Mock Test 1 — Binomial Theorem

The coefficient of $x^4$ in the expansion of $\left(\frac{x}{2} - \frac{3}{x^2}\right)^{10}$ is:

The total number of terms in the expansion of $(x + y + z)^{15}$ after combining like terms is:

The middle term in the expansion of $\left(2x - \frac{1}{2x}\right)^{12}$ is exactly:

The value of the coefficient sum series $C_1 + 2C_2 + 3C_3 + \dots + 10C_{10}$ for $n=10$ is:

The term independent of $x$ in the expansion of $\left(x^2 + \frac{1}{x}\right)^{9}$ occurs at which index value $r$?

If the expression $(1+x)^n = C_0 + C_1x + \dots + C_nx^n$, the value of the fractional sum $\frac{C_0}{1} + \frac{C_1}{2} + \dots + \frac{C_n}{n+1}$ is:

The coefficient of $a^3 b^2 c^1$ in the multinomial expansion of $(a + b + c)^6$ is:

For a small value of $x$ ($x \ll 1$), the rational binomial expression $\frac{1}{(1-x)^2}$ can be linearly approximated as:

The sum of the squares of the binomial coefficients $\sum_{r=0}^{5} \binom{5}{r}^2$ is equal to:

The value of the alternating sum $C_0 - C_1 + C_2 - C_3 + \dots + (-1)^n C_n$ is:

Find the value of $n$ if the coefficient of the second, third, and fourth terms in the expansion of $(1+x)^n$ form an Arithmetic Progression (AP).

Determine the remainder when the number $5^{99}$ is divided by 13.

Evaluate the value of the index $r$ that yields the greatest binomial coefficient in the expansion of $(1+x)^{12}$.

Assertion (A): The total number of terms in the expansion of $(x+y)^{20}$ is exactly 21.
Reason (R): The binomial expansion of $(a+b)^n$ starts at $r=0$ and ends at $r=n$, which creates exactly $n+1$ distinct terms.

Assertion (A): The sum of the coefficients of the odd terms equals the sum of the coefficients of the even terms in any positive integer binomial expansion.
Reason (R): Setting $x = -1$ in the identity equation $(1+x)^n = C_0 + C_1x + C_2x^2 + \dots$ forces the total alternating sum to equal exactly zero.