Binomial Theorem

Q. Find an approximation of (0.99) 5 using the first three terms of its expansion.

Q.

Prove ${41}^n-{14}^n\text{is a multiple of 27.}$

Q. The sum of the coefficients of even powers of x in the expansion of ${(1+x+x^2+x^3)}^5$ is

1. 510
2. 512
3. 521
4. 522

Q.

The first 3 terms in the expansion of $(1+ax{)}^n$ (n ≠ 0) are 1, 6x and 16$x^2$. Then the value of a and n are respectively

1. 3 and 2

2. 2/3 and 9

3. 2 and 9

4. 3/2 and 6

Q.

The value of $\frac{({18}^3+7^3+\mathrm{3.18.7.25})}{3^6+\mathrm{6.243.2}+\mathrm{15.81.4}+\mathrm{20.27.8}+\mathrm{15.9.16}+\mathrm{6.3.32}+64}$ is

1. 1
2. 5
3. 25
4. 100

Q.

If the coefficients of $x^{r-1},x^r$ and $x^{r+1}$ in the binomial expansion of $(1+x{)}^n$ are in AP, prove that
$n^2-n(4r+1)+4r^2-2=0$

Q. Expand the following expression:
${(\frac{2}{x}-\frac{x}{2})}^5$

Q.

If $(5+2\sqrt{6}{)}^n=m+f$, where n and m are positive integers and 0 $\le$ f < 1, then $\frac{1}{1-f}-f$ is equal to:

1. m

2. $m+\frac{1}{m}$

3. $\frac{1}{m}$

4. $m-\frac{1}{m}$

Q.

$1+\frac{2}{3}.\frac{1}{2}+\frac{2.5}{3.6}{\left(\frac{1}{2}\right)}^2.+\frac{\mathrm{2.5.8}}{\mathrm{3.6.9}}{\left(\frac{1}{2}\right)}^3$ + .... =

1. 2^1/3

2. 3^1/4

3. 4^1/3

4. 3^1/3

Q. Expand the following expression:
$(1-2x{)}^3$

Q.

The sum of coefficients of the last six terms in the expansion of $(1+x{)}^{11}$ is

1. 512

2. 256

3. 2048

4. 1024

Q.

If P is a prime number, then $n^p$ - n is divisible by p when n is a

1. Natural number greater than 1

2. Irrational number

3. Odd number

4. Complex number

Q. 14. Prove that 2 "c-4"r n

Q.

The expression $(2+\sqrt{2}{)}^4$ has value, lying between

1. 134 and 135

2. None of these

3. 135 and 136

4. 136 and 137

Q. Show that ${\{i^{23}+{\left(\frac{1}{i}\right)}^{29}\}}^2=-4$.

Q.

The coefficient of x4 in the expansion of ${(\frac{x}{2}-\frac{3}{x^2})}^{10}$ is

1. 

2. 

3. 

4. 

Q.

The sum of the coefficients of even powers of x in the expansion of $(1+x+x^2+x^3{)}^5$ is equal to:

1. 256

2. 512

3. 1024

4. None

Q.

For positive integers n₁, n₂ the value of the expression $(1+i{)}^{n_1}$+$(1+i^3{)}^{n_1}$+$(1+i^5{)}^{n_2}$ + $(1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

1. =-1

2. =+1

3. =

4. >0, >0

Q.

If is a prime number, then $n^p$ - n is divisible by p

when n is a

1. Natural number greater than 1

2. Irrational number

3. Complex number

4. Odd number

Q.

If $(1+ax{)}^n$ = 1 + 8x + 24 $x^2$ + ......, then the value of a

and n is

1. 3, 6

2. 1, 2

3. 2, 4

4. 2, 3

Q.

For positive integers n₁, n₂ the value of the expression $(1+i{)}^{n_1}$+$(1+i^3{)}^{n_1}$+$(1+i^5{)}^{n_2}$ + $(1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

1. =+1

2. =-1

3. =

4. >0, >0

Q.

If $(1+ax{)}^n$ = 1 + 8x + 24 $x^2$ + ......, then the value of a and n is

1. 2, 4

2. 1, 2

3. 2, 3

4. 3, 6

Q.

Let $S=C_0^2+1.\frac{C_1^2}{C_0}+2.\frac{C_2^2}{C_1}+3.\frac{C_3^2}{C_2}+.........+n.\frac{C_n^2}{C_{n-1}}$ Where $C_r={}^n{C}_r$ then which of the following is wrong

1. 2^n-1 divides (S+n)

2. 2ⁿ divides 'S'

3. 'S' is a positive integer

4. (n+2) divides (S+n)

Q.

If |x| < 1, then in the expansion of

$(1+2x+3x^2+4x^3+.......{)}^{\frac{1}{2}}$, the coefficient of $x^n$ is

1. n

2. 1

3. 

4. 

Q.

If is a prime number, then $n^p$ - n is divisible by p when n is a

1. Natural number greater than 1

2. Irrational number

3. Complex number

4. Odd number

Q. 6. (96)*

Q.

Co-efficient of $x^5$ in the expansion of $(1+x^2{)}^5(1+x{)}^4$ is ___________.

1. 50

2. 40

3. 30

4. 60

Q. 9. (99)

Q. $\sum _{r=0}^n(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\cdots \infty ]$ is equal to

1. $\frac{1}{2^n-1}$
2. $\frac{1}{1-2^n}$
3. $\frac{1}{2^n+1}$
4. $\frac{2^n}{2^n+1}$

Q. Given that $\frac{d}{dx}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=Ax+B$.

1. $-1$
2. $1$
3. $2$
4. $4$