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Let the domain of the function \( f(x)=\log_{3}\log_{5}\!\big(7-\log_{2}(x^{2}-10x+85)\big)+\sin^{-1}\!\left(\left|\dfrac{3x-7}{17-x}\right|\right) \) be \( (\alpha,\beta] \). Then \( \alpha+\beta \) is equal to:

Let the domain of the function \( f(x) = \log_{3}\!\big(\log_{5}\!\big(\log_{7}(9x - x^{2} - 13)\big)\big) \) be the interval \( (m, n) \). Let the hyperbola \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) have eccentricity \( \frac{n}{3} \) and the length of the latus rectum \( \frac{8m}{3} \). Then \( b^{2} - a^{2} \) is equal to:

Let \( f(x)=\ (x)^2-[ x+3] - 3,\ x\in \mathbb{R} \), where \([  . ]\) is the greatest integer function. Then

Let \( f : R \to R \) be defined as \( f(x)=\dfrac{2x^2-3x+2}{3x^2+x+3} \). Then \( f \) is:

If the domain of the function \( f(x) = \cos^{-1}\!\left(\frac{2x - 5}{11 - 3x}\right) + \sin^{-1}\!\left(2x^2 - 3x + 1\right) \) is the interval \( [\alpha, \beta] \), then \( \alpha + 2\beta \) is equal to:

Let \( R = \{(x,y) \in \mathbb{N} \times \mathbb{N} : \log_{a}(x+y) \le 2\} \). Then the minimum number of elements, required to be added in \( R \) to make it a transitive relation, is __________.

If the domain of the function \( f(x)=\sin^{-1}\!\left(\dfrac{1}{x^{2}-2x-2}\right) \) is \( (-\infty,\alpha] \cup [\beta,\gamma] \cup (\delta,\infty) \), then \( \alpha+\beta+\gamma+\delta \) is equal to:

Consider two sets \( A=\{x\in\mathbb{Z}:\,|(|x-3|-3)|\le 1\} \) and \( B=\left\{x\in\mathbb{R}-\{1,2\}:\,\dfrac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|)=0\right\} \). Then the number of onto functions \( f\!:\!A\to B \) is equal to:

If \( g(x) = 3x^2 + 2x - 3 \), \( f(0) = -3 \) and \( 4g(f(x)) = 3x^2 - 32x + 72 \), then \( f(g(2)) \) is equal to:

If the domain of the function \( f(x) = \sin^{-1}\!\left(\dfrac{5 - x}{3 + 2x}\right) + \dfrac{1}{\log\_{e}(10 - x)} \) is \( (-\infty, \alpha] \cup [\beta, \gamma) - \{\delta\} \), then \( 6(\alpha + \beta + \gamma + \delta) \) is equal to:

The number of points in the interval \( [2,4] \), at which the function \( f(x)=\left[x^{2}-x-\frac{1}{2}\right] \), where \( [.] \) denotes the greatest integer function, is _________.

If the domain of the function \( f(x)=\sqrt{\log_{0.6}\!\left(\left|\frac{2x-5}{x^{2}-4}\right|\right)} \) is \( (-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty) \), then the value of \( a+b+c+d+e \) is _______.

Let \( f \) be a function such that \( 3 f(x) + 2 f\!\left( \frac{m}{19x} \right) = 5x, \, x \neq 0 \), where \( m = \sum_{i=1}^{9} (i^2) \). Then \( f(5)-f(2) \) is equal to:

Let \( f \) and \( g \) be functions satisfying \( f(x + y) = f(x) f(y),\; f(1) = 7 \) and \( g(x + y) = g(xy),\; g(1) = 1 \), for all \( x, y \in \mathbb{N} \). If \( \sum\limits_{x=1}^{n} \left( \frac{f(x)}{g(x)} \right) = 19607 \), then \( n \) is equal to:

Let [.] denote the greatest integer function. If the domain of the function \( f(x)=\sin^{-1}\!\left(\frac{x+[x]}{3}\right) \) is \( [\alpha,\beta] \), then \( \alpha^{2}+\beta^{2} \) is equal to: