NCERT Exemplar Problems Class 11 Mathematics Chapter 8 Binomial Theorem

Short Answer Type Questions:
Q1. Find the term independent of x, where x≠0, in the expansion of ${ \left( \frac { 3{ x }^{ 2 } }{ 2 } -\quad \frac { 1 }{ 3x } \right) }^{ 15 }$

Q2. If the term free from x is the expansion of ${ \left( \sqrt { x } -\frac { k }{ { x }^{ 2 } } \right) }^{ 10 }$ is 405, then find the value of k.

Sol: Given expansion is ${ \left( \sqrt { x } -\frac { k }{ { x }^{ 2 } } \right) }^{ 10 }$

Q3. Find the coefficient of x in the expansion of (1 – 3x + 1x²)( 1 -x)¹⁶.

Sol: (1 – 3x + 1x²)( 1 -x)¹⁶

Q4. Find the term independent of x in the expansion of ${ \left( 3x-\frac { 2 }{ { x }^{ 2 } } \right) }^{ 15 }$

Sol: Given Expression ${ \left( 3x-\frac { 2 }{ { x }^{ 2 } } \right) }^{ 15 }$

Q5. Find the middle term (terms) in the expansion of

Q6. Find the coefficient of x¹⁵ in the expansion of ${ \left( x-{ x }^{ 2 }\quad \right) }^{ 10 }$

Sol: Given expression is ${ \left( x-{ x }^{ 2 }\quad \right) }^{ 10 }$

Q7. Find the coefficient of $\frac { 1 }{ { x }^{ 17 } }$ in the expansion of ${ \left( { x }^{ 4 }-\frac { 1 }{ { x }^{ 3 } } \quad \right) }^{ 15 }$

Q8. Find the sixth term of the expansion (y^1/2 + x^1/3)ⁿ, if the binomial coefficient of the third term from the end is 45.

Q9. Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)¹⁸ are equal.

Q10. If the coefficient of second, third and fourth terms in the expansion of (1 + x)²” are in A.P., then show that 2n² – 9n + 7 = 0.

Q11. Find the coefficient of x⁴ in the expansion of (1 + x + x² + x³)¹¹.

Long Answer Type Questions

Q12. If p is a real number and the middle term in the expansion ${ \left( \frac { p }{ 2 } +2\quad \right) }^{ 8 }$ is 1120, then find the value of p.

Q15. In the expansion of (x + a)ⁿ, if the sum of odd term is denoted by 0 and the sum of even term by Then, prove that

Q17. Find the term independent ofx in the expansion of (1 +x + 2x³) ${ \left( \frac { 3 }{ 2 } { x }^{ 2 }-\frac { 1 }{ 3x } \quad \quad \right) }^{ 9 }$

Objective Type Questions

Q18. The total number of terms in the expansion of (x + a)¹⁰⁰ + (x – a)¹⁰⁰ after simplification is

(a) 50
(b) 202
(c) 51
(d) none of these

Q19. If the integers r > 1, n > 2 and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)²ⁿ are equal, then
(a) n = 2r
(b) n = 3r
(c) n = 2r + 1
(d) none of these

Q20. The two successive terms in the expansion of (1 + x)²⁴ whose coefficients are in the ratio 1 : 4 are
(a) 3^rd and 4^th
(b) 4^th and 5^th
(c) 5^th and 6^th
(d) 6^th and 7^th

Q21. The coefficients of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)²ⁿ ~¹ are in the ratio
(a) 1 : 2
(b) 1 : 3
(c) 3 : 1
(d) 2:1

Q22. If the coefficients of 2^nd, 3^rd and the 4^th terms in the expansion of (1 + x)ⁿ are in A.P., then the value of n is
(a) 2
(b) 7
(c) 11
(d) 14