1
JEE Main 2021 (Online) 18th March Evening Shift
Numerical
+4
-1
Out of Syllabus
Let $${}^n{C_r}$$ denote the binomial coefficient of xr in the expansion of (1 + x)n. If $$\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R$$, then $$\alpha$$ + $$\beta$$ is equal to ___________.
2
JEE Main 2021 (Online) 17th March Evening Shift
Numerical
+4
-1
Let the coefficients of third, fourth and fifth terms in the expansion of $${\left( {x + {a \over {{x^2}}}} \right)^n},x \ne 0$$, be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to ___________.
3
JEE Main 2021 (Online) 17th March Morning Shift
Numerical
+4
-1
If (2021)3762 is divided by 17, then the remainder is __________.
4
JEE Main 2021 (Online) 16th March Evening Shift
Numerical
+4
-1
Out of Syllabus
Let n be a positive integer. Let

$$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^k} + {{\left( {{{31} \over {32}}} \right)}^k}} \right]}$$. If

$$63A = 1 - {1 \over {{2^{30}}}}$$, then n is equal to _____________.