Binomial Theorem for Any Index

Q.

The fourth term in the expansion of $(1-2x{)}^{\frac{3}{2}}$ will be

1. - $\frac{3}{4}$$x^4$

2. $\frac{x^3}{2}$

3. -$\frac{x^3}{2}$

4. $\frac{3}{4}$$x^4$

Q. If the expansion of $\frac{1}{(1-ax)(1-bx)}=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n+\cdots$, then $a_n$ is (where $a\ne b,|ax|,|bx|<1$)

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

Q. The approximate value of $(1.0002{)}^{3000}$ is

1. $1.6$
2. $1.4$
3. $1.8$
4. $1.2$

Q. If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected and $\frac{(1+x{)}^{3/2}-{(1+\frac{1}{2}x)}^3}{(1-x{)}^{1/2}}$ may be approximated as $a+bx+c{x}^2$, then

1. $ab>c$
2. $a=b$
3. $bc=a$
4. $c>b=a$

Q. If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected and $\frac{(1+x{)}^{3/2}-{(1+\frac{1}{2}x)}^3}{(1-x{)}^{1/2}}$ may be approximated as $a+bx+c{x}^2$, then

1. $ab>c$
2. $a=b$
3. $bc=a$
4. $c>b=a$

Q. The coefficient of $x^{18}$ in the product $(1+x)(1-x{)}^{10}(1+x+x^2{)}^9$

1. $84$
2. $126$
3. $-126$
4. $-84$

Q. The approximate value of $(1.0002{)}^{3000}$ is

1. $1.2$
2. $1.4$
3. $1.6$
4. $1.8$

Q.

Find the cube root of 126 to 5 places of decimals.___

Q. The approximate value of $(1.0002{)}^{3000}$ is

1. $1.6$
2. $1.2$
3. $1.4$
4. $1.8$

Q. The coefficient of $x^{18}$ in the product $(1+x)(1-x{)}^{10}(1+x+x^2{)}^9$

1. $84$
2. $126$
3. $-126$
4. $-84$

Q. If ${10}^m$ divides ${101}^{100}-1$, then the greatest value of $m$ is

Q.

The expansion ${(1+x)}^n$ = 1 + nx + $\frac{n(n-1)}{2!}$${\left(x\right)}^2$........... is valid if |x| > 1.

1. True

2. False

Q.

If $\frac{(1-3x{)}^{\frac{1}{2}}+(1-x{)}^{\frac{5}{3}}}{\sqrt{4-x}}$ is approximately equal to a + bx for small values of x, then (a, b) =

1. (1, $\frac{35}{24}$)

2. (1, - $\frac{35}{24}$)

3. (2, $\frac{35}{12}$)

4. (2, - $\frac{35}{12}$)

Q. If the expansion of $\frac{1}{(1-ax)(1-bx)}=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n+\cdots$, then $a_n$ is (where $a\ne b,|ax|,|bx|<1$)

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