Binomial Theorem & its Simple Applications Test

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Binomial Theorem & its Simple Applications Test - 1

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Question 1
2 / -0.83

The number of terms in the expansion of (2x - 3y)⁸ is

A
6

B
9

C
8

D
5

Solution

Since this binomial is to the power 8, there will be nine terms in the expansion.
Question 2
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The middle term in the expansion of

A
254

B
252

C
250

D
251

Solution

n = 10
Middle term = (n/2) + 1
= (10/2) + 1
= 6th term
T(6) = T(5+1)
= ¹⁰ C₅ [(2x² )/3]⁵ [(3/2x² )]⁵
= ¹⁰ C₅
= 252
Question 3
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In the expansion of (a+b)ⁿ , N the number of terms is:

A
n + 1

B
n

C
n –1

D
aⁿ

Solution

The total number of terms in the binomial expansion of (a + b)ⁿ is n + 1, i.e. one more than the exponent n.
Question 4
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Find the value of r, if the coefficients of (2r + 4)^th and (r –2)^th terms in the expansion of (1 + x)¹⁸ are equal.

A
6

B
5

C
4

D
3

Solution
Question 5
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The sixth term in the expansion of is

A
252

B
-896/27

C
4580/17

D
5580/17

Solution
Question 6
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If the coefficients of 7^th and 13^th terms in the expansion of (1 + x)ⁿ are equal, then n is equal to

A
10

B
15

C
18

D
20

Solution
Question 7
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The 6^th term in the expansion of is

A
-5040 / x

B
5040 x

C
5040 / x

D
- 5040 x

Solution
Question 8
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What is the coefficient of x⁵ in the expansion of (1-x)^-6 ?

A
252

B
250

C
-252

D
251

Solution

(1-x)^-6
=>(1-x)^(-6/1)
It is in the form of (1-x)^(-p/q) , p =6, q=1
(1-x)^(-p/q) = 1+p/1!(x/q)¹ + p(p+q)/2!(x/q)² + p(p+q)(p+2q)/3!(x/q)³ + p(p+q)(p+2q)(p+3q)/4!(x/q)⁴ ........
= 1+6/1!(x/1)¹ + 6(7)/2!(x/1)² + 6(7)(8)/3!(x/1)³ + 6(7)(8)(9)/4!(x/1)⁴ +.......................
So, coefficient of x⁵ is (6*7*8*9*10)/120
= 252
Question 9
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In the expansion of the binomial expansion (a + b)ⁿ , which of the following is incorrect ?

A
The exponent of a decreases from n to 0.

B
The number of terms is n+1.

C
The exponent of b increases from 0 to n.

D
The binomial coefficients in (a+b)ⁿ , which are equidistant from beginning and end are not equal.

Solution

Correct Answer: d
Explanation:- The coefficient of terms (x+a)ⁿ equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients.
ⁿ C_r = ⁿ C_{n –r} , r = 0,1,2,…,n.
Question 10
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The middle term in the expansion of (x + y)¹⁰ is the

A
6^th term

B
5^th term

C
average of 5^th and 6^th terms

D
7^th term

Solution

Number of terms(n) = 10
Middle term = (n/2) + 1
= (10/2) + 1
= 5 + 1
= 6th term
Question 11
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In a binomial expansion with power 13

A
one of the middle term is the 7^th term

B
the middle term is the average of the 6^th and the 7^th terms.

C
the middle term is obtained by subtracting the 6^th term from the 7^th term.

D
there are two middle terms, 6^th and 7^th

Solution
Question 12
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If in the expansion of (1+x)²⁰ , the coefficients of r^th and (r+4)^th terms are equal, then the value of r is equal to:

A
9

B
7

C
10

D
8

Solution

Coefficients of the rth and (r+4)th terms in the given expansion are C_{r −120} and ²⁰ C_r+3 .
Here,C_{r −120} = ²⁰ C_r+3
⇒r −1+r+3 = 20
[∵if ⁿ C_x = ⁿ C_y ⇒x = y or x+y = n]
⇒r = 2 or 2r = 18
⇒r = 9
Question 13
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The number of terms in the expansion of (x –y + 2z)⁷ are:

A
35

B
38

C
36

D
37

Solution

Here the number of terms can be calculated by:
= ((n+ 1) * (n+2)) /2
where, n =7
∴Number of terms = 36
Question 14
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The number of terms in the expansion of (a + b + c)ⁿ are:

A

B

C

D

Solution

No. of terms is ⁿ⁺² C₂
Question 15
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The general term in the expansion of (a - b)ⁿ is

A
T_r+1 = (-1)^r ⁿ C_r a^n-r b^r

B
T_r+1 = ⁿ C_r a^n-r b^r

C
T_r+1 = ⁿ C_r a^r b^r

D
T_r+1 = (-1)^r ⁿ C_r a^r b^r

Solution

If a and b are real numbers and n is a positive integer, then:
(a - b)ⁿ = ⁿ C₀ aⁿ + ⁿ C₁ a^{(n –1)} b¹ + ⁿ C₂ a^{(n –2)} b² + ...... + ⁿ C_r a^{(n –r)} b^{r+ ... +}ⁿ C_n bⁿ ,
The general term or (r + 1)th term in the expansion is given by:
T_{r + 1} = (-1)C_r a^{(n –r)} b^r