Assertion Reasoning Quiz

Assertion (A): The value of \((1.01)^{1000000}\) is strictly greater than 10,000.
Reason (R): \((1+x)^n = 1 + nx + \text{other positive terms}\) for any positive real \(x\) and positive integer \(n\).

Assertion (A): The expansion of \((a+b)^4\) requires calculating \({}^4C_0\), \({}^4C_1\), \({}^4C_2\), \({}^4C_3\), and \({}^4C_4\), which map directly to the numbers 1, 4, 6, 4, 1.
Reason (R): The addition of elements in Pascal's triangle row for index \(n\) generates the row for index \(n+1\) using the property \({}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r\).

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): The value of \({}^7C_3\) is equal to the value of \({}^7C_4\).
Reason (R): In Pascal's triangle, the elements are symmetric because \({}^nC_r = {}^nC_{n-r}\).

Assertion (A): For distinct integers \(a\) and \(b\), \((a-b)\) is a factor of \(a^n - b^n\) for every positive integer \(n\).
Reason (R): \(a^n\) can be written as \(((a-b) + b)^n\), and upon expansion using the Binomial Theorem, every term except the last one contains \((a-b)\) as a factor.

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.