1
JEE Main 2023 (Online) 29th January Evening Shift
+4
-1 Let K be the sum of the coefficients of the odd powers of $$x$$ in the expansion of $$(1+x)^{99}$$. Let $$a$$ be the middle term in the expansion of $${\left( {2 + {1 \over {\sqrt 2 }}} \right)^{200}}$$. If $${{{}^{200}{C_{99}}K} \over a} = {{{2^l}m} \over n}$$, where m and n are odd numbers, then the ordered pair $$(l,\mathrm{n})$$ is equal to

A
(50, 101)
B
(50, 51)
C
(51, 101)
D
(51, 99)
2
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1 If $$a_r$$ is the coefficient of $$x^{10-r}$$ in the Binomial expansion of $$(1 + x)^{10}$$, then $$\sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}}$$ is equal to

A
3025
B
4895
C
5445
D
1210
3
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1 If $${({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}$$ then $$\alpha$$ is equal to :

A
30
B
10
C
15
D
60
4
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1 The value of $$\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}}$$ is

A
$${}^{44}{C_{23}}$$
B
$${}^{45}{C_{23}}$$
C
$${}^{44}{C_{22}}$$
D
$${}^{45}{C_{24}}$$
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