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Let \( f : \mathbb{R} - \left\{ \dfrac{\alpha}{6} \right\} \to \mathbb{R} \) be defined by \( f(x) = \dfrac{5x + 3}{6x - \alpha} \). Then the value of \( \alpha \) for which \( (f \circ f)(x) = x \), for all \( x \in \mathbb{R} - \left\{ \dfrac{\alpha}{6} \right\} \), is:

The domain of the function \( cosec^{-1}\!\left(\frac{1+x}{x}\right) \) is:

If \( f(x)=\sin\!\left(\cos^{-1}\!\left(\dfrac{1-2^{2x}}{1+2^{2x}}\right)\right) \) and its first derivative with respect to \( x \) is \( -\dfrac{b}{a}\log_e 2 \) when \( x=1 \), where \( a \) and \( b \) are integers, then the minimum value of \( \left|a^{2}-b^{2}\right| \) is __________.

Let \( f:\mathbb{R}\to\mathbb{R} \) and \( g:\mathbb{R}\to\mathbb{R} \) be defined as where \( a, b \) are non-negative real numbers. If \( (g\circ f)(x) \) is continuous for all \( x\in\mathbb{R} \), then \( a+b \) is equal to ____________.

Let \( f:\mathbb{N} \to \mathbb{N} \) be a function such that \( f(m+n) = f(m) + f(n) \) for every \( m,n \in \mathbb{N} \). If \( f(6) = 18 \), then \( f(2)\cdot f(3) \) is equal to:

If the functions are defined as \( f(x)=\sqrt{x} \) and \( g(x)=\sqrt{1-x} \), then what is the common domain of the following functions: \( f+g,\, f-g,\, \frac{f}{g},\, \frac{g}{f} , g-f\) where \( (f \pm g)(x)=f(x)\pm g(x) \), \( (f/g)(x)=\frac{f(x)}{g(x)} \)

The inverse of \( y = 5^{\log x} \) is:

The range of the function \( f(x) = \log_{\sqrt{5}}\!\left( 3 + \cos\!\left(\frac{3\pi}{4} + x\right) + \cos\!\left(\frac{\pi}{4} + x\right) + \cos\!\left(\frac{\pi}{4} - x\right) - \cos\!\left(\frac{3\pi}{4} - x\right) \right) \) is:

Let \(f:\mathbb{R}\to \mathbb{R}\) be defined as:

Let \(A=\{x\in \mathbb{R}:f \text{ is increasing}\}\). Then \(A\) is equal to:

Let \( g : \mathbb{N} \to \mathbb{N} \) be defined as
\( g(3n+1) = 3n + 2, \)
\( g(3n+2) = 3n + 3, \)
\( g(3n+3) = 3n + 1, \\ \text{for all } n \ge 0. \)
Then which of the following statements is true?

If \( a+\alpha=1, \; b+\beta=2 \) and \( a f(x)+\alpha f\!\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, \; x\neq 0 \), then the value of the expression
\( \displaystyle \frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}\) is __________.

The number of real roots of the equation \( e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^{x}+1=0 \) is:

Let \( f : \mathbb{R} - \{3\} \to \mathbb{R} - \{1\} \) be defined by \( f(x) = \dfrac{x - 2}{x - 3} \). Let \( g : \mathbb{R} \to \mathbb{R} \) be given as \( g(x) = 2x - 3 \). Then, the sum of all the values of \( x \) for which \( f^{-1}(x) + g^{-1}(x) = \dfrac{13}{2} \) is equal to:

Consider functions \( f: A \to B \) and \( g: B \to C \) \( (A,B,C \subseteq R) \) such that \( (g \circ f)^{-1} \) exists, then:

If \( A = \{ x \in \mathbb{R} : |x - 2| > 1 \} \), \( B = \{ x \in \mathbb{R} : \sqrt{x^2 - 3} > 1 \} \), \( C = \{ x \in \mathbb{R} : |x - 4| \ge 2 \} \) and \( Z \) is the set of all integers, then the number of subsets of the set \( (A \cap B \cap C)^{c} \cap Z \) is \( \_\_\_\_\_\_\_\_\_\_ \).

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JEE Main 2021 25th July Morning Shift

Let \( g : \mathbb{N} \to \mathbb{N} \) be defined as
\( g(3n+1) = 3n + 2, \)
\( g(3n+2) = 3n + 3, \)
\( g(3n+3) = 3n + 1, \\ \text{for all } n \ge 0. \)
Then which of the following statements is true?