Special Terms of Binomial Expansion MCQ Quiz - Objective Question with Answer for Special Terms of Binomial Expansion - Download Free PDF

7Concept:

Binomial Expansion:

• The binomial theorem is used to expand expressions of the form .
• The general term in the expansion of is given by , where ⁿC_k is the binomial coefficient.
• To find the coefficient of , we identify the corresponding terms from the expansion and set the coefficient equal to 35.

Calculation:

Given the expansion of , we have:

The coefficient of x³ in  the expansion is 35.

We use the binomial expansion formula for the term x³

⇒ (p+ q)_c3 = 35 = ⁷C₃

⇒ p+q =7

∴ the correct answer is Option C

Concept:

• The general term in a binomial expansion of (a + b)n is given by: 
• If sin θ = sin α ⇒ θ = 

Calculation:

Given, middle term of is equal to .

Since, n = 10

⇒ Middle term =  term = 6^th term.

∴ T₆ = T₅₊₁

= 

⇒  = 

⇒  =  sin⁵x

⇒  = 252 sin5x

⇒ sin5x = 

⇒ sin x =  = sin

∴ x =

Calculation

Given:

A = Coefficient of 30th term in (1 + x)2n-1 = 2n–1C29

B = Coefficient of 12th term in (1 + x)2n-1 = 2n–1C11 

2A = 5B

⇒ 

⇒ 

⇒ 

⇒ 2n – 12 = 30

⇒ n = 21

Hence option 2 is correct

Concept:

General term: General term in the expansion of (a + b)n is given by

Calculation:

We know that Tr+1 = Cr an-r br.

In the given binomial expression , n = 10, a = x and b = .

∴ Tr+1 = ¹⁰Cr x^10-r  = 10Cr (-1)^r x10-2r

For the term to be independent of x, we must have 10 - 2r = 0.

⇒ r = 5.

The required term is:

10C5 (-1)⁵ = - ¹⁰C₅.

Concept:

General term: General term in the expansion of (a + b)n is given by

1. When n is even, the middle term = 

2. When n is odd, the middle terms are  and 

∴ The middle term will be  = T₅, which is the 5th term.

Concept:

General term: General term in the expansion of (x + y)n is given by

Middle terms: The middle terms is the expansion of (x + y) n depends upon the value of n.

• If n is even, then total number of terms in the expansion of (x + y) n is n +1. So there is only one middle term i.e. \(\rm \left( {\frac{n}{2} + 1} \right){{\rm{\;}}^{th}}\) term is the middle term.
• If n is odd, then total number of terms in the expansion of (x + y) n is n + 1. So there are two middle terms i.e. and  are two middle terms.

Here, we have to find the middle terms in the expansion of 

Here n = 8 (n is even number)

∴ Middle term = 

T5 = T (4 + 1) = 8C4 × (2x) (8 - 4) × 

T5 =  8C₄ × 2⁴

CONCEPT:

In the expansion of (a + b)n the general term  is given by: Tr + 1 = nCr ⋅ an – r ⋅ br

Note: In the expansion of (a + b)n , the rth term from the end is [(n + 1) – r + 1] = (n – r + 2)th term from the beginning.

In the expansion of (a + b)n , the middle term is  term if n is even.

In the expansion of (a + b)n , if n is odd then there are two middle terms which are given by:. 

CALCULATION:

Given: (x + 3)6 

Here, n = 6

∵ n = 6 and it as even number.

As we know that, in the expansion of (a + b)n the middle term is  term if n is even.

Concept:

General term: General term in the expansion of (x + y)n is given by

Middle terms: The middle terms is the expansion of (x + y) n depends upon the value of n.

• If n is even, then total number of terms in the expansion of (x + y) n is n +1. So there is only one middle term i.e. \(\rm \left( {\frac{n}{2} + 1} \right){{\rm{\;}}^{th}}\) term is the middle term.
• If n is odd, then total number of terms in the expansion of (x + y) n is n + 1. So there are two middle terms i.e. and  are two middle terms.

Calculation:

Here, we have to find the middle terms in the expansion of 

Here n = 5 (n is odd number)

∴ Middle term =  and  = 3^rd and 4^th

T3 = T (2 + 1) = 5C2 × (2x) (5 - 2) ×   and T4 = T (3 + 1) = 5C3 × (2x) (5 - 3) ×  

T3 =  5C2 × (2³x) and T4 = 5C3 × 2² × 

T3 = 80x and T4 = 

Hence the middle term of expansion is 80x and 

Concept:

General term: General term in the expansion of (x + y) ⁿ is given by

Calculation:

Given expansion is

General term =  

For the term independent of x the power of x should be zero 

i.e 

⇒ r = 2

Concept:

We have (x + y) ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^n-1 . y + ⁿC₂ x^n-2. y² + …. + ⁿC_n yⁿ

General term: General term in the expansion of (x + y) ⁿ is given by: 

Calculation:

We have to find term independent of x in 

We know that,

For the term independent of x, power of x should be zero

Therefore, 20 – 5r = 0

⇒ r = 4

Concept:

General term: General term in the expansion of (a + b)n is given by

Calculation:

We know that Tr+1 = ⁿCr an-r br.

In the given binomial expression (3 + ax)9, n = 9, a = 3 and b = ax.

∴ Tr+1 = 9Cr 39-r (ax)r = 9Cr 3⁹  xr.

For the coefficients of x2 and x³, we must have r = 2 and 3 respectively.

⇒ 9C2 39  = 9C3 39 

⇒ a = .

Concept:

General term: General term in the expansion of (x + y)n is given by

Calculation:

We have to find term independent of x in 

As we know, 

⇒  

= ⁹C_r x^{18 - 2r} (-1)^r x^-r

= (-1)^r 9C_r x^{18 - 3r}

For the term independent of x, power of x should be zero 

Therefore, 18 - 3r = 0

∴ r = 6

Hence the value is (-1)^6 9C₆ = 84

Concept:

General term: General term in the expansion of (x + y)n is given by

Middle terms: The middle terms is the expansion of (x + y) n depends upon the value of n.

• If n is even, then the total number of terms in the expansion of (x + y) n is n +1. So there is only one middle term i.e.  term is the middle term.

• If n is odd, then the total number of terms in the expansion of (x + y) n is n + 1. So there are two middle terms i.e. and  are two middle terms.

(1−x)n=∑k=0n(nk)1n−k(−x)k(1−x)n=∑k=0n(nk)1n−k(−x)k(1−x)n=∑k=0n(nk)1n−k(−x)

Calculation:

Given:

(1 + 4x + 4x2)5

⇒ [(1 + 2x)²]⁵

⇒ (1+ 2x)¹⁰ 

Here n = 10 (n is even number)

∴ Middle term =  = 6th term 

Middle Term, T₆ =  T_{5 + 1} = ¹⁰C₅ (1)⁵ (2x)⁵

⇒ × 32x⁵

⇒ 8064 x⁵

∴ The coefficient of the middle term in the expansion of (1 + 4x + 4x2)5  is 8064.

Concept:

General term: General term in the expansion of (x + y)n is given by

Middle terms: The middle terms is the expansion of (x + y) n depends upon the value of n.

• If n is even, then there is only one middle term i.e. \(\rm \left( {\frac{n}{2} + 1} \right){{\rm{\;}}^{th}}\) term is the middle term.
• If n is odd, then there are two middle terms i.e. and  are two middle terms.

Calculation:

Here, we have to find the middle terms in the expansion of 

Here n = 10 (n is even number)

∴ Middle term = 

T₆ = T (5 + 1) = ¹⁰C₅ × (x) (10 - 5) × 

T₆ =  ¹⁰C5

Concept:

General term: General term in the expansion of (x + y)n is given by

In the expansion of (x + y)n the number of terms is (n + 1)

From the end (n + 1)^th term is 1st term and n^th term is 2 term 

Calculation:

In the expansion of  ,n^th term from the end of an expansion is 2nd term 

T_r+1 = ⁿC_r (2x)^{(n - r)} 

T₂ =  ⁿC₁.(2x)^{(n - 1)} 

= 

= 

=