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Let and \( g(x) = f(|x|) + |f(x)| \). If the number of points where \( g \) is not continuous and is not differentiable are \( \alpha \) and \( \beta \) respectively, then \( \alpha + \beta \) is equal to ______.

If \( f \) is continuous at \( x=\dfrac{\pi}{2} \), then the value of \( \displaystyle \int\limits_{0}^{3b-6} \left|x^2+2x-3\right|\,dx \) is:

Let \( \lim_{x\to 2}\frac{(\tan(x-2))(r x^{2}+(p-2)x-2p)}{(x-2)^{2}}=5 \) for some \( r,p\in \mathbb{R} \). If the set of all possible values of \( q \), such that the roots of the equation \( r x^{2}-p x+q=0 \) lie in \( (0,2) \), be the interval \( (\alpha,\beta] \), then \( 4(\alpha+\beta) \) equals:

Let $\alpha,\beta\in\mathbb{R}$ be such that the functionbe differentiable for all $x\in\mathbb{R}$. Then $34(\alpha+\beta)$ is equal to:

The number of points, at which the function \( f(x) = \max \{ 6x,\; 2 + 3x^2 + |x-1|\cos|x^2 - \tfrac{1}{4}| \} \), \( x \in (-\pi,\pi) \), is not differentiable is, is ________.

is continuous at \( x=0 \), then \( a+b \) is equal to:

be continuous at \( x = -\dfrac{3}{2} \). If \( f of (x) = \dfrac{7}{5} \), then \( x \) is equal to:

If \( \displaystyle \lim_{x\to 0} \frac{e^{(a-1)x} + 2\cos bx + (c-2)e^{-x}}{x\cos x - \log_e(1 + x)} = 2 \), then \( a^2 + b^2 + c^2 \) is equal to:

Let \([t]\) denote the greatest integer less than or equal to \( t \). If the function

is continuous at \( x=0 \), then \( a^{2}+b^{2} \) is equal to:

If the function \( f(x) = \frac{e^{x}\left(e^{\tan x - x}-1\right)+\log_{9}(\sec x + \tan x)-x}{\tan x - x} \) is continuous at \( x = 0 \), then the value of \( f(0) \) is equal to:

If \( y=\tan^{-1}\!\left(\dfrac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2\tan^{-1}\!\left(\dfrac{x}{1+\sqrt{1-x^{2}}}\right) \), then \( \dfrac{dy}{dx} \) at \( x=\dfrac{\sqrt{3}}{2} \) is equal to:

The value of \( \displaystyle \lim_{x\to 0} \left( \dfrac{x^{2}\sin^{2}x}{x^{2}-\sin^{2}x} \right) \) is:

The value of \( \displaystyle \lim_{x \to 0} \frac{\log_e \!\big( \sec(e{x}) \cdot \sec(e^{2}x) \cdots \sec(e^{10}x) \big)}{e^{2} - e^{2 \cos x}} \) is equal to:

The product of all possible values of \( \alpha \), for which \( \displaystyle \lim_{x \to 0}\left(\dfrac{1 - \cos(\alpha x)\cos((\alpha + 1)x)\cos((\alpha + 2)x)}{\sin^{2}((\alpha + 1)x)}\right) = 2 \), is:

If \( \lim_{x\to 2}\frac{\sin(x^3-5x^2+ax+b)}{(\sqrt{x-1}-1)\log_e(x-1)}=m \), then \( a+b+m \) is equal to:

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JEE Main 2026 23rd Jan Evening Shift

is continuous at \( x=0 \), then \( a+b \) is equal to: