1
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of $${\left( {2{x^3} + {3 \over x}} \right)^{10}}$$ is $${5^{10}} - \beta \,.\,{3^9}$$, then $$\beta$$ is equal to ____________.

2
JEE Main 2022 (Online) 25th June Morning Shift
Numerical
+4
-1
Out of Syllabus

Let Cr denote the binomial coefficient of xr in the expansion of $${(1 + x)^{10}}$$. If for $$\alpha$$, $$\beta$$ $$\in$$ R, $${C_1} + 3.2{C_2} + 5.3{C_3} +$$ ....... upto 10 terms $$= {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\left( {{C_0} + {{{C_1}} \over 2} + {{{C_2}} \over 3} + \,\,.....\,\,upto\,10\,terms} \right)$$ then the value of $$\alpha$$ + $$\beta$$ is equal to ___________.

3
JEE Main 2022 (Online) 24th June Evening Shift
Numerical
+4
-1

The remainder on dividing 1 + 3 + 32 + 33 + ..... + 32021 by 50 is _________.

4
JEE Main 2021 (Online) 1st September Evening Shift
Numerical
+4
-1
Out of Syllabus
If the sum of the coefficients in the expansion of (x + y)n is 4096, then the greatest coefficient in the expansion is _____________.