Assertion Reasoning Quiz

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: \[ 98^5<100^5. \]
Reason: In the binomial expansion of \((100-2)^5\), every term after \(100^5\) is negative.

Assertion: For every positive integer \(n\), \[ {}^{n}C_0-{}^{n}C_1+{}^{n}C_2-\cdots+(-1)^n{}^{n}C_n=0. \]
Reason: Substituting \(x=-1\) in the expansion of \((1+x)^n\) gives \((1-1)^n=0\).

Assertion: Every term in the expansion of \[ \left(x+\frac{1}{x}\right)^6 \] contains a non-negative integral power of \(x\).
Reason: The power of \(x\) in the \((r+1)\)th term is \(6-2r\).

Assertion (A): In the expansion of \((2x+3y)^5\), the sum of the powers of \(x\) and \(y\) in every term is exactly 5.
Reason (R): In the expansion of \((a+b)^n\), the sum of the indices of \(a\) and \(b\) is \(n\) in every term of the expansion.

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