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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