Binomial Theorem MCQ Quiz - Objective Question with Answer for Binomial Theorem - Download Free PDF

Concept:

The general term of the binomial (a + b)n is given by T_r+1 = ⁿC_ra^n-rb^r.

Calculation:

Given, 

∴ General term, 

⇒ 

For constant term, power of x = 0

⇒ 2r - 12 = 0

⇒ r = 6

∴  = α × 28 ×  

⇒ α = 

⇒ 25α = 9 × 11 × 7 = 693.

∴ The value of 25α is equal to 693.

The correct value is Option 3.

Concept:

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

Calculation:

We have to find coefficient of x³ in 

General term: 

For coefficient of x³;

⇒ 9 – 3r = 3

⇒ 6 = 3r

∴ r = 2

Now, Coefficient of x³ 

= 3² × 2² × 2⁷ × 3² = 2⁹ × 3⁴

∴ Option 3 is correct.

Explanation:

Given:

⇒ Mean x = np = 6....(i)

⇒ SD = ...(ii)

Dividing (ii) by (i), we get

⇒ 

⇒ q= 1/3

Now

⇒ p = 1 – q = 1- 1\3 = 2/3

From (i)

⇒

⇒n =9

∴ Option (d) is correct.

Explanation:

Given:

(8 + 3√7) 20  = U + V...(i)

(8 - 3√7) 20  = W...(ii)

⇒ (U + V)W =] (8 + 3√7) 20][(8 - 3√7) 20]

= (64 – 63)²⁰ = 1²⁰ = 1

∴ Option (b) is correct.

Explanation:

Given:

(8 + 3√7) 20  = U + V...(i)

(8 - 3√7) 20  = W...(ii)

Here, 0

Now, adding Eqs. (i) and (ii), we get

U + V +W = (8 + 3√7) 20 + (8 - 3√7) 20

= 2[²⁰C₀8²⁰+ ²⁰C₂8¹⁸. (3√7 )²+........+(3√7)²⁰

⇒ It is an even number.

Also, 0

Thus,  V + W is an integer

V + W = 1

∴ Option (d) is correct.

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:

(1 + x)n = nC0 × 1(n-0) × x 0+ nC1 × 1(n-1) × x 1 + nC2  × 1(n-2) × x2 + …. + nCn  × 1(n-n) × xn

n^th  term of the G.P. is an = arⁿ⁻¹

Sum of n terms = s = ; where r >1

Sum of n terms = s = ; where r

Calculation:

C(n, 1) + C(n, 2) + _ _ _ _  _ + C(n, n) 

⇒ ⁿC₁ + ⁿC₂ + ... + ⁿC_n 

⇒ ⁿC₀ + nC1 + nC2 + ... + nCn - ⁿC₀

⇒ (1 + 1)ⁿ - ⁿC_o 

⇒ 2ⁿ - 1 =  = 1 × 

Comparing it with a G.P sum = a × , we get a = 1 and r = 2

∴ 2n - 1 = 1 + 2 + 2² + ... +2^n-1 which will give us n terms in total.

Concept:

(1 + x)ⁿ = ⁿC₀ × 1^(n-0) × x ⁰+ ⁿC₁ × 1^(n-1) × x ¹ + ⁿC₂  × 1(n-2) × x² + …. + ⁿC_n  × 1(n-n) × xⁿ

Calculation:

Given expansion is (1 + x)²ⁿ

⇒ ²ⁿC₀ ×1^(2n-0) × x⁰ +  2nC₁ ×1(2n-1) × x¹ + ... +  2nC_2n ×1(2n-2n) × x²ⁿ

First term = 2nC0 ×1 × 1 = 1

Last term =  2nC_2n ×1 × x²ⁿ = 1 × x²ⁿ = x²ⁿ

⇒ Sum = 1 + x²ⁿ

Coefficient of 1 = 1, coefficient of x²ⁿ = 1

∴ sum of the coefficients = 1 + 1 = 2.

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:

Expansion of (1 + x)n:

Calculation:

Given: the third term in the binomial expansion of (1 + x)m is (-1/8)x²

So, the third term in the binomial expansion of (1 + x)m is 

 = (-1/8)x²

⇒ 

⇒ 4m² - 4m + 1 = 0

⇒ (2m - 1)² = 0

⇒ 2m - 1 = 0

∴ m = 

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:

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.

Calculation:

Here, we have to find the coefficient of the middle term in the binomial expansion of (2 + 3x) ⁴

Here n = 4 (n is even number)

∴ Middle term =

T₃ = T _{(2 + 1)} = ⁴C₂ × (2) ^{(4 - 2)} × (3x) ²

T₃ = 6 × 4 × 9x² = 216 x²

∴ Coefficient of the middle term = 216

Concept:

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

Expansion of (1 + x)ⁿ:

Calculation:

To Find: coefficient of x² in the expansion of 

Now, the coefficient of x² in the expansion = 

Formula used:

(1 + x)n  = [nC0 + nC1 x + nC2 x2 + … +nCn xn]

• C₀ + C₁ + C₂ + … + C_n = 2ⁿ
• C₀ + C₂ + C₄ + … =  2n-1
• C₁ + C₃ + C₅ + … = 2^n-1

Calculation:

(1 + x)⁵⁰  = [⁵⁰C0 + ⁵⁰C1 x + ⁵⁰C2 x2 + … +⁵⁰Cn x⁵⁰]    ----(1)

Here, n = 50

Using the above formula, the sum of odd terms of the coefficient is

S = (50C1 + 50C3­ + 50C5 + ……. + 50C49)

⇒ S = 2^50-1 = 2⁴⁹

∴ Sum of odd terms of the coefficient = 249