Let f(x) and g(x) be twice differentiable functions satisfying f′′(x)=g′′(x) for all x∈R, f′(1)=2g′(1)=4 and g(2)=3f(2)=9. Then f(25)−g(25) is equal to :
Held on 5 Apr 2026 · 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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