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JEE Advance | PYQ

Let \( f:(0,1)\to \mathbb{R} \) be the function defined as \( f(x)=[4x]\left(x-\frac{1}{4}\right)^{2}\left(x-\frac{1}{2}\right) \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then which of the following statements is(are) true?

Consider the function \( f : \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \rightarrow (-\infty, \infty) \) defined by
\( f(x) = (|x| + |x - 1|)\sin x + [ x\sin x ]\), where \( [x\sin x ]\) is the greatest integer less than or equal to \( x\sin x \). Let \( \alpha \) be the total number of points in the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) at which \( f \) is NOT continuous and let \( \beta \) be the total number of points in the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) at which \( f \) is NOT differentiable. Then the value of \( \alpha + \beta \) is __________.

For a real number \( \alpha \), let \( [\alpha] \) denote the greatest integer less than or equal to \( \alpha \). For a finite set \( S \), let \( |S| \) denote the number of elements in the set \( S \).
Consider the functions \( f:(-3,3)\rightarrow(-\infty,\infty) \) and \( g:(-3,3)\rightarrow(-\infty,\infty) \) defined by \( f(x)=[x^{3}] \log_{e}\!\big(1+\sin^{2}(\pi(x-[x]))\big) \) and \( g(x)=x^{3}\sin^{2}\!\big(\pi\log_{e}(1+x-[x])\big) \).
Let \( A=\{x\in(-3,3): f \text{ is discontinuous at } x\} \) and \( B=\{x\in(-3,3): g \text{ is discontinuous at } x\} \). Then the value of \( |A|+2|B|-|A\cap B| \) is ____.

Let \( \alpha \) be a positive real number. Let \( f:\mathbb{R}\rightarrow\mathbb{R} \) and \( g:(\alpha,\infty)\rightarrow\mathbb{R} \) be the functions defined by \( f(x)=\sin\!\left(\dfrac{\pi x}{12}\right) \) and \( g(x)=\dfrac{2\log_{e}(\sqrt{x}-\sqrt{\alpha})}{\log_{e}(e^{\sqrt{x}}-e^{\sqrt{\alpha}})} \). Then the value of \( \lim\limits_{x\to \alpha^{+}} f(g(x)) \) is __________.

Let \( S \) be the set of all \( (\alpha,\beta) \in \mathbb{R} \times \mathbb{R} \) such that
\( \displaystyle \lim_{x \to \infty} \frac{\sin(x^2)(\log_e x)^{\alpha} \sin\!\left(\frac{1}{x^2}\right)}{x^{\alpha\beta}\,(\log_e(1+x))^{\beta}} = 0 \)
The which of the following is (are) correct?

Let \( f(x) \) be a continuously differentiable function on the interval \( (0, \infty) \) such that\( f(1) = 2 \) and
\( \displaystyle \lim_{t \to x} \frac{t^{10} f(x) - x^{10} f(t)}{t^{9} - x^{9}} = 1 \)
for each \( x > 0 \). Then, for all \( x > 0 \), \( f(x) \) is equal to:

Let \( \mathbb{R} \) denote the set of all real numbers. Define the function \( f:\mathbb{R}\to\mathbb{R} \) by

Then which one of the following statements is TRUE?

If \( \beta=\displaystyle \lim_{x\to 0}\frac{e^{x^{3}}-(1-x^{3})^{1/3}+((1-x^{2})^{1/2}-1)\sin x}{x\sin^{2}x} \), then the value of \( 6\beta \) is __________.

Let \( k\in \mathbb{R} \). If \( \displaystyle \lim_
{x\to 0^{+}}\left(\sin(\sin kx)+\cos x+x\right)^{\frac{2}{x}}=e^{6} \), then the value of \( k \) is:

Let \( \mathbb{R} \) denote the set of all real numbers. For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Let \( n \) denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f : \mathbb{R} \to \mathbb{R} \) be an arbitrary function and let \( g : \mathbb{R} \to \mathbb{R} \) be the function defined by \( g(x) = x f(x) \), for all \( x \in \mathbb{R} \).
Then which of the following statements is (are) TRUE?

Let \( x_0 \) be the real number such that \( e^{x_0} + x_0 = 0 \). For a given real number \( \alpha \), define
\( g(x) = \dfrac{3x e^{x} + 3x - \alpha e^{x} - \alpha x}{3(e^{x} + 1)} \) for all real numbers \( x \).
Then which one of the following statements is TRUE?

Let \( \alpha \) and \( \beta \) be the real numbers such that \( \displaystyle\lim_{x \to 0} \dfrac{1}{x^{3}}\left( \dfrac{a}{2}\int_{0}^{x}\dfrac{1}{1-t^{2}}\,dt + \beta x \cos x \right)=2 \).
Then the value of \( \alpha + \beta \) is________.

Let \( f : (0,1) \to \mathbb{R} \) be the function defined as \( f(x) = \sqrt{n} \) if \( x \in \left[ \frac{1}{n+1}, \frac{1}{n} \right) \) where \( n \in \mathbb{N} \). Let \( g : (0,1) \to \mathbb{R} \) be a function such that \( \int_{x^2}^{x} \sqrt{\frac{1-t}{t}} \, dt < g(x) < 2\sqrt{x} \) for all \( x \in (0,1) \). Then \( \displaystyle \lim_{x \to 0} f(x) g(x) \)