Number of Terms in Binomial Expansion

Q.

The total number of terms in the expansion of $(x+a{)}^{100}+(x-a{)}^{100}$ after simplification is

1. 202

2. 51

3. None of these

4. 50

Q.

Find the number of terms in the expansion of $(a+b+c{)}^n.$

Q.

The coefficient of $\frac{1}{x}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

1. $\frac{n!}{\left\{\right(n-1)!(n+1)!\}}$

2. $\frac{\left(2n\right)!}{\left[\right(n-1)!(n+1)!]}$

3. None of these

4. $\frac{\left(2n\right)!}{(2n-1)!(2n+1)!}$

Q. The number of integral terms in the expansion of $(\sqrt{3}{+}^8\sqrt{5}{)}^{256}$ is ?

1. 35
2. 33
3. 34
4. 32

Q. The total number of terms which are dependent on the value of $x$, in the expansion ${(x^2-2+\frac{1}{x^2})}^n,n\in N$ is equal to

1. $2n+1$
2. $2n$
3. $n+1$
4. $n$

Q.

Wrtie the number of terms in the expansion of $\left[\right(2x+y^3{)}^4{]}^7$

Q.

Write the number of terms in the expansion of $(1-3x+3x^2-x^3{)}^8.$

Q.

The number of irrational terms in the expansion of ${(5^{\frac{1}{3}}+2^{\frac{1}{4}})}^{100}$ is

1. 94

2. 92

3. 91

4. 93

Q.

Write the number of terms in the expansion of $(2+\sqrt{3}x{)}^{10}+(2-\sqrt{3}x{)}^{10}.$

Q.

The number of irrational terms in the expansion of $(4^{1/5}+7^{1/10}{)}^{45}$ is

1. 40

2. 5

3. 41

4. None of these

Q. The number of terms in the expansion of ${(9x^2+12x+4)}^{30}$ is

Q.

The Coefficient of $x^n$ in the expansion of ${\log }_a\left(1+x\right)$ is

1. $\frac{{\left(-1\right)}^{n-1}}{n}$

2. $\left(\frac{-1^{n-1}}{n}\right){\log }_{a}e$

3. $\left(\frac{-1^{n-1}}{n}\right){\log }_{e}a$

4. $\left(\frac{-1^n}{n{\log }_{a}e}\right)$

Q. The number of terms in the expansion of $(x^3+9x^2+27x+27{)}^{25}$ is

1. $26$
2. $25$
3. $75$
4. $76$

Q. The natural number $m,$ for which the coefficient of $x$ in the binomial expansion of ${(x^m+\frac{1}{x^2})}^{22}$ is $1540,$ is

Q. $\mathrm{Number}\mathrm{of}\mathrm{terms}\mathrm{in}\mathrm{the}\mathrm{expansion}\mathrm{of}(a+b{)}^n\mathrm{is}n+1$

1. False
2. True

Q. If $\left[x\right]$ denotes the greatest integer function of $x$, then $\left[\right(6\sqrt{6}+14{)}^{2n+1}]$ is

1. an integer divisible by $2$
2. an integer divisible by $3$
3. an integer divisible by$5$
4. an odd prime number

Q. The number of distinct terms in the expansion of $(x+y^2{)}^{13}+(x^2+y{)}^{14}$ is

1. 27
2. 29
3. 28
4. 25

Q.

If |x| < 1 then the coefficient of $x^n$ in the expansion

of $(1+x+x^2+.......{)}^2$ will be

1. None of these

2. 1

3. n

4. n+1

Q. The function $f\left(x\right)=[x{]}^2-[x^2]$ (where $\left[y\right]$ is the greatest integer function less than or equal to $y$), is discontinuous at

1. All integers
2. All integers except $0$ and $1$
3. All integers except $0$
4. All integers except $1$

Q. The set of all real number $x$ for which $x^2-|x+2|+x>0$ is

1. $(-\infty ,-2)\cup (2,\infty )$
2. $(-\infty ,-\sqrt{2})\cup (\sqrt{2},\infty )$
3. $(-\infty ,-1)\cup (1,\infty )$
4. $(\sqrt{2},\infty )$

Q.

One of the terms in the expansion of $(x+y{)}^n$ is $k{x}^8{y}^{13}$. Find the number of terms in the expansion.

__

Q. The sum of an infinite geometric progression is $243$ and the sum of its first five terms is $275$. Find the progression.

Q. The number of irrational terms in the expansion of ${(4^{1/5}+7^{1/10})}^{45}$ is

1. 40
2. 5
3. none of these
4. 41

Q. The number of irrational terms in the expansion of ${(3^{\frac{1}{8}}+5^{\frac{1}{4}})}^{84}$ is

1. $73$
2. $74$
3. $75$
4. $76$

Q. ${\int }_{\pi }^{2\pi }\left[\sqrt{2}\cos x\right]dx=$ , where [.] is greatest integer function.

1. $-\frac{\pi }{2}$
2. $\frac{\pi }{2}$
3. $\pi$
4. $\frac{\pi }{4}$

Q. The total number of terms in the expansion of $(x+a{)}^{100}+(x-a{)}^{100}$ after simplification is

1. 202
2. 51
3. 50
4. None of these

Q. ${\int }_0^2\left(\right[x{]}^2-\left[x^2\right])dx$ is equal to (where $[.]$ denotes the greatest integer function)

1. $4+2\sqrt{3}+\sqrt{2}$
2. $-4+\sqrt{3}+\sqrt{2}$
3. $4+\sqrt{3}-\sqrt{2}$
4. $-2+\sqrt{3}+\sqrt{2}$

Q. In the expansion of $(7^{\frac{1}{3}}+{11}^{\frac{1}{9}}{)}^{6561}$,

1. there are exactly $730$ rational terms
2. there are exactly $5832$ irrational terms
3. the term which involves greatest binomial coefficients is rational
4. the term which involves greatest binomial coefficient is irrational

Q. The value of ${\int }_0^2[x^2-1]dx$ dx, where [x] denotes the greatest integer function is given by:

1. $3-\sqrt{3}-\sqrt{2}$
2. 1
3. 2
4. -1

Q. Let a, b be two positive numbers, where a>b and $4\times GM=5\times HM$ for the numbers. Then a is

1. 4b
2. $\frac{1}{4}b$
3. b
4. 2b