Sum of binomial coefficients with alternate signs

Q.

If the eccentricities of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1and\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ be e and $e_1$, then $\frac{1}{e^2}+\frac{1}{e_1^2}=$

1. 1

2. 2

3. 3

4. None of these

Q. In the expansion of $(x-1)(x-2).....(x-18)$, the coefficient of $x^{17}$ is

1. $684$
2. $-171$
3. $171$
4. $-342$

Q.

If in the expansion of $(1+x{)}^n$, a, b, c are three

consecutive coefficients, then n =

1. 

2. None of these

3. 

4. 

Q.

Find the imaginary part of the complex number $(1+i{)}^{(1-i)}$

1. 

2. 

3. 

4. 

Q. In the expansion of $(1+x{)}^{50}$, the sum of the coefficient of odd powers of x is

1. 0
2. 2⁴⁹
3. 
4. 

Q. Find the value of ${}^n{C}_1{+}^{n}C5{+}^{n}C9{+}^n{C}_{13}.....$

1. $2^{\frac{n-1}{2}}sin\frac{n\pi }{4}+2^{n-2}$
2. $2^{\frac{n}{2}}sin\frac{n\pi }{4}+2^{n-1}$
3. $2^{n-2}(1+sin\frac{n\pi }{4})$
4. $2^{\frac{n-2}{2}}sin\frac{n\pi }{4}+2^{n-2}$

Q.

$9^n$ + 7 is divisible by ________.

1. 4

2. 8

3. 3

4. 6

Q. If $n$ is a positive integer, then $(\sqrt{3}+1{)}^{2n}-(\sqrt{3}-1{)}^{2n}$ is :

1. an odd positive interger
2. an even positive interger
3. a rational number other than positive integers
4. an irrational number

Q.

If $2\times {}^n{C}_5$ = $9\times {}^{n-2}{C}_5$, then the value of n will be

1. 7

2. 10

3. 9

4. 5

Q.

A set contains (2n+1) elements. The number of sub-sets of the set which

contain at most n elements is

1. 

2. 

3. 

4. 

Q.

The value of $\sum _{r=0}^n(2r+1)({}^n{C}_r{)}^2$ is equal to :

1. (n+1). ²ⁿC_n

2. (2n+1). ²ⁿC_n

3. n. ²ⁿC_n

4. (2n+2). ²ⁿC_n

Q.

Find the sum to indicated number of terms in each of the geometric progressions in $\sqrt{7},\sqrt{21},3\sqrt{7},.....$ n terms.

Q. The sum of the rational terms in the expansion of ${(\sqrt{2}+\sqrt[5]{53})}^{10}$ is

1. 32
2. 9
3. 41
4. None of these

Q.

Q.

If ${}^n{C}_x{=}^n{C}_y$ then

1. x=y

2. n = x + y

3. n = x

4. n = y

Q. ${}^{10}{C}_1{+}^{10}{C}_3{+}^{10}{C}_5{+}^{10}{C}_7{+}^{10}{C}_9=$

1. 2¹⁰
2. 2⁹
3. 2¹⁰-1
4. None of these

Q.

${10}^{th}$ term of ${(3-\sqrt{\frac{17}{4}+3\sqrt{2}})}^{20}$ is

1. a rational number

2. a positive integer

3. an irrational number

4. a negative integer

Q. The sum of coefficients of intergral power of $x$ in the binomial expansion ${(1-2\sqrt{x})}^{50}$is:

1. $\frac{1}{2}(2^{50}+1)$
2. $\frac{1}{2}(3^{50}+1)$
3. $\frac{1}{2}\left(3^{50}\right)$
4. $\frac{1}{2}(3^{50}-1)$

Q.

If ${}^n{C}_r=C_r$ then $C_0+(C_0+C_1)+(C_0+C_1+C_2)+....+(C_0+C_1+.....+C_n)$ =

1. n.2^n-1

2. (n+2)2ⁿ

3. 2ⁿ

4. (n+2)2^n-1

Q.

Find the ${20}^{th}$ term in the binomial expansion of $(1+x{)}^{20}$ when $x=-5$

1. $-20\times 5^{19}$

2. $20\times 5^{20}$

3. $20\times 5^{19}$

4. $-20\times 5^{20}$

Q. If $a(\frac{1}{b}+\frac{1}{c}),b(\frac{1}{c}+\frac{1}{a}),c(\frac{1}{a}+\frac{1}{b})$ are in A.P. prove that $a,b,c$ are in A.P.

Q.

The value of $\sum _{0\le i<}^{}\sum _{j\le n}^{}({}^n{C}_i+{}^n{C}_j)is:$

1. 

2. 

3. 

4.