1
JEE Main 2023 (Online) 8th April Morning Shift
Numerical
+4
-1 The largest natural number $$n$$ such that $$3^{n}$$ divides $$66 !$$ is ___________.

2
JEE Main 2023 (Online) 6th April Evening Shift
Numerical
+4
-1 If $$(20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}$$, then $$k$$ is equal to ___________.

3
JEE Main 2023 (Online) 6th April Morning Shift
Numerical
+4
-1 The coefficient of $$x^{18}$$ in the expansion of $$\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$$ is __________.

4
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1 Let the sixth term in the binomial expansion of $${\left( {\sqrt {{2^{{{\log }_2}\left( {10 - {3^x}} \right)}}} + \root 5 \of {{2^{(x - 2){{\log }_2}3}}} } \right)^m}$$ in the increasing powers of $$2^{(x-2) \log _{2} 3}$$, be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is __________.

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