1
JEE Main 2024 (Online) 29th January Morning Shift
Numerical
+4
-1

The area (in sq. units) of the part of the circle $$x^2+y^2=169$$ which is below the line $$5 x-y=13$$ is $$\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)$$, where $$\alpha, \beta$$ are coprime numbers. Then $$\alpha+\beta$$ is equal to __________.

2
JEE Main 2024 (Online) 29th January Morning Shift
Numerical
+4
-1

If the points of intersection of two distinct conics $$x^2+y^2=4 b$$ and $$\frac{x^2}{16}+\frac{y^2}{b^2}=1$$ lie on the curve $$y^2=3 x^2$$, then $$3 \sqrt{3}$$ times the area of the rectangle formed by the intersection points is _________.

3
JEE Main 2024 (Online) 27th January Evening Shift
Numerical
+4
-1

If the area of the region $$\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}$$ is $$\mathrm{A}$$, then $$12 \mathrm{~A}$$ is equal to ________.

4
JEE Main 2024 (Online) 27th January Morning Shift
Numerical
+4
-1
Let the area of the region $\left\{(x, y): x-2 y+4 \geqslant 0, x+2 y^2 \geqslant 0, x+4 y^2 \leq 8, y \geqslant 0\right\}$ be $\frac{\mathrm{m}}{\mathrm{n}}$, where $\mathrm{m}$ and $\mathrm{n}$ are coprime numbers. Then $\mathrm{m}+\mathrm{n}$ is equal to _____________.