Binomial Theorem

The number of terms in the expansion of ${(2 + \sqrt{3} x)}^{10} + {(2 - \sqrt{3} x)}^{10}$ is:
1. 11
2. 6
3. 12
4. 22
The middle term in the expansion of ${(x + \frac{1}{x})}^{10}$ is:
1. ${}^{11}C_{5}$
2. ${}^{11}C_{6}$
3. None of these
4. ${}^{10}C_{5}$
If a and b denote the sum of the coefficients in the expansion of ${(1 - 3 x + 10 x^{2})}^{n}$ and ${(1 + x^{2})}^{n}$ respectively, then the relation between a and b is:
1. $a = b$
2. $a = b^{3}$
3. $a = 2 b$
4. $a = b^{2}$
The coefficient of the middle term in the expansion of ${(1 + x)}^{2 n}$ is:
1. ${}^{2 n}C_{1}$
2. ${}^{n}C_{1}$
3. ${}^{2 n}C_{n}$
4. None of these
The term without x in the expansion of $(2 x - \frac{1}{2 x^{2}})$ is:
1. 495
2. - 495
3. - 7920
4. 7920
If $r^{t h}$ term in the expansion of ${(2 x^{2} - \frac{1}{x})}^{12}$ is without x, then r is equal to:
1. 8
2. 7
3. 9
4. 10
If A and B are the sums of odd and even terms respectively in the expansion of ${(x + a)}^{n}$ , then ${(x + a)}^{2 n} - {(x - a)}^{2 n}$ is equal to:
1. 4 (A + B)
2. 4 (A - B)
3. AB
4. 4 AB
The number of irrational terms in the expansion of ${(4^{\frac{1}{5}} + 7^{\frac{1}{10}})}^{45}$ is:
1. 40
2. 5
3. 41
4. None of these
The coefficient of $x^{- 17}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is:
1. 1365
2. - 1365
3. 3003
4. - 3003
In the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ , $x^{- 17}$ occurs in $r^{t h}$ term, then:
1. r = 10
2. r = 11
3. r = 12
4. r = 13
In the expansion of ${(x - \frac{1}{3 x^{2}})}^{9}$ , the term independent of x is:
1. $T_{3}$
2. $T_{4}$
3. $T_{5}$
4. None of these
If in the expansion of ${(1 + y)}^{n}$ , the coefficients of 5th, 6th and 7th terms are in A.P., then n is equal to:
1. 7, 14
2. 7, 11
3. 8, 16
4. None of these
If the coefficient of x in ${(x^{2} + \frac{λ}{x})}^{5}$ is 270, then $λ$ is:
1. 3
2. 4
3. 5
4. 2
The coefficient of $x^{4}$ in ${(\frac{x}{2} - \frac{3}{x^{2}})}^{10}$ is:
1. $\frac{405}{256}$
2. $\frac{504}{259}$
3. $\frac{450}{263}$
4. None of these
The coefficient of the term independent of x in the expansion of ${(a x + \frac{b}{x})}^{14}$ is:
1. $14! a^{7} b^{7}$
2. $\frac{14!}{7!} a^{7} b^{7}$
3. $\frac{14!}{{(7!)}^{2}} a^{7} b^{7}$
4. $\frac{14!}{{(7!)}^{3}} a^{7} b^{7}$
If n is a rational number, which is not a whole number, then the number of terms in the expansion of ${(1 + x)}^{n}$ , $|x| < 1$ , is:
1. n - 1
2. n + 1
3. n
4. Infinitely many
If n is a positive integer, then the binomial coefficients equidistant from the beginning and the end in the expansion of ${(x + a)}^{n}$ are:
1. Reciprocal to each other
2. Multiplicative inverse of each other
3. Additive inverse of each other
4. Equal
If the $r^{t h}$ term in the expansion of ${(\frac{x^{3}}{3} - \frac{2}{x^{2}})}^{10}$ contains $x^{20}$ , then r is:
1. 4
2. 3
3. 5
4. 2
If the coefficients of $x^{- 7}$ and $x^{- 8}$ in the expansion of ${(2 + \frac{1}{3 x})}^{n}$ are equal, then n = ?
1. 56
2. 15
3. 45
4. 55
The largest term in the expansion of ${(1 + x)}^{19}$ when $x = \frac{1}{2}$ is:
1. $8^{t h}$
2. $7^{t h}$
3. $5^{t h}$
4. $6^{t h}$
f coefficients of three successive terms in the expansion of ${(x + 1)}^{n}$ are in the ratio 1 : 3 : 5, then n is equal to:
1. 8
2. 7
3. 6
4. 9
The exponent of x occurring in the $7^{t h}$ term of expansion of ${(\frac{3 x}{2} - \frac{8}{7 x})}^{9}$ is:
1. - 5
2. 3
3. 5
4. - 3
The term independent of x in the expansion of ${(x - \frac{3}{x^{2}})}^{18}$ is:
1. ${}^{18}C_{12}$
2. ${}^{18}C_{6}$
3. ${}^{18}C_{6} . 3^{6}$
4. $3^{6}$
The number of dissimilar terms in the expansion of ${(a + b + c)}^{12}$ is:
1. 78
2. 39
3. 13
4. 91
The term containing $x^{3}$ in the expansion of ${(x - 2 y)}^{7}$ is:
1. Fourth
2. Sixth
3. Fifth
4. Third
The coefficients of $x^{p}$ and $x^{q}$ (p, q are positive integers) in the binomial expansion of ${(1 + x)}^{p + q}$ are:
1. Reciprocal to each other
2. Additive inverse of each other
3. Unequal
4. Equal
If 2nd, 3rd and 4th terms in the expansion of ${(x + a)}^{n}$ are 240, 720 and 1080 respectively, then the value of n is:
1. 5
2. 10
3. 15
4. 20
If the first three terms in the expansion of ${(x + a)}^{n}$ are 729, 7290 and 30375 respectively, then the value of n is:
1. 5
2. 8
3. 6
4. 9
The coefficient of $\frac{1}{x^{17}}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is:
1. - 455
2. - 3003
3. - 1365
4. 1365
The two consecutive terms in the expansion of ${(3 + 2 x)}^{74}$ , which have equal coefficients, are:
1. 28th and 29th
2. 11th and 12th
3. 30th and 31st
4. 27th and 28th
The remainder when $7^{103}$ is divided by 25 is:
1. 0
2. 18
3. 16
4. 9
The number $101^{100} - 1$ is divisible by:
1. 10
2. 9
3. 8
4. 7
The greater of two numbers $300!$ and $\sqrt{300^{300}}$ is:
1. 100
2. 200
3. 300
4. 500
If the second, third and fourth terms in the expansion of ${(a + b)}^{n}$ are 135, 30 and 10/3 respectively, then:
1. $a = 3$
2. $b = \frac{1}{3}$
3. $n = 5$
4. All the above
The number of irrational terms in the expansion of ${(2^{\frac{1}{2}} + 3^{\frac{1}{10}})}^{55}$ is:
1. 47
2. 56
3. 50
4. 48
The number of terms in the expansion of ${(a + b + c)}^{n}, n \in N$ is:
1. $\frac{(n + 1) (n + 2)}{2}$
2. $(n + 1)$
3. $(n + 2)$
4. $n (n + 1)$
The ratio of the coefficient of $x^{15}$ to the term independent of x in ${(x^{2} + \frac{2}{x})}^{15}$ is:
1. 12 : 32
2. 1 : 32
3. 32 : 12
4. 32 : 1
If A and B are coefficient of $x^{n}$ in the expansions of ${(1 + x)}^{2 n}$ and ${(1 + x)}^{2 n - 1}$ respectively, then $\frac{A}{B}$ equals:
1. 1
2. 2
3. $\frac{1}{2}$
4. $\frac{1}{n}$
If the middle term of ${(\frac{1}{x} + x \sin x)}^{10}$ is equal to $7 \frac{7}{8}$ , then value of x is:
1. $2 n π + \frac{π}{6}$
2. $n π + \frac{π}{6}$
3. $n π + {(- 1)}^{n} \frac{π}{6}$
4. $n π + {(- 1)}^{n} \frac{π}{3}$
Given the integers r > 1, n > 2, and coefficients of ${(3 r)}^{t h}$ and ${(r + 2)}^{n d}$ terms in the binomial expansion of ${(1 + x)}^{2 n}$ are equal, then:
1. n = 2r
2. n = 3r
3. n = 2r + 1
4. None of these

Binomial Theorem

Quiz