JEE Main 2026 (Online) 6th April Evening Shift | Binomial Theorem Question 2 | Mathematics

If $\left(1-x^3\right)^{10}=\sum\limits_{\mathrm{r}=0}^{10} \mathrm{a}_{\mathrm{r}} x^{\mathrm{r}}(1-x)^{30-2 \mathrm{r}}$, then $\frac{9 \mathrm{a}_9}{\mathrm{a}_{10}}$ is equal to $\_\_\_\_$ .

Your input ____

JEE Main 2026 (Online) 5th April Morning Shift

Numerical

-1

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\frac{1}{x^3}-x^4\right)^n, x \neq 0$, is zero, then the value of $n$ is $\_\_\_\_$ .

Your input ____

JEE Main 2026 (Online) 21st January Evening Shift

Numerical

-1

If $\left(\frac{1}{{ }^{15} \mathrm{C}_0}+\frac{1}{{ }^{15} \mathrm{C}_1}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_1}+\frac{1}{{ }^{15} \mathrm{C}_2}\right) \ldots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right)=\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_0{ }^{14} \mathrm{C}_1 \cdots{ }^{14} \mathrm{C}_{12}}$, then $30 \alpha$ is equal to $\_\_\_\_$ .

Your input ____