Relations & Functions Questions - JEE

The solution of differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2 + 1}{2xy}$ satisfying $y(1) = 0$ is given by:

Let f(x) be a function on ℝ and f(x − 2) = f(x + 2). If f(x) = 0 has only three real roots in [0, 4] and one of them is 4, then the number of real roots of f(x) = 0 in (−8, 10] is

Let $R=\{(a,b): a/b$ is a prime number$\}$ on first twenty natural numbers. Consider: (I) $R$ is reflexive and symmetric but not transitive; (II) Range of $R^{-1}$ has 20 elements; (III) Domain of $R^{-1}$ has 10 elements. Which statements are true?

If $f(x) = \log_e \left( \frac{1 - x}{1 + x} \right) |x| < 1$, then $f \left( \frac{2x}{1 + x^2} \right)$ is equal to :

The number of functions f from {1, 2, 3, …, 20} onto {1, 2, 3, …, 20} such that f(k) is a multiple of 3 whenever k is a multiple of 4, is

If the function f : ℝ − {1, −1} → A defined by f(x) = $\frac{x^2}{1-x^2}$ is surjective, then A is equal to

Let $ |x - 2| = y $. The solution set of $ \dfrac{y-1}{y-2} \leq 0 $ is:

Let $f(x) = \ln(x^2 + ax + 1)$. If $f(x)$ is defined $\forall\, x \in R$, then the number of integers in the range of 'a' is:

135. The function $f:[0,\infty)\to[0,\infty)$ defined by $f(x)=\dfrac{2x}{1+2x}$ is:

Given: f : A → B be a function defined as $ f(x) = \dfrac{x-1}{x-2} $, where $ A = R - \{2\} $ and $ B = R - \{1\} $. So, f(x) is bijective function or invertible function. Then $ f^{-1}(x) $ equals:

Example 90: If $f(x) = x^3 - 3x^2 - 4x + b\sin x + c\cos x$ for all $x \in \mathbb{R}$ is a one-one function, find the value of $b^2 + c^2$.

Let function $f(x) = \sqrt{e^x + x - a}$ for $a \in \mathbb{R}$. If there exists $x_0 \in [-1, 1]$ such that $f(f(x_0)) = x_0$, then the range of $'a'$ is:

The domain of the function $ f(x) = \dfrac{1}{4-x^2} + \log_{10}(x^3 - x) $ is

JM Q28.

JM Q21 -- counting problem.

JM Q27.

JM Q24.

JM Q31.

$f(x+y)=f(x)f(y)$, $f(1)=3$. If $\sum_{i=1}^n f(i)=363$, find $n$.

Let $f$ be a degree-3 polynomial with $f(k)=-2/k$ for $k=2,3,4,5$. Find $52-10f(10)$.

JM Q32.

JM Q30.

If the function f(x) = [3.5 + b sin x] (where [·] denotes the greatest integer function) is an even function, find the complete set of values of b.

The functions f₁(x) = log(x + √(x² + 1)) and f₂(x) = x · ((aˣ - 1)/(aˣ + 1)) are respectively:

If $f :( -\infty , 2 ] \to ( -\infty , 4 ]$, where $f ( x ) = x ( 4 - x )$, then $f ^{-1}( x )$ is given by:

Let $f: \mathbb{R} \to \mathbb{R}$, where $f(x) = \frac{x^2 + ax + 1}{x^2 + x + 1}$. Then the complete set of values of $a$ such that $f(x)$ is onto is:

Let f(x) = 4x(1 − x), 0 ≤ x ≤ 1 and y = f[f{f(x)}], 0 ≤ x ≤ 1. Find the number of points where the graph of y meets the line y = x.

Let $g(x)$ be a function defined on $[-1, 1]$. If the area of equilateral triangle with two of its vertices at $(0, 0)$ and $(x, g(x))$, is $\frac{3}{4}$ sq unit, the function $g(x)$ may be

A function $f(x)$ which is invertible must be

The domain of the definition of the function f(x) = 1/(4 − x2) + log₁₀(x3 − x) is

Let f(x) = 8x + 3 and g(x) = x² + 4. Find (g∘f)(2).

Find $f(x) = \frac{1}{2}|x+2| - 1\bigg|\frac{1}{2}(x-2)\bigg| + 1$. Hence find the range of $f(x)$.

100. If $f(x) = x^3 - 3x + 1$, then minimum number of real roots of $f(f(x)) = 0$ is:

Domain of the function f(x), if 3^x + 3^{f(x)} = \text{minimum of } F(t), where F(t) = \min\{2t^3 - 15t^2 + 36t - 25, 2|\sin t|\} is

Let f(x) = 7x² + x - 8 and g(x) = |x|. Find (g∘f)(0) + (f∘g)(-3).

Let $f(x) = \frac{1}{1-x}$, then {$f \circ (f \circ f)$}(100) is equal to

Let f : \mathbb{R} \to \mathbb{R} and g : \mathbb{R} \to \mathbb{R} be two one-one and onto functions, such that they are the mirror images of each other about the line y = a. If h(x) = f(x) + g(x), then h(x) is

Let max{|x + y|, |x - y|} = 1 and y = x - [x] be two equations which x and y satisfy, then the number of ordered pairs (x, y) is

If the relation R : A → B, where A = {1, 2, 3, 4} and B = {1, 3, 5} is defined by R = {(x, y) : x y, x ∈ A, y ∈ B}, then RoR−1 is

136. Let $f:R\to R$ be given as \(f(x)=\begin{cases}2x+\alpha^2, & x\geq 2\\ \dfrac{\alpha x}{2}+10, & x

50. Let $f$ be an invertible function from $R \to R$ satisfying the equation \[f^3(x) - (x^3 + 2)f^2(x) + (2x^3 + 1)f(x) - x^3 = 0.\] Then the value of $f'(8) \times (f^{-1})'(8)$ is:

The domain of $f(g(x))$ is

For any x = a ≥ 5, f(a) = $\sqrt{a-5}$ ≥ 0. What is the range of the function?

Range of the function f(x) = log2(2 − log2(16 sin2x + 1)) is:

The value of a and b for which |e|x−b| − a| = 2 has four distinct solutions, are:

If $f(x) + 2f\left(\dfrac{1}{x}\right) = 3x$, $x \neq 0$ and $S = \{x \in \mathbb{R} : f(x) = f(-x)\}$, then $S$

Let $\sum_{k=1}^{10} f(a+k) = 16(2^{10}-1)$, where the function $f$ satisfies $f(x+y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2$. Then the natural number $a$ is ______.

135. The function $f:[0,\infty)\to[0,\infty)$ defined by $f(x)=\dfrac{2x}{1+2x}$ is:

Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A → B | 2 ∈ f(A) and f is not one-one}

The function $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x - 5\left[\dfrac{x}{5}\right]$, where $\mathbb{N}$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to $x$, is

Relations & Functions Questions (810)