Relations and Functions

8 solved problems from JEE previous year papers.

Relations and Functions

Let A = {(x, y) : 2x + 3y = 23, x, y ∈ N} and B = {x : (x, y) ∈ A}. Then the Number of One-One Functions From A to B Is Equal To

JEE Main 2024 PYQ: A is the set of natural-number solutions to 2x+3y=23, and B is the set of their x-values. Find the number of…
Functions

If f(x) + 6f(1/x) = 35/(3x) − 7/2 for x ∈ R − {0}, and the Limit of (1/(αx) + f(x)) as x → 0 Exists Finitely and Is Equal to β, Then α − 2β Is

JEE Main 2025 PYQ: f(x) satisfies f(x)+6f(1/x)=35/(3x)-7/2. If lim(x→0)[1/(αx)+f(x)] exists finitely and equals β, find α-2β by…
Functions

Let f(x) = 5ˣ/(5ˣ + 5). Then the Sum of the Series f(1/20) + f(2/20) + f(3/20) + ⋯ + f(39/20) Is Equal To

JEE Main 2021 PYQ: for f(x)=5^x/(5^x+5), find the sum f(1/20)+f(2/20)+…+f(39/20) using the pairing identity f(x)+f(2-x)=1.
Functions

Let the Range of f(x) = 1/(2 + sin 3x + cos 3x), x ∈ R, Be [a, b]. If α and β Are Respectively the A.M. and the G.M. of a and b, Then α/β Is Equal To

Find the range [a,b] of f(x)=1/(2+sin3x+cos3x), then find α/β where α, β are the AM and GM of a and b, using the fact that…
Relations and Functions

Given a Non-Empty Set X, Consider P(X), the Power Set of X. Define A R B if and Only if A ⊆ B. Is R an Equivalence Relation on P(X)? Justify Your Answer

Given a non-empty set X, define R on P(X), the power set of X, by ARB iff A⊆B. Determine, with justification, whether R is an…
Functions

f(x)=2x/√(1+9x²) Composed 10 Times, Find √(3α+1)

Consider f:R→R defined by f(x)=2x/√(1+9x²). If (f∘f∘⋯∘f, 10 times)(x)=2^10x/√(1+9αx²), find the value of √(3α+1).
Relations and Functions

Is f(x)=(2^x−2^−x)/(2^x+2^−x) One-One, Onto, Both or Neither?

The function f:(−∞,∞)→(−∞,1), defined by f(x)=(2^x−2^−x)/(2^x+2^−x), is: neither one-one nor onto, onto but not one-one, both, or…
Functions

Domain of log5(18x−x²−77) Is (α,β), Find α²+β²+γ²

If the domain of log5(18x−x²−77) is (α,β) and the domain of log(x−1)[(2x²+3x−2)/(x²−3x−4)] is (γ,δ), then α²+β²+γ² is: A)195…