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Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(0) = 1 \) and \( f(2x) - f(x) = x \) for all \( x \in \mathbb{R} \). If \( \displaystyle \lim_{n \to \infty} \Big[ f(x) - f\!\left(\dfrac{x}{2^n}\right) \Big] = G(x) \), then \( \displaystyle \sum_{r=1}^{10} G(r^2) \) is equal to:

Let \( A \) be the set of all functions \( f : Z \to Z \) and \( R \) be a relation on \( A \) such that \( R=\{(f,g): f(0)=g(1) \text{ and } f(1)=g(0)\} \). Then \( R \) is:

The number of real solution(s) of the equation \( x^2 + 3x + 2 = m\ \, in\{\,|x - 3|, \, |x + 2|\,\} \) is :

Let \( [ x ] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x)=\sec^{-1}(2[ x ] + 1) \) is:

Let \( f(x) = \dfrac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32} \). Then the value of \( 8\!\left( f\!\left(\tfrac{1}{15}\right) + f\!\left(\tfrac{2}{15}\right) + \cdots + f\!\left(\tfrac{59}{15}\right) \right) \) is equal to:

Let \( f:[0,3]\!\to\!A \) be defined by \( f(x)=2x^{3}-15x^{2}-36x+7 \) and \( g:[0,\infty)\!\to\!B \) be defined by \( g(x)=\dfrac{x^{2025}}{x^{2025}+1} \). If both the functions are onto and \( S=\{x\in \mathbb{Z}:x\in A \text{ or } x\in B\} \), then \( n(S) \) is equal to:

If \( f(x) = \frac{2^x}{2^x + \sqrt{2}} , \ x \in \mathbb{R} \), then \( \displaystyle \sum_{k=1}^{81} f\!\left( \frac{k}{82} \right) \) is equal to:

For an integer \( n \ge 2 \), if the arithmetic mean of all coefficients in the binomial expansion of \( (x + y)^{2n - 3} \) is \( 16 \), then the distance of point \( P(2n - 1, n^2 - 4n) \) from the line \( x + y = 8 \) is:

Let the domain of the function \( f(x)=\cos^{-1}\!\Big(\dfrac{4x+5}{3x-7}\Big) \) be \( [\alpha,\beta] \) and the domain of \( g(x)=\log_{2}\!\big(2-6\log_{27}(2x+5)\big) \) be \( (\gamma,\delta) \). Then \( \big|7(\alpha+\beta)+4(\gamma+\delta)\big| \) is equal to ______.

Let \( f: R \rightarrow R \) be a function defined by \( f(x)=(2+3a)x^{2}+\left(\frac{a+2}{a-1}\right)x+b, \, a\neq 1 \). If \( f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy \), then the value of \( 28\sum_{i=1}^{5} |f(i)| \) is:

If the domain of the function \( f(x) = \log_{9}\!\left( \frac{2x-3}{5+4x} \right) + \sin^{-1}\!\left( \frac{4+3x}{2-x} \right) \) is \( (\alpha, \beta] \), then \( \alpha^{2} + 4\beta \) is equal to:

Let \( f \) be a real valued continuous function defined on the positive real axis such that \(\displaystyle g(x) = \int_{0}^{x} t\, f(t)\, dt \). If \( g(x^{3}) = x^{6} + x^{7} \), then value of \(\displaystyle \sum_{r=1}^{15} f(r^{3}) \) is:

Let \( f(x) = \log_e x \) and \( g(x) = \dfrac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( fog \) is:

Let \( f:\mathbb{R}-\{0\}\rightarrow(-\infty,1) \) be a polynomial of degree \( 2 \), satisfying \(\displaystyle f(x)f\!\left(\frac{1}{x}\right)=f(x)+f\!\left(\frac{1}{x}\right) \). If \( f(k)=-2k \), then the sum of squares of all possible values of \( k \) is:

If the range of the function \( f(x)=\frac{5-x}{x^{2}-3x+2},\ x\neq 1,2, \) is \( (-\infty,\alpha]\cup[\beta,\infty) \), then \( \alpha^{2}+\beta^{2} \) is equal to:

If the domain of the function \( f(x) = \log_{7} \left( 1 - \log_{4} (x^{2} - 9x + 18) \right) \) is \( (\alpha, \beta) \cup (\gamma, \delta) \), then \( \alpha + \beta + \gamma + \delta \) is equal to:

Let the domains of the functions \( f(x)=\log_{4}\log_{3}\log_{7}\left(8-\log_{2}\left(x^{2}+4x+5\right)\right) \) and \( g(x)=\sin^{-1}\!\left(\dfrac{7x+10}{x-2}\right) \) be \( (\alpha,\beta) \) and \( [\gamma,\delta] \), respectively. Then \( \alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2} \) is equal to:

If the domain of the function \( f(x)=\dfrac{1}{\sqrt{10+3x-x^{2}}}+\dfrac{1}{\sqrt{x+|x|}} \) is \( (a,b) \), then \( (1+a^{2})+b^{2} \) is equal to:

If the domain of the function \( \log_{5}(18x-x^{2}-77) \) is \( (\alpha,\beta) \) and the domain of the function \(\displaystyle \log_{(x-1)}\!\left(\frac{2x^{2}+3x-2}{x^{2}-3x-4}\right) \) is \( (\gamma,\delta) \), then \( \alpha^{2}+\beta^{2}+\gamma^{2} \) is equal to:

Let \( f \) be a function such that \( f(x)+3\,f\!\left(\dfrac{24}{x}\right)=4x,\; x\neq 0 \). Then \( f(3)+f(8) \) is equal to:

The function \( f : (-\infty,\infty) \to (-\infty,1) \) defined by \( f(x) = \dfrac{2^x - 2^{-x}}{2^x + 2^{-x}} \) is:

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JEE Main 2025 23rd Jan Morning Shift

Let \( f(x) = \log_e x \) and \( g(x) = \dfrac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( fog \) is: