Assertion Reasoning Quiz

Assertion (A): The sum \(\sum_{r=0}^{n} 3^r \cdot {}^nC_r\) is equal to \(4^n\).
Reason (R): The general expression \((1+a)^n\) is equal to \(\sum_{r=0}^{n} {}^nC_r a^r\).

Assertion (A): The number of terms in the expansion of \((x+y)^{10}\) is 11.
Reason (R): The total number of terms in the expansion of \((a+b)^n\) is \(n+1\).

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: The expansion of \[ \left(x^2+\frac{3}{x}\right)^4 \] contains a term independent of \(x\).
Reason: The exponent of \(x\) in the \((r+1)\)th term is \(8-3r\), and no integral value of \(r\) between \(0\) and \(4\) makes this exponent zero.

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): \(6^n - 5n\) 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.