Let f be a differentiable function satisfying f(x)=1−2x+∫0xe(x−t)f(t)dt,x∈R and let g(x)=∫0x(f(t)+2)15(t−4)6(t+12)17dt,x∈R. If p and q are respectively the points of local minima and local maxima of g, then the value of ∣p+q∣ is equal to ____.
Held on 28 Jan 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 __________.
Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a twice differentiable function such that $f(3)=18, f^{\prime}(3)=0$ and $f^{\prime \prime}(3)=4$. Then $\lim _{x \rightarrow 1}\left(\log _{\mathrm{e}}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$ is equal to :
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Work through every JEE Main Calculus PYQ, year by year.