1
JEE Main 2021 (Online) 26th February Morning Shift
Numerical
+4
-1
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 $$\left( {\matrix{ n \cr k \cr } } \right) = {}^n{C_k}$$)
2
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 __________.
3
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 __________.
4
JEE Main 2021 (Online) 24th February Evening Shift
Numerical
+4
-1
For integers n and r, let $$\left( {\matrix{ n \cr r \cr } } \right) = \left\{ {\matrix{ {{}^n{C_r},} & {if\,n \ge r \ge 0} \cr {0,} & {otherwise} \cr } } \right.$$ The maximum value of k for which the sum $$\sum\limits_{i = 0}^k {\left( {\matrix{ {10} \cr i \cr } } \right)\left( {\matrix{ {15} \cr {k - i} \cr } } \right)} + \sum\limits_{i = 0}^{k + 1} {\left( {\matrix{ {12} \cr i \cr } } \right)\left( {\matrix{ {13} \cr {k + 1 - i} \cr } } \right)}$$ exists, is equal to _________.
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