1st Year Mathematics Chapter 7 Test7. Mathematical Induction and Binomial Theorem

Q1. Multiple choice questions (Attempt all · 10 × 1 = 10)

1. If n is even positive integer then the middle term in the expansion of (a + b)ⁿ is.
   1. $$\left( \frac{n}{2} \right)^{\text{th}} \text{term}$$
   2. $$\left( \frac{n + 1}{2} \right)^{\text{th}} \text{term}$$
   3. $$\left( \frac{n - 1}{2} \right)^{\text{th}} \text{term}$$
   4. $$\left( \frac{n}{2} + 1 \right)^{\text{th}} \text{term}$$
2. $\left( \begin{array}{c} 11 \\ 0 \end{array} \right) + \left( \begin{array}{c} 11 \\ 2 \end{array} \right) + \left( \begin{array}{c} 11 \\ 4 \end{array} \right) + \cdots + \left( \begin{array}{c} 11 \\ 10 \end{array} \right)$ is equal to
   1. 2¹¹
   2. 2¹²
   3. 2¹⁰
   4. 2¹¹- 1
3. Middle term (s) in the expansion of $\left( 2x^2 - \frac{1}{5x} \right)^{11}$ is :
   1. T₅ , T₆
   2. T₄ , T₅
   3. T₆ , T₇
   4. T₇ , T₈
4. In the binomial expansion of ( a + x ) ⁿ the exponent of x _________ from zero to index .
   1. Remain Same
   2. Increases
   3. Decrease
   4. Varies
5. $\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \cdots + n\binom{n}{n} = ?$
   1. $n.2^n$
   2. $n.2^{n -1}$
   3. $n.2^{n+1}$
   4. $2^{n+1}$
6. In the expansion of $(a + b)^n$, $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ is ______ term:
   1. $r^{th}$
   2. $( r + 1 )^{ th}$
   3. $( r - 1 )^{th}$
   4. $( r + 2 )^{th}$
7. $(a + x)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} x^r \text{ where } n \in \mathbb{Z}$
   1. Integers
   2. Positive integers
   3. Non - negative
   4. Real numbers
8. The coefficient of the 3rd last term in the expansion of (1 + x)³⁰⁰ is:
   1. 277
   2. 44850
   3. 303
   4. 4305600
9. $T_{r+1} = \frac{n(n-1)(x-2) \cdots (n-r+1)}{r!} x^r$ is the ______ of binomial series.
   1. Binomial term
   2. General term
   3. $n^{th}$ term
   4. $7^{th}$ term
10. 1² + 2² + 3²+......+ n² = _____________
    1. $$\frac{n(n+1)(n+2)}{6}$$
    2. $$\frac{n(n+1)(2n+1)}{3}$$
    3. $$\frac{n(n+1)(2n+1)}{6}$$
    4. $$\frac{n(n+1)(n+2)}{3}$$