1
JEE Main 2025 (Online) 28th January Morning Shift
Numerical
+4
-1
Change Language

If $\alpha=1+\sum\limits_{r=1}^6(-3)^{r-1} \quad{ }^{12} \mathrm{C}_{2 r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x-\sqrt{3} y+1=0$ is ________.

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2
JEE Main 2025 (Online) 23rd January Morning Shift
Numerical
+4
-1
Change Language

The sum of all rational terms in the expansion of $\left(1+2^{1 / 3}+3^{1 / 2}\right)^6$ is equal to _________.

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3
JEE Main 2025 (Online) 22nd January Evening Shift
Numerical
+4
-1
Change Language

If $\sum_\limits{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to _________.

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4
JEE Main 2025 (Online) 22nd January Morning Shift
Numerical
+4
-1
Change Language

If $\sum_\limits{r=0}^5 \frac{{ }^{11} C_{2 r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to __________.

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