JEE ADVANCED 2025 PAPER 2 ONLINE

Let

$$\alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}}$$

Then the value of

$$\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2$$

is _____________.

The solubility of barium iodate in an aqueous solution prepared by mixing 200 mL of 0.010 M barium nitrate with 100 mL of 0.10 M sodium iodate is $\boldsymbol{X} \times 10^{-6} \mathrm{~mol} \mathrm{dm}^{-3}$. The value of $\boldsymbol{X}$ is ____________.

Use: Solubility product constant $\left(K_{\mathrm{sp}}\right)$ of barium iodate $=1.58 \times 10^{-9}$

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument of $z$, with $-\pi<\arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0<\arg (\omega)<\pi$. Let

$$\alpha=\arg \left(\sum\limits_{n=1}^{2025}(-\omega)^n\right)$$

Then the value of $\frac{3 \alpha}{\pi}$ is ________________.