On account of the disruption in education due to the corona pandemic, we're opening up our platform for teachers, free of cost. Know More →

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.

Select Subject

Physics

Chemistry

Mathematics

Binomial Theorem

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

A

64

B

36

49

D

25

Explanation

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

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

A

1

B

6

C

5

3

Explanation

Correct Marks 4

Incorrectly Marks -1

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

Correct Marks 4

Incorrectly Marks -1

Q 19. The coefficient of in the polynomial

A

B

D

Explanation

There are total factors, let
Let

Clearly,
and roots of the equation are
Sum of roots

Correct Marks 4

Incorrectly Marks -1

Q 20.

A

0

B

C

Explanation



Correct Marks 4

Incorrectly Marks -1

Q 21. is equal to

A

B

C

Explanation

Now, coefficient of in
coefficient of in
coefficient of in




Correct Marks 4

Incorrectly Marks -1

Q 22. Then which one of the following holds for all , by the principle of mathematical induction?

A

B

C

Correct Marks 4

Incorrectly Marks -1

Q 23. If and be positive, then the coefficients of and in the expansion of will be

Equal

B

Equal in magnitude but opposite in sign

C

Reciprocal to each other

D

None of the above

Explanation

Coefficient of is and coefficient of is
Both the coefficients are equal

Correct Marks 4

Incorrectly Marks -1

Q 24. The expression is a polynomial of degree

A

5

B

6

7

D

8

Explanation

We know that,
.]
Here,



Given expression is a polynomial of degree 7.

Correct Marks 4

Incorrectly Marks -1

Q 25. If and , then

B

C

D

None of these

Explanation

Since,
 
Now,