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If $ y = y(x) $ is an implicit function of $ x $ such that $ \log_{e}(x + y) = 4xy $, then $ \frac{d^{2}y}{dx^{2}} $ at $ x = 0 $ is equal to ________.

Let $\displaystyle f(x) = \sqrt{\lim_{r \to x} \left\{ \frac{2r^2 \Big[ (f(r)^2 - f(x)f(r) \Big]}{r^2 - x^2} - r^3 e^{\dfrac{f(r)}{r}} \right\}}$ be differentiable in $(-\infty,0)\cup(0,\infty)$ and $ f(x)=1 $. Then the value of $ e a $, such that $ f(a)=0 $, is equal to: _________.

Let $y = f(x) = \sin^3 \left( \frac{\pi}{3} \left( \cos \left( \frac{\pi}{3\sqrt{2}} \left( -4x^3 + 5x^2 + 1 \right)^{\frac{3}{2}} \right) \right) \right)$. Then, at $x = 1$.

If $\displaystyle \lim_{n \to \infty} \frac{(n+1)^{k-1}}{n^{k+1}} \left[(nk+1) + (nk+2) + \ldots + (nk+n)\right] = 33 \cdot \lim_{n \to \infty} n^\dfrac{1}{{k+1}} \left[1^k + 2^k + 3^k + \ldots + n^k\right] $, Then the integral value of $ k $ is equal to ________.

Let $ f(x) = ax^3 + bx^2 + cx + 41 $ be such that $ f(1) = 40 $ , $ f'(1) = 2 $ and $ f''(1) = 4 $. Then $ a^2 + b^2 + c^2 $ is equal to:

, Where$ [\alpha] $ denotes the greatest integer less than or equal to $ \alpha $. Then the number of points in $ \mathbb{R} $ where $ f $ is not differentiable is ________.

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JEE Main 2022 25th July Morning Shift

, Where$ [\alpha] $ denotes the greatest integer less than or equal to $ \alpha $. Then the number of points in $ \mathbb{R} $ where $ f $ is not differentiable is ________.

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ