1
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1

If the coefficient of $$x^{15}$$ in the expansion of $$\left(\mathrm{a} x^{3}+\frac{1}{\mathrm{~b} x^{1 / 3}}\right)^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $$\left(a x^{1 / 3}-\frac{1}{b x^{3}}\right)^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$(\mathrm{a}, \mathrm{b})$$ :

A
a = 3b
B
ab = 1
C
ab = 3
D
a = b
2
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1
Out of Syllabus

The coefficient of $${x^{301}}$$ in $${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$$ is :

A
$${}^{500}{C_{300}}$$
B
$${}^{501}{C_{200}}$$
C
$${}^{500}{C_{301}}$$
D
$${}^{501}{C_{302}}$$
3
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)
4
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1
Out of Syllabus

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
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