Number of Terms in Binomial Expansion

Q. The coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3{)}^6.$ is?

1. 132
2. 144
3. -132
4. -144

Q.

In the expansion of ${(4^{\frac{1}{3}}+6^{\frac{-1}{4}})}^{20}$, consider the following statements:

(i) the number of irrational terms = 18

(ii) the number of rational terms = 2

(iii) the middle term is irrational.

Then the correct statements are:

1. (i) and (iii)

2. (ii) and (iii)

3. All three

4. (i) and (ii)

Q. The number of integral terms in the expansion of $(\sqrt{6}+\sqrt{7}{)}^{32}$ is:

1. 16

2. 18

3. 20

4. 17

Q.

Find the coefficients of $x^{32}$ and $\frac{1}{x^{17}}$ in the expansion of ${(x^4-\frac{1}{x^3})}^{15}$

Q. Write the general term in the expansion of $(x^2-yx{)}^{12},x\ne 0$.

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

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

Q.

Find the first integral term in the expansion of $(\sqrt{3}+\sqrt[3]{32}{)}^9$

1. 

2. 

3. 

4. 

Q.

Using conradiction method, check the validity of the following statement if n is a real number with $n>3,then,n^2>9.$

Q. Let $n\in N$ and n < $(5\sqrt{3}+8{)}^4$. Then the greatest value of n is

1. 77040
2. 77042
3. 77041
4. none of these

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

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

Q.

Find the number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{85}{)}^{256}$

__

Q. The greatest integer less than or equal to ${(\sqrt{2}+1)}^6$ is?

1. 196
2. 197
3. 199
4. 198

Q. The term independent of x expansion of ${(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}})}^{10}$ is

1. 210
2. 4
3. 120
4. 310

Q. For $2\le r\le n,\left(\begin{array}{c}n\\ r\end{array}\right)+2\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{c}n\\ r-2\end{array}\right)$ is equal to

1. $\left(\begin{array}{c}n+1\\ r-1\end{array}\right)$
2. $\left(\begin{array}{c}n+1\\ r+1\end{array}\right)$
3. $\left(\begin{array}{c}n+2\\ r\end{array}\right)$
4. $2\left(\begin{array}{c}n+2\\ r\end{array}\right)$

Q.

Find all possible values of $\sqrt{x^2+25}$

1. [0, ∞)

2. (0, ∞)

3. [5, ∞)

4. [25, ∞)

Q.

The interval that gives all possible values of the expression $\sqrt{x^2-4x+20}$ is

1. [4, )

2. [16, )

3. (, -4)

4. (16, )

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

Q.

Find the greatest common factor of $6x^2{y}^2,9x{y}^3,3x^4{y}^6$.

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.

The number of integral terms in the expansion of ${(\sqrt{3}-\sqrt[8]{85})}^{256}$ is

1. 32

2. 33

3. 34

4. 35

Q. $3^{2n}$ when divided by $8$ will leave the remainder.

1. $5$
2. $1$
3. $0$
4. A multiple of $3$