Term independent of x MCQ Quiz in मराठी - Objective Question with Answer for Term independent of x - मोफत PDF डाउनलोड करा

Calculation:

⇒ - 2x³ + 3x² + 9 + 14x - 21 - 63/x²

⇒ - 2x³ +  3x2 + 9 + 14x - 21 - 63x^-2

Here, coefficients of x2 and x  are 3 and 14 respectively,

Their product = 3 × 14 = 42

∴ The correct answer is 42

Concept:

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

Calculation:

The general term of  is given by, 

Now, the term is free from x (independent term), then power of x is zero

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

9 - 3r = 0 

⇒r = 3

⇒r + 1 = 4

∴ 

=  -84k³

∴ -84 = -84k³ 

⇒ k = 1

Hence, option (3) is correct. 

Concept:

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

Calculation:

Given binomial is 

∴ General term = Tr+1 =  = ...(i)

Now, for the coefficient of term containing x¹⁵, 30 – 3r = 15 ⇒ r = 5

∴ coefficient of x¹⁵  = ¹⁵C₅ 2⁵  [ using (i) ] 

For the term independent of x, put 30 – 3r = 0 ⇒ r = 10

∴ Constant term = ¹⁵C₁₀ 2¹⁰ [ using (i) ] 

∴ ratio of the coefficient of x15 to the term independent of x

= 

=  [∵ ]

= 

∴ Required ratio is 1 : 32.

Concept:

(1 + x)ⁿ = 

Calculation:

Given, 

⇒ 

The term independent of x in this expression is obtained when

x^k is multiplied by 1/x^k .

⇒ The term independent of x = 

∴ The correct answer is option (3).

Concept:

The (r + 1)th term of (a + b)n is given by Tr+1 = nCran-rbr 

Calculation:

Given, second, third and fourth terms in the expansion of (a + b)n are 135, 30 and  

∴ 

and, 

and, 

By 

By  

By  

⇒ 

⇒ 4n – 8 = 3n – 3 

⇒ n = 5

∴ The value of n is 5.

Concept:

The (r + 1)^th term of (a + b)ⁿ is given T_r+1 = ⁿC_ra^n-rb^r 

Calculation:

Given, 

The general term of 

= 

Put r = 6 to get coeff. of 

Put r = 7 to get coeff. of 

= 

∴ 

= 

⇒ p = 

∴ 108p = 180 × 

= 54

∴ The value of 108p is 54.

Concept: Binomial expansion.

The binomial (a + b) can be expanded to the power n (n is non-negative) in the following form:

(a + b)ⁿ = 

where , denotes the (r + 1)^th term.

Solution:

We have = 

⇒ a = 2x², b = , n = 12

∴ 

⇒ 

⇒ 

Now, for the term to be independent of x, the power of x should be zero

⇒ 24 - 3r = 0

⇒ 3r = 24

⇒ r = 8

∴  is independent of x.

∴ The term independent of x is the 9^th term.

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, 

⇒  

= ⁷C_r x^3-r

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

Therefore, 3 - r = 0

⇒ r = 3

Hence the term is T₃₊₁ and the coefficient is ⁷C₃

Calculation:

Given,

We are given the equation:

1 \)

Simplify the terms inside the bracket:

The general term T{r+1}\) is given by:

For the term independent of x, the exponent of x must be zero:

The required term is T₅:

∴ The term independent of x is 210.

Concept:

• Binomial Theorem: Used to expand expressions of the form (a + b)ⁿ into a sum involving binomial coefficients.
• Term Independent of x: A term in an expansion where the power of x is 0. For any expression raised to a power, to find the constant term, we solve for the power of x becoming 0 in the general term.
• Given expression is of the form (x^1/3 − x^−1/2)¹⁰.
• Let the general term be: T_r+1 = C(10, r) × (x^1/3)^10−r × (−x^−1/2)^r
• Total power of x in the term = (10 − r)/3 − r/2
• Set this power equal to 0 to find the constant term.

Calculation:

Given expression: [(x + 1)/(x^2/3 − x^1/3 + 1) − (x − 1)/(x − x^1/2)]¹⁰

Simplifying and rewriting: = [(x^1/3 + 1)/(x^2/3 − x^1/3 + 1)] − [(√x − 1)/(√x − 1)] = (x^1/3 − x^−1/2)¹⁰

Let T_r+1 be the general term:

T_r+1 = C(10, r) × (x^1/3)^10−r × (−x^−1/2)^r

⇒ T_r+1 = C(10, r) × (−1)^r × x^{(10−r)/3 − r/2}

For term independent of x:

⇒ (10 − r)/3 − r/2 = 0

⇒ Multiply entire equation by 6:

⇒ 2(10 − r) − 3r = 0

⇒ 20 − 2r − 3r = 0

⇒ 20 − 5r = 0

⇒ r = 4

Now compute the coefficient:

T₅ = C(10, 4) × (−1)⁴ = C(10, 4) = 210

∴ The term independent of x is 210.