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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7516

For $k\in\mathbf{N}$, let $\dfrac{1}{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+20)}=\displaystyle\sum_{k=0}^{20}\dfrac{A_k}{\alpha+k}$, where $\alpha>0$. Then the value of $100\left(\dfrac{A_{14}+A_{15}}{A_{13}}\right)^2$ is equal to $\_\_\_\_\_\_\_\_$ .

Sequence and Series · Miscellaneous

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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7517

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}\left[[x]-\sin x\right]dx$ is equal to :

Definite Integration

$-\pi$$\pi$$0$$1$