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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.

Solution:

QUESTION: 2

The sixth term in the expansion of is

Solution:

(2x^{2} - 1/3x^{2})^{10}

T(6) = T(5+1) = 10C5(2x^{2})^{5} (-1/3x^{2})^{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

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

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).[4^{4}.x^{4}]/5^{4}.[5^{5}/(2^{5}.x^{5})]

=−(9.8.7.6.5)/(5.4.3.2.1).[(4^{4}.x^{4})/5^{4}].[5^{5}/(2^{5}.x^{5})]

=−126(2^{8}).(5/2^{5}.x)

=−(630×2^{3})/x

=−(5040)/x

QUESTION: 5

What is the coefficient of x^{5} in the expansion of (1-x)^{-6 }?

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 x^{5} 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 ?

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

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

Solution:

QUESTION: 9

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

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:

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

Solution:

n = 10

Middle term = (n/2) + 1

= (10/2) + 1

= 6th term

T(6) = T(5+1) = 10C5[(2x^{2})/3]^{5} [(3/2x^{2})]^{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:

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

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

Solution:

QUESTION: 14

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

Solution:

No. of terms is ^{n+2}C_{2}

QUESTION: 15

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

Solution:

If a and b are real numbers and n is a positive integer, then

(a - b)^{n} = nC0 a^{n} + nC1 a^{(n – 1)} b^{1} + nC2 a^{(n – 2)} b^{2}+ ...... + nCr a^{(n – r)} b^{r+ ... +} nCnb^{n},

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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