Binomial Theorem MCQ Quiz in मल्याळम - Objective Question with Answer for Binomial Theorem - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

General Term in the expansion (x - y)ⁿ = T_r+1 = (-1)^r × ⁿC_r × x^n-r ×  y^r

Calculation:

The general term in the expansion of  is given by, 

) 

For independent terms, the power of x is zero.

On equating power of x in (1) with zero, we get,

⇒ 3r = 12

⇒ r = 4

⇒ r + 1 = 5

So, the 5th term in the given expansion is independent of x.

Hence, option (3) is correct.

Formula used:

n! = n × (n - 1) × (n - 2).......3 × 2 × 1 

Calculation:

Given that,

4 × nC5 = 9 × n-1C5

Using the above formula

⇒ 

⇒ 

⇒ 9n - 45 = 49

⇒ 5n = 4n

⇒ n = 9

Concept:

• ⁿC_n = 1
• ⁿC₁ = n

Calculation:

We have to find the value of 

⇒ ⁿ⁺⁰C_n + ⁿ⁺¹C_n

⇒ nCn + n+1Cn 

⇒ 1 + (n + 1)

⇒ (n + 2)

∴  The value of  is ^{n + 2}C_1.

Concept:

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

• General term: General term in the expansion of (x + y) n is given by
• In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th term.

Note: 

• In the binomial expansion of (x + y)n, the rth term from end = In the binomial expansion of (y + x)n, the rth term from the start.
• If we interchange the term x → y, it will give rth term from the beginning.

Calculation:

We have to find 9th term from the end in (x – 1/x) 12

We know that rth term from end means (n – r + 2)th term from start.

So. 9th term from the end = [12 – 9 + 2]th term from start = 5th term from start

General term: 

= ¹²C₄ x⁴

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

In the binomial expansion of (x + y)ⁿ , the r^th term from end is (n – r + 2)^th term.

Calculation:

We have to find 4th term in the expansion of 

We know that, 

= 220 x^3/2

∴ Option 3 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.  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 = 10 (n is even number)

∴ Middle term = 

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

T5 = -2⁵ × 10C5

Calculation:

For the given expression (x2 + x)10  ,  n = 10

(r + 1)th term

If this term is independent of x then the index of x must be zero.

i.e.  20 - r = 0,   r = 20.

Here r > n

So no term is independent of x.

Concept:

In the binomial expansion of (a + b)ⁿ, the term which does not involve any variable is said to be an independent term.

The general term in the binomial expansion of (a + b)ⁿ is given by:

Calculation:

Given:

Let (r + 1)^th be the independent term in the expansion of .

We know that the general term in the binomial expansion of (a + b)ⁿ is given by:

Here, a = x², n = 9 and b = 1 / x.

∵ The (r + 1)^th term is the independent term in the expansion of  ⇒ x^{18 – 3r} = x⁰

⇒ 18 – 3r = 0 ⇒ r = 6

⇒ 7^th term in the expansion of  is the independent term.

We have to find the value of the 7^th term in the expansion of

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 = 8 (n is even number)

∴ Middle term = 

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

T₅ =  8C₄

Concept:

Binomial Expansion:

• (a + b)n = C0 a0 bn + C1 a1 bn-1 + C2 a2 bn-2 + … + Cr ar bn-r + … + Cn-1 an-1 b1 + Cn an b0, where C0, C1, …, Cn are the Binomial Coefficients defined as Cr = nCr = .
• The total number of terms in the expansion is n + 1.
• The (r + 1)^th term in the expansion is T_r+1 = C_r a^r b^n-r.

Calculation:

For the given expression , n = 7.

There will be 7 + 1 = 8 terms in the expansion of the above expression.

Expanding by reversing the order of the terms (i.e. a = -3y and b = ), we have:

T₅ = T₄₊₁ = ⁷C₄  (-3y)⁴  = 35  (3⁴y⁴)  = 105x³y⁴.