Let f:(−2,2)→R be defined by f(x)={\begin{matrix}x[x] & , & -2<x<0 \\ (x-1)[x] & , & 0\leq x<2\end{matrix} where [x] denotes the greatest integer function. If m and n respectively are the number of points in (–2,2) at which y=∣f(x)∣ is not continuous and not differentiable, then m+n is equal to ________.
Held on 10 Apr 2023 · Verified 6 Jul 2026.
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Let $y = y(x)$ be the solution of the differential equation $x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx$, $y(1) = \dfrac{\pi}{2}$ and let $\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is __________.
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