Test: Binomial Theorem For Positive Index

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QUESTION: 1

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: 2

The sixth term in the expansion of is

A.
252
B.
-896/27
C.
4580/17
D.
5580/17

Solution:

(2x² - 1/3x²)¹⁰
T(6) = T(5+1) = 10C5(2x²)⁵ (-1/3x²)⁵
= - 10!/(5!)(5!) * (2)⁵ * (1/(3)⁵)
= - 10!/(5!)(5!)]32/243
= - 896/27

QUESTION: 3

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:

Coefficent of 7th term is nC6 and 13th term is nC12
nC6 = nC12
⇒ nC(n-6) = nC12
n-6 = 12
n = 18

QUESTION: 4

The 6^th term in the expansion of is

A.
-5040/X
B.
5040 X
C.
5040/X
D.
- 5040 X

Solution:

(4x/5−5/2x)⁹
Tr+1 in the expansion of (x+a)ⁿ is given by:
Tr+1=nCr x^(n−r).a^r
Hence 6th term will be given by:
T6=T5+1 = 9C5 (4x/5)⁴(−5/2x)⁵
=−(9.8.7.6.5)/(5.4.3.2.1).[4⁴.x⁴]/5⁴.[5⁵/(2⁵.x⁵)]
=−(9.8.7.6.5)/(5.4.3.2.1).[(4⁴.x⁴)/5⁴].[5⁵/(2⁵.x⁵)]
=−126(2⁸).(5/2⁵.x)
=−(630×2³)/x
=−(5040)/x

QUESTION: 5

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: 6

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 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 and

nCr = nCn – r, r = 0,1,2,…,n.

QUESTION: 7

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: 8

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: 9

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: 10

In the expansion of (a+b)ⁿ, nN 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: 11

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) = 10C5[(2x²)/3]⁵ [(3/2x²)]⁵
= 10C5
= 252

QUESTION: 12

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 ²⁰C_r+3.
Here,C_r−120 = ²⁰C_r+3⇒ r−1+r+3=20 [∵ if ⁿC_x=ⁿC_y ⇒ x=y or x+y=n]
⇒r=2 or 2r=18
⇒r=9

QUESTION: 13

The number of terms in the expansion of (x – y + 2z)⁷ are:

A.
35
B.
38
C.
36
D.
37

Solution:

QUESTION: 14

The number of terms in the expansion of (a + b + c)ⁿ are:

Solution:

No. of terms is ⁿ⁺²C₂

QUESTION: 15

The general term in the expansion of (a - b)ⁿ is

A.
T_r+1 = (-1)^r ⁿC_r a^n-rb^r
B.
T_r+1 = ⁿC_r a^n-rb^r
C.
T_r+1 = ⁿC_r a^rb^r
D.
T_r+1 = (-1)^r ⁿC_r a^rb^r

Solution:

If a and b are real numbers and n is a positive integer, then
(a - b)ⁿ = nC0 aⁿ + nC1 a^{(n – 1)} b¹ + nC2 a^{(n – 2)} b²+ ...... + nCr a^{(n – r)} b^{r+ ... +} nCnbⁿ,
The general term or (r + 1)th term in the expansion is given by
Tr + 1 = (-1)Cr a^(n–r) b^r

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Test: Binomial Theorem For Positive Index

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