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JEE Main

Explore popular questions from Binomial Theorem for JEE Main. This collection covers Binomial Theorem previous year JEE Main questions hand picked by experienced teachers.

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Binomial Theorem

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Q 1. If {tex} ( 27 ) ^ { 999 } {/tex} is divided by {tex} 7 , {/tex} then the remainder is

6

B

1

C

1

D

3

Explanation

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Q 2. {tex} 2 ^ { 3 n } - 7 n - 1 {/tex} is divisible by

A

64

B

36

49

D

25

Explanation

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Q 3. For each {tex} n \in N , 2 ^ { 3 n } - 1 {/tex} is divisible by

A

8

B

16

C

32

None of these

Explanation

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Q 4. If the binomial expansion of {tex} ( a + b x ) ^ { - 2 } {/tex} is {tex} \frac { 1 } { 4 } - 3 x + \dots , {/tex} where {tex} a > 0 , {/tex} then {tex} ( a , b ) {/tex} is

{tex} ( 2,12 ) {/tex}

B

{tex} ( 2,8 ) {/tex}

C

{tex} ( - 2,12 ) {/tex}

D

None of these

Explanation

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Q 5. The number of terms in the expansion of {tex} ( 1 + x ) \left( 1 + x ^ { 3 } \right) \left( 1 + x ^ { 6 } \right) {/tex} {tex} \left( 1 + x ^ { 12 } \right) \left( 1 + x ^ { 24 } \right) \dots \left( 1 + x ^ { 3 \times 2 ^ { 2 } } \right) {/tex} is

A

{tex} 2 ^ { n + 3 } {/tex}

B

{tex} 2 ^ { n + 4 } {/tex}

C

{tex} 2 ^ { n + 5 } {/tex}

None of these

Explanation

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Q 6. The number of terms in {tex} ( 1 + x ) ^ { 101 } \left( 1 + x ^ { 2 } - x \right) ^ { 100 } {/tex} is

A

302

B

301

202

D

101

Explanation

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Q 7. If coefficient of {tex} x ^ { 2 } y ^ { 3 } z ^ { 4 } {/tex} in {tex} ( x + y + z ) ^ { n } {/tex} is {tex} A , {/tex} then coefficient of{tex} x ^ { 4 } y ^ { 4 } z {/tex} is

A

{tex}\mathrm{2A}{/tex}

B

{tex} \frac { n A } { 2 } {/tex}

{tex} \frac { A } { 2 } {/tex}

D

None of these

Explanation

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Q 8. Let {tex} r ^ { \text { th } } {/tex} term of a series be given by {tex} T _ { r } = \frac { r } { 1 - 3 r ^ { 2 } + r ^ { 4 } } {/tex} . Then {tex} \lim\limits _ { n \rightarrow \infty } \sum\limits _ { r = 1 } ^ { n } T _ { r } {/tex} is

A

{tex}3 / 2 {/tex}

B

{tex} 1/ 2 {/tex}

{tex} - 1 / 2 {/tex}

D

{tex} -3/ 2 {/tex}

Explanation

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Q 9. The coefficient of {tex} a ^ { 4 } b ^ { 5 } {/tex} in the expansion of {tex} ( a + b ) ^ { 9 } {/tex} is

{tex} \frac { 9 ! } { 4 ! 5 ! } {/tex}

B

{tex} \frac { 9 ! } { 6 ! 3 ! } {/tex}

C

{tex} \frac { 4 ! 5 ! } { 9 ! } {/tex}

D

None of these

Explanation

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Q 10. If the coefficient in the third term of the expansion of {tex} \left( x ^ { 2 } + \frac { 1 } { 4 } \right) ^ { n } {/tex} when expanded in decreasing powers of {tex} x {/tex} is 31 , then {tex} n {/tex} is equal to

A

16

B

20

C

30

32

Explanation

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Q 11. The sum of coefficients in the expansion of {tex} \left( 1 + x - 3 y ^ { 2 } \right) ^ { 2163 } {/tex} is

A

1

-1

C

{tex} 2 ^ { 2163 } {/tex}

D

None of these

Explanation

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Q 12. The sum of the rational terms in the expansion of {tex} \left( \sqrt { 2 } + 3 ^ { 1 / 5 } \right) ^ { 10 } {/tex} is

A

20

B

21

C

40

41

Explanation

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Q 13. In the expansion of {tex} ( 1 + x ) ^ { 50 } {/tex} , let S be the sum of coefficients of odd power of {tex} x , {/tex} then S is

A

0

{tex} 2 ^ { 49 } {/tex}

C

{tex} 2 ^ { 50 } {/tex}

D

{tex} 2 ^ { 51 } {/tex}

Explanation

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Q 14. The coefficient of {tex} x ^ { 53 } {/tex} in {tex} \sum\limits _ { r = 0 } ^ { 100 } 100 \mathrm { C } _ { r } ( x - 3 ) ^ { 100 - r } 2 ^ { r } {/tex} is

A

{tex} ^ { 100 } \mathrm { C } _ { 51 } {/tex}

B

{tex}^{100} \mathrm { C } _ { 52 } {/tex}

{tex} ^{-100} \mathrm { C } _ { 53 } {/tex}

D

{tex} ^ { 100 } \mathrm { C } _ { 54 } {/tex}

Explanation

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Q 15. The coefficient of {tex} x ^ { m } {/tex} in {tex} ( 1 + x ) ^ { r } + ( 1 + x ) ^ { r + 1 } + ( 1 + x ) ^ { r + 2 } + \ ... {/tex} {tex} + ( 1 + x ) ^ { n } , r \leq m \leq n {/tex} is

{tex} ^ { n + 1 } C _ { m + 1 } {/tex}

B

{tex} ^ { n - 1 } C _ { m - 1 } {/tex}

C

{tex} ^ { n } \mathrm { C } _ { m } {/tex}

D

{tex} ^ { n } \mathrm { C } _ { m + 1 } {/tex}

Explanation

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Q 16. If {tex} \frac { 1 } { \sqrt { 2 x + 1 } } \times \left\{ \left( \frac { 1 + \sqrt { 2 x + 1 } } { 2 } \right) ^ { n } - \left( \frac { 1 - \sqrt { 2 x + 1 } } { 2 } \right) ^ { n } \right\} {/tex} is a polynomial of degree {tex} 5 , {/tex} then {tex} n {/tex} is equal to

A

9

B

10

11

D

None of these

Explanation

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Q 17. {tex} 3 ^ { 51 } {/tex} when divided by 8 leaves the remainder,

A

1

B

6

C

5

3

Explanation

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Q 18. If {tex} \frac { T _ { 2 } } { T _ { 3 } } {/tex} in the expansion of {tex} ( a + b ) ^ { n } {/tex} and {tex} \frac { T _ { 3 } } { T _ { 4 } } {/tex} in the expansion of {tex} ( a + b ) ^ { n + 3 } {/tex} are equal, then {tex} n {/tex} is equal to

A

3

B

4

5

D

6

Explanation

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Q 19. If (1 + x)15 = C0 + C1x + C2x2 + ... + C15x15, then C2 + 2C3 + 3C4 + ... + 14C15 is equal to

A

14.214

13.214 + 1

C

13.214 − 1

D

None of these

Explanation

We have, (1 + x)15 = C0 + C1x + C2x2 + ... + C15x15

$\Rightarrow \frac{\left( 1 + x \right)^{15} - 1}{x} = C_{1} + C_{2}x + ...C_{15}x^{14}$

On differentiating both sides w.r.t. x, we get

$\frac{x \bullet 15{(1 + x)}^{14} - {(1 + x)}^{15} + 1}{x^{2}} = C_{2} + 2C_{3}x + ... + 14C_{15}x^{13}$

On putting x = 1, we get

C2 + 2C3 + ... + 14C15 = 15.214 − 215 + 1

= 13.214 + 1

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Q 20. If the coefficients of second, third and fourth terms in the expansion of (1+x)2n are in A.P., then

A

2n2 + 9 n + 7 = 0

2 n2 − 9 n + 7 = 0

C

2 n2 − 9 n − 7 = 0

D

None of these

Explanation

It is given that

2nC1, 2nC2 and 2nC3 are A.P.

∴ 2 • 2nC2 = 2nC1 + 2nC3

$2 \bullet \frac{\left( 2\ n \right)\ !}{\left( 2\ n - 2 \right)\ !2\ !} = \frac{\left( 2\ n \right)!}{\left( 2\ n - 1 \right)!} + \frac{\left( 2\ n \right)!}{(2\ n - 3\ !3\ !)}$

$\Rightarrow 2\frac{(2\ n)(2\ n - 1)}{2} = 2\ n + \frac{\left( 2\ n \right)\left( 2\ n - 1 \right)(2\ n - 2)}{3\ !}$

⇒ 6(2 n−1) = 6 + (2 n − 1)(2 n − 2)

⇒ 12 n − 6 = 6 + 4 n2 − 6 n + 2

⇒ 4 n2 − 18 n + 14 = 0 ⇒ 2 n2 − 9 n + 7 = 0

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Q 21. If $\left| x \right| < \frac{1}{2}$, then the coefficient of xr in the expansion of $\frac{1 + 2x}{{(1 - 2x)}^{2}}$, is

A

r2r

B

(2r − 1)2r

C

r22r + 1

(2r + 1)2r

Explanation

$\frac{1 + 2x}{{(1 - 2x)}^{2}} = (1 + 2x){(1 - 2x)}^{- 2}$

$= \left( 1 + 2x \right)\left( 1 + \frac{2}{1!}\left( 2x \right) + \frac{2 \bullet 3}{2!}\left( 2x \right)^{2} + ... + \frac{2 \bullet 3\ldots r}{\left( r - 1 \right)!}\left( 2x \right)^{r - 1} + \frac{2 \bullet 3 \bullet 4\ldots\left( r + 1 \right)\left( 2x \right)^{r}}{r!} \right)$

The coefficient of $x^{r} = 2\frac{r!}{\left( r - 1 \right)!}2^{r - 1} + \frac{\left( r - 1 \right)!}{r!}2^{r}$

= r2r + (r+1)2r = 2r(2r + 1) = = !)

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Q 22.
$\begin{pmatrix} 30 \\ 0 \\ \end{pmatrix}\begin{pmatrix} 30 \\ 10 \\ \end{pmatrix} - \begin{pmatrix} 30 \\ 1 \\ \end{pmatrix}\begin{pmatrix} 30 \\ 11 \\ \end{pmatrix} + ...\begin{pmatrix} 30 \\ 20 \\ \end{pmatrix}\begin{pmatrix} 30 \\ 30 \\ \end{pmatrix}\text{is\ equal\ to}$

A

30C11

B

60C10

30C10

D

65C55

Explanation

Given, A = 30C030C1030C130C11 + 30C230C12 + … + 30C2030C30

= coefficient of x20 in (1 + x)30(1 − x)30

= coefficient of x20 in (1+x2)30

= coefficient of $x^{20}\text{\ in\ }\sum_{r = 0}^{30}\text{\ \ }\left( - 1 \right)^{r}\ \ ^{30}C_{r}(x^{2})^{r}$

= ( − 1)1030C10{for coefficient of x20, let r = 10}

= 30C10

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Q 23. If (1−x+x2)n = a0 + a1x + a2x2 + … + a2nx2n, then a0 + a2 + a4 + … + a2n is equal to

$\frac{3^{n} + 1}{2}$

B

$\frac{3^{n} - 1}{2}$

C

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

D

$\frac{3^{n - 1} - 1}{2}$

Explanation

We have,

a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + … + a2n x2n = (1−x+x2)n

Putting x = 1 and − 1, we get

(a0+a2+a4+…) + (a1+a3+a5+…) = 1 …(i)

And,

(a0+a2+a4+…) − (a1+a3+a5…) = 3n …(ii)

Adding (i) and (ii), we get

$a_{0} + a_{2} + a_{4} + \ldots = \frac{3^{n} + 1}{2}$

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Q 24.

If C0, C1, C2, ……Cn denote the binomial coefficient in the expansion of (1 + x)n, then

$C_{0}\frac{C_{1}}{2} + \frac{C_{2}}{3} + ... + \frac{C_{n}}{n + 1}\ \text{is\ equal\ to}$

$\frac{2^{n + 1} - 1}{n + 1}$

B

$\frac{2^{n} - 1}{n}$

C

$\frac{2^{n - 1} - 1}{n - 1}$

D

$\frac{2^{n + 1} - 1}{n + 2}$

Explanation

We know that ,

(1+x)n = C0 + C1x + C2x2 + … + Cnxn

On integrating both sides, from 0 to 1, we get

$\left\lbrack \frac{\left( 1 + x \right)^{n + 1}}{n + 1} \right\rbrack_{0}^{1} = \left\lbrack {C_{0}x + \frac{C_{1}x^{2}}{2} + \frac{C_{2}x^{3}}{3} + \ldots + \frac{C_{n}x^{n + 1}}{n + 1}}^{\ } \right\rbrack_{0}^{1}$

$\Longrightarrow \frac{2^{n + 1} - 1}{n + 1} = C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + ... + \frac{C_{n}}{n + 1}$

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Q 25. If the ratio of the 7th term from the beginning to the seventh term from the end in the expansion of $\left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^{x}$ is $\frac{1}{6}$ then x, is

9

B

6, 15

C

12, 9

D

None of these

Explanation

7th term from the beginning in the expansion of $\left( 2^{1/3} + \frac{1}{3^{1/3}} \right)^{x}$ is given by

$T_{7} = \ ^{x}C_{6}\left( 2^{1/3} \right)^{x - 6}\left( \frac{1}{3^{1/3}} \right)^{6}$

7th term from the end in the expansion of $\left( 2^{1/3} + \frac{1}{3^{1/3}} \right)^{x}$ is the (x+1−7+1)th = (x−5)th term from the beginning. Therefore,

$T_{x - 5} = \ ^{x}C_{x - 6}\left( 2^{1/3} \right)^{6}\left( \frac{1}{3^{1/3}} \right)^{x - 6}$

We have,

$\frac{T_{7}}{T_{x - 5}} = \frac{1}{6}$

⇒ 6 T7 = Tx − 5

$\Rightarrow 6\ \ ^{x}C_{6}\ 2^{\frac{x - 6}{3}}3^{- 2} = \ ^{x}C_{x - 6}\ 2^{2}3^{- \left( \frac{x - 6}{3} \right)}$

$\Rightarrow 2^{\frac{x - 9}{3}} = 3^{- \left( \frac{x - 9}{3} \right)}$

$\Rightarrow 6^{\frac{x - 9}{3}} = 1\ \Rightarrow x - 9 = 0\ \Rightarrow x = 9$