Relations & Functions Questions (810)

The converse of the statement 'If sun is not shining, then sky is filled with clouds' is

Set of values of \(a\) for which the function \(f : \mathbb{R} \to \mathbb{R}\), given by \(f(x) = x^3 + (a+2)x^2 + 3ax + 10\) is one-one is given by:

Let \(f: R \to R\) be defined by \(f(x) = \dfrac{x}{1+x^2}\), \(x \in R\). Then the range of \(f\) is:

If elements in B_1, B_2, \ldots, B_n are not repeated, the total number of elements is 3n but each element is repeated 9 times. If S = 15 (from equation (i)), find n.

The objective function \(z\) occurs maximum at (15, 15) and (0, 20). If \(z = 115p + 15q\), find \(q\) in terms of \(p\) such that \(z\) is maximized subject to the conditions at both corner points being equal.

A positive integer is a natural number also.

Product of two rational numbers is

●14. A − B equals

The conditional statement of 'You will get a sweet dish after the dinner' is

Let A, B and C be finite sets such that \(A \cap B \cap C = \phi\) and each one of the sets \(A \Delta B\), \(B \Delta C\) and \(C \Delta A\) has 100 elements. Given that \(A \Delta B = (A \cup B) - (A \cap B)\), the number of elements in \(A \cup B \cup C\) is __________.

If \(a\) is an irrational number which is divisible by \(b\), then the number \(b\)

Sum of two irrational numbers is

The proposition \(\sim(p \vee \sim q) \vee \sim(p \vee q)\) is logically equivalent to

If \(f: [1, \infty) \to [2, \infty)\) is given by \(f(x) = x + \frac{1}{x}\), then \(f^{-1}(x)\) equals

Given statement is p: If it is raining, then I will not come. The contrapositive of statement p is:

Let the statements be expressed as \(r = {\sim}p \leftrightarrow q\). Statement-1: \(r_1 = (p \land {\sim}q) \lor ({\sim}p \land q)\) and Statement-2: \(r_2 = {\sim}(p \leftrightarrow q) = (p \land q) \lor ({\sim}q \land {\sim}p)\). Which of the following is correct?

The number of functions \(f : \{a_1, a_2, \ldots, a_{10}\} \to \{b_1, b_2, \ldots, b_5\}\) is

Sum of two natural numbers is a rational number.

For α ∈N, consider R = {(x, y) : 7 | (3x + αy)} on N. R is an equivalence relation if and only if:

If X = \{4^n - 3n - 1 : n \in \mathbb{N}\} and Y = \{9(n-1) : n \in \mathbb{N}\}, where \mathbb{N} is the set of natural numbers, then X \cup Y is equal to

\(\sim(p \vee (\sim q))\) is equal to

If \(f: \mathbb{R} \to \mathbb{R}\) satisfies \(f(x + y) = f(x) + f(y)\), for all \(x, y \in \mathbb{R}\) and \(f(1) = 7\), then \(\displaystyle\sum_{r=1}^{n} f(r)\) is

All the integers are irrational also.

Zero is a natural number.

Let \(f(x)=\sqrt{x-2}+\sqrt{4-x}\). Choose domain \(X\) and codomain \(Y\) so that \(f:X\to Y\) is bijective.

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

All the rational numbers are irrational also.

Irrational numbers are real numbers also.

If n(A) = 4, n(B) = 3, n(A \times B \times C) = 24, then n(C) equals

From the relation \(2x + y = 41\), the number of elements in the range of \(R = \{1, 3, 5, 7, \ldots, 37, 39\}\) is:

Let $A=\{1,2,\ldots,10\}$ and $B=\{0,1,2,3,4\}$. The number of elements in the relation $R=\{(a,b)\in A\times A:\ 2(a-b)^2+3(a-b)\in B\}$ is __________.

Let f (x) = log x and g(x) = 4 3 2 e x -2x +3x -2x+2 2 . Then the domain of f ∘ g is 2x -2x+1

If \(f(x) = \sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cdot \cos\left(x + \frac{\pi}{3}\right)\) and \(g\left(\frac{5}{4}\right) = 1\), then \((g \circ f)(x)\) is

The number of solutions of $3\cos^4\theta-5\cos^2\theta-2\sin^6\theta+2=0$ in $[0,2\pi]$

Let $f: \mathbb{R} \to (2, \infty)$ be a function defined as $f(x) = x^3 - 12ax + 15 - 2a + 3b(x)$. If $f(x)$ is surjective on $\mathbb{R}$, then the value of $a$ is equal to

Let $f:\mathbb{R}-\left\{-\dfrac{1}{2}\right\}\to\mathbb{R}$ and $g:\mathbb{R}-\left\{-\dfrac{5}{2}\right\}\to\mathbb{R}$ be defined as $f(x)=\dfrac{2x+3}{2x+1}$ and $g(x)=\dfrac{|x|+1}{2x+5}$. Then the domain of the function $f\circ g$ is:

Find the inverse of \(f(x)=\dfrac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}}\), \(x\in(-1,1)\).

The range of $f(x)=4\sin^{-1}\!\left(\dfrac{x^2}{x^2+1}\right)$ is

For the differentiable function $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$, let $3f(x)+2f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10$, then $\left|f(3)+f'\!\left(\dfrac{1}{4}\right)\right|$ is equal to

The period of \(f(x) = 2\cos\frac{x-\pi}{3}\) is

The number of elements in $\{n\in\mathbb{Z}:\ |n^2-10n+19|<6\}$ is _______.

The set \((A \cap B')' \cup (B \cap C)\) is equal to

Let \(f : A \to B\) be a function defined as \(f(x) = \dfrac{x-1}{x-2}\), where \(A = R - \{2\}\) and \(B = R - \{1\}\). Then f is

If \(f(x) = ax + b\) and the equation \(f(x) = f^{-1}(x)\) is satisfied by every real value of \(x\), then which of the following is true?

If f(x) = 2x + |x|, g(x) = \(\frac{1}{3}\)(2x − |x|), and h(x) = f(g(x)), find the domain of \(\sin^{-1}(h(h(h(...h(x)...))))\) (n times).

Let the range of the function f (x) = 6 + 16 cos x ⋅ cos( \pi - x) ⋅ cos( \pi + x) ⋅ sin 3x ⋅ cos 6x, x \in R be [\alpha, \beta]. 3 3 Then the distance of the point (\alpha, \beta) from the line 3x + 4y + 12 = 0 is :

If the domain of $f(x)=\ln_e(4x^2+11x+6)+\sin^{-1}(4x+3)+\cos^{-1}\!\left(\dfrac{10x+6}{3}\right)$ is $(\alpha,\beta]$, then $36|\alpha+\beta|$ is equal to

Find the domain of the function f(x) = 1/(4x - |x2 - 10x + 9|)

Find the domain of the function f(x) = log₃ log₁/₃(x² + 10x + 25) + 1/([x] + 5) (where [·] denotes the greatest integer function).

Ex. 11 The graph of \(f(x)\) is given. Which of the following represents the graph of \(-f(x) + 2\)?