Short Quiz

Assertion (A): The expansion of \((x + \frac{1}{x})^5\) has exactly one term that is independent of \(x\).
Reason (R): A term is independent of \(x\) if the sum of the powers of \(x\) evaluates to zero.

Assertion (A): The sum of the coefficients in the expansion of \((1+x)^{10}\) is 1024.
Reason (R): Substituting \(x=1\) in the expansion of \((1+x)^n\) gives \(\sum_{r=0}^{n} {}^nC_r = 2^n\).

Assertion (A): \(6^n - 5n\) always leaves a remainder of 1 when divided by 25 for any positive integer \(n\).
Reason (R): \(6^n\) can be written as \((1+5)^n\), which expands to \(1 + 5n + 25k\) where \(k\) is a natural number.

Assertion: The sum of all numerical coefficients in the expansion of \((2x+3y)^5\) is \(3125\).
Reason: Substituting \(x=1\) and \(y=1\) gives \[ (2+3)^5=5^5=3125. \]