2. How can I effectively prepare for the Sets, Relations, and Functions section in JEE Advanced?
Ans. To prepare effectively, start by thoroughly understanding the basic concepts and definitions. Practice solving problems from previous years' papers, utilize standard textbooks, and take mock tests. Additionally, focus on time management and problem-solving techniques to enhance your speed and accuracy during the exam.
3. Are previous years' questions on Sets, Relations, and Functions available for practice?
Ans. Yes, previous years' questions from JEE Advanced (2018-2024) are available in various formats, including online resources and study guides. These questions provide valuable insights into the exam pattern and types of problems that may be asked, helping you to prepare more effectively.
4. What are some common mistakes students make in Sets, Relations, and Functions problems?
Ans. Common mistakes include misinterpreting set operations, overlooking the properties of relations and functions, and confusion between different types of functions. Additionally, students may neglect to properly determine the domain and range, leading to incorrect answers. Careful reading of the questions and clear understanding of concepts can help avoid these pitfalls.
5. How important is the topic of Sets, Relations, and Functions in the JEE Advanced exam?
Ans. The topic of Sets, Relations, and Functions is quite important in JEE Advanced as it forms the foundation for many other concepts in mathematics. Questions from this area can appear in various forms, contributing to problem-solving and analytical skills necessary for the exam. A strong grasp of these concepts can significantly aid in scoring well in the mathematics section.
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