1
JEE Main 2025 (Online) 3rd April Evening Shift
Numerical
+4
-1
Change Language

Let $\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots+a_{20} x^{20}$. If $\left(a_1+a_3+a_5+\ldots+a_{19}\right)-11 a_2=121 k$, then $k$ is equal to_________ .

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2
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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3
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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4
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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