1
JEE Main 2025 (Online) 7th April Evening Shift
Numerical
+4
-1
Change Language
The sum of the series $2 \times 1 \times{ }^{20} \mathrm{C}_4-3 \times 2 \times{ }^{20} \mathrm{C}_5+4 \times 3 \times{ }^{20} \mathrm{C}_6-5 \times 4 \times{ }^{20} \mathrm{C}_7+\cdots \cdots+18 \times 17 \times{ }^{20} \mathrm{C}_{20}$, is equal to ____________.
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2
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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3
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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4
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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