Practice question of Binomial Theorem for iitjee , nda and airforce (Set 1)

Progression

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Question no. 1

What is equal to ?

looks_one ⁿ⁺²C₁

looks_two ⁿ⁺²C_n

looks_3 ⁿ⁺³C_n

looks_4 ⁿ⁺²C_n+1

Option looks_one ⁿ⁺²C₁

Solution :

Question no. 2

What is the sum of coefficients in the expansion of (1+x)ⁿ ?

looks_one 2ⁿ

looks_two 2ⁿ - 1

looks_3 2ⁿ + 1

looks_4 (n+1)

Option looks_one 2ⁿ

Solution :
(1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ........ + ⁿC_nxⁿ
put x = 1
2ⁿ = ⁿC₀ + ⁿC₁ + ⁿC₂ + ........ + ⁿC_n
sum of coefficient = ⁿC₀ + ⁿC₁ + ⁿC₂ + ........ + ⁿC_n = 2ⁿ

Question no. 3

The value of the term independent of x in the expansion of is

looks_one 9

looks_two18

looks_3 48

looks_4 84

Option looks_4 84

Solution :
T_r+1 = ⁿC_r(x²)^n-r(-1/x)^r
= ⁿC_r(x)^2n-2r-r(-1)^r = ⁿC_r(x)^2n-3r(-1)^r
For independent of x :
2n -3r = 0
3r = 2 *9
r = 6
The term independent of x = ⁹C₆(-1)⁶ = 84

Question no. 4

What is the ratio of coefficient of x¹⁵ to the term independent of x in ?

looks_one 1/64

looks_two 1/32

looks_3 1/16

looks_4 1/4

Option looks_two 1/32

Solution :

Question no. 5

What is the middle term in the expansion of ?

looks_one 35x⁴/8

looks_two 17x⁵/8

looks_3 35x⁵/8

looks_4 none of these

Option looks_one 35x⁴/8

Solution :
n is even , number of terms will be odd.

Question no. 6

What is the coefficient of x⁴ in the expansion of ?

looks_one -16

looks_two 16

looks_3 8

looks_4 -8

Option looks_two 16

Solution :

Question no. 7

What is the number of terms in the expansion of (a+b+c)ⁿ ?

looks_one n+1

looks_two n+2

looks_3 n(n+1)

looks_4 (n+1)(n+2)/2

Option looks_4 (n+1)(n+2)/2

Solution :
(a+b+c)ⁿ
Number of variable (r) = 3
Number of terms = ^n+r-1C_r-1
Number of terms = ^n+3-1C_3-1 = ⁿ⁺²C₂
Number of terms = ⁿ⁺²C₂ = (n+2)!/(n!2!) = (n+1)(n+2)/2

Question no. 8

What is the coefficient of x¹⁷ in the expansion of ?

looks_one 189/8

looks_two 567/2

looks_3 21/16

looks_4 None of these

Option looks_one 189/8

Solution :

Question no. 9

What is the sum of coefficient of all the terms in the expansion of (45x - 49)⁴ ?

looks_one -256

looks_two -100

looks_3 100

looks_4 256

Option looks_4 256

Solution :
Sum of coefficient
put x = 1 in expression
Sum of coefficient = (45-49)⁴ = (4)⁴ = 256

Question no. 10

What is the sum of the coefficient of x⁴ in the expansion of (1 + 2x +3x² + 4x³ +.... ) ^1/2 ?

looks_one 1/4

looks_two 1/16

looks_3 1

looks_4 1/128

Option looks_3 1

Solution :
(1 + 2x +3x² + 4x³ +.... ) = (1-x)^-2
(1 + 2x +3x² + 4x³ +.... ) ^1/2 = (1-x)^{-2 * 1/2} = (1-x)^-1
(1-x)^-1 = 1 + x + x² + x³ + x⁴ ......
sum of the coefficient of x⁴ = 1

Question no. 11

If t_r is the rth term in the expansion (1+x)¹⁰¹, then what i the ratio equal to?

looks_one 20x/19

looks_two 83x

looks_3 19x

looks_4 83x/19

Option looks_4 83x/19

Solution :

Question no. 12

What is the coefficient of x³y⁴ in (2x+ 3y²)⁵ ?

looks_one 240

looks_two 360

looks_3 720

looks_4 1080

Option looks_3 720

Solution :
(2x+ 3y²)⁵
T_r+1 = ⁵C_r(2x)^5-r(3y²)^r
x³ = x^5-r ⇒ 5-r = 3 ⇒ r = 2
coefficient of x³y⁴ = ⁵C₂(2)^5-2(3)² = ⁵C₂(2)³(3)²
coefficient of x³y⁴ = 720

Question no. 13

If the coefficients of 5^th , 6^th and 7^th terms are in the expansion of (1+x)ⁿ be in A.P. , then n =

looks_one 7 only

looks_two 14 only

looks_3 7 or 14

looks_4 None of these

Option looks_3 7 or 14

Solution :
T₅ = ⁿC₄(x)⁴
T₆ = ⁿC₅(x)⁵
T₇ = ⁿC₆(x)⁶
According to Question
T₅ + T₇ = 2T₆
ⁿC₄ + ⁿC₆ = 2ⁿC₅

Question no. 14

The coefficient of 1/x in the expansion of is

looks_one

looks_two

looks_3

looks_4 None of these

Option looks_two

Solution :

Question no. 15

The greatest coefficient in the expansion of (1+x)²ⁿ⁺¹ is

looks_one

looks_two

looks_3

looks_4

Option looks_one

Solution :
The greatest coefficient in the expansion of (1+x)²ⁿ⁺¹ is :
Number of terms is odd
therefore , greatest coefficient = ²ⁿ⁺¹C_(2n+1+1)/2 = ²ⁿ⁺¹C_n+1

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