1
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 _____________.
2
JEE Main 2021 (Online) 26th February Morning Shift
Numerical
+4
-1
Out of Syllabus
Let m, n$$\in$$N and gcd (2, n) = 1. If $$30\left( {\matrix{ {30} \cr 0 \cr } } \right) + 29\left( {\matrix{ {30} \cr 1 \cr } } \right) + ...... + 2\left( {\matrix{ {30} \cr {28} \cr } } \right) + 1\left( {\matrix{ {30} \cr {29} \cr } } \right) = n{.2^m}$$, then n + m is equal to __________.

(Here $$\left( {\matrix{ n \cr k \cr } } \right) = {}^n{C_k}$$)
3
JEE Main 2021 (Online) 25th February Evening Shift
Numerical
+4
-1
If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is __________.
4
JEE Main 2021 (Online) 25th February Evening Shift
Numerical
+4
-1
The total number of two digit numbers 'n', such that 3n + 7n is a multiple of 10, is __________.