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Let \( f \) be a polynomial function such that \( \log_{2}(f(x))=\big[\log_{2}\!\big(2+\frac{2}{3}+\frac{2}{9}+\ldots \infty\big)\big]\cdot \log_{3}\!\left(1+\frac{f(x)}{f(1/x)}\right) \), \( x>0 \) and \( f(6)=37 \). Then \( \displaystyle\sum_{n=1}^{10} f(n) \) is equal to ______.

Consider the relation \( R \) on the set \( \{ -2, -1, 0, 1, 2 \} \) defined by \( (a, b) \in R \) if and only if \( 1 + ab > 0 \). Then, among the statements:
I. The number of elements in \( R \) is \( 17 \).
II. \( R \) is an equivalence relation.

Let \([.]\) denote the greatest integer function. If the domain of the function \( f(x) = \cos^{-1}\!\left(\frac{4x + 2[x]}{3}\right) \) is \([\alpha,\beta]\), then \( 12(\alpha + \beta) \) is equal to:

Let \( A=\{1,2,3,4,5,6\} \). The number of one-one functions \( f:A\to A \) such that \( f(1)\ge 3 \), \( f(3)\le 4 \) and \( f(2)+f(3)=5 \), is ______.

For the function \( f : [1,\infty) \to [1,\infty) \) defined \( f(x) = (x-1)^4 + 1 \), among the two statements:
(I) The set \( S=\{x \in [1,\infty) : f(x)=f^{-1}(x)\} \) contains exactly two elements, and
(II) The set \( S=\{x \in [1,\infty) : f(x)=f^{-1}(x+1)\} \) is an empty set,

Let for some \( \alpha\in \mathbb{R} \), \( f:\mathbb{R}\to\mathbb{R} \) be a function satisfying \( f(x+y)=f(x)+2y^{2}+y+\alpha xy \) for all \( x,y\in\mathbb{R} \). If \( f(0)=-1 \) and \( f(1)=2 \), then the value of is:

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:

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JEE Main 2026 4th April Evening Shift

Let for some \( \alpha\in \mathbb{R} \), \( f:\mathbb{R}\to\mathbb{R} \) be a function satisfying \( f(x+y)=f(x)+2y^{2}+y+\alpha xy \) for all \( x,y\in\mathbb{R} \). If \( f(0)=-1 \) and \( f(1)=2 \), then the value of is: