Question 1

The coefficient of x^2 term in the binomial expansion of ( 1 3 x^ 1 2 +x^ -1 4 )^ 10 is

Accepted Answer

General term of given binomial expansion is aligned T_ r+1 & = ^ 10 C_r ( 1 3 x^ 1 2 )^ 10-r (x^ - 1 4 )^r \ & = ^ 10 C_r ( 1 3 )^ 10-r x^ 10-r 2 - r 4 aligned We have to find coefficient of x^2 aligned 10-r 2 - r 4 & =2 r=4 \ T_ 4+1 & = ^ 10 C_4 ( 1 3 )^6 x^2 \ Coefficient of x^2 & = ^ 10 C_4 ( 1 3 )^6 \ & = 10 9 8 7 4 3 2 1 1 3^6 = 70 243 aligned

Question 2

The coefficient of x^n in the expansion of e^ 7 x +e^x e^ 3 x is

Accepted Answer

Given, e^ 7 x +e^x e^ 3 x e^ 4 x +e^ -2 x Series of e^a=1+ a 1! + a^2 2! + a^3 3! aligned e^ 4 x + & e^ -2 x = (1+ 4 x 1! + (4 x)^2 2! + (4 x)^3 3! + ) \ & + (1+ (-2 x) 1! + (-2 x)^2 2! + ) aligned Here, coefficient of aligned & x^2 4^2 2! + (-2)^2 2! \ & x^3 4^3 3! + (-2)^3 3! \ & \ & x^n= 4^n n! + (-2)^n n! aligned

Question 3

The coefficient of the highest power of x in the expansion of (x+ x^2-1 )^8+ (x- x^2-1 )^8

Accepted Answer

Since (x+ x^2-1 )^8+ (x- x^2-1 )^8 array r =2 \ ^8 C_0 x^8+ ^8 C_2 x^6 (x^2-1 )+ ^8 C_4 x^4 (x^2-1 )^2 . \ .+ ^8 C_6 x^2 (x^2-1 )^3+ ^8 C_8 x^0 (x^2-1 )^4 \ array So coefficient of highest power of x =2 \ ^8 C_0+ ^8 C_2+ ^8 C_4+ ^8 C_6+ ^8 C_8 \ =(1+1)^8+(1-1)^8=2^8=256

Question 4

If the 17^ th and the 18^ th terms in the expansion of (2+a)^ 50 are equal, then the coefficient of x^ 35 in the expansion of (a+x)^ -2 is

Accepted Answer

Given, 17^ th and 18^ th terms in the expansion (2+a)^ 50 are equal T_ 17 =T_ 18 ^ 50 C_ 16 (2)^ 34 (a)^ 16 = ^ 50 C_ 17 (2)^ 33 (a)^ 17 a= ^ 50 C_ 16 ^ 50 C_ 17 2=1 Now, coefficient of x^ 35 in the expansion (1+x)^ -2 =-36 Coefficient of x^r is (r+1) in (1-x)^ -2 .

Question 5

If x^ 18 occurs in the rth term in the expansion of (x^4+ 1 x^3 )^ 15 , then what is the value of r ?

Accepted Answer

In the expansion of (x^4+ 1 x^3 )^ 15 , let T_r is the r^ th term aligned T_r & = ^ 15 C_ r-1 (x^4 )^ 15-r+1 ( 1 x^3 )^ r-1 \ & = ^ 15 C_ r-1 x^ 64-4 r-3 r+3 =15_ C_ r-1 x^ 67-7 r aligned x ^ 18 occurs in this term 18=67-7 r 7 r=49 r=7 .

Question 6

The middle term in the expansion of ( 10 x + x 10 )^ 10 is

Accepted Answer

General term =T_ r+1 = ^ 10 C _r ( 10 x )^ 10-r ( x 10 )^r Here n=10, which is an even number. Now, [ 10 2 +1 ]^ th term i.e. 6^ th term is the middle term. Hence, middle term =T_6 T _ 5+1 = ^ 10 C _5 ( 10 x )^ 10-5 ( x 10 )^5 = ^ 10 C _5 ( 10 x )^5 ( x 10 )^5= ^ 10 C _5.

Question 7

The number of terms in the expansion of (1+5 2 x)^9+(1-5 2 x)^9, is

Accepted Answer

We have, (1+5 2 x)^9+(1-5 2 x)^9 =2 \ ^2 C_0+ ^9 C_2(5 2 x)^2+ + ^9 C_8(5 2 x)^8 \ Clearly, it has 5 terms.

Question 8

If the sum of the coefficients in the expansion of (x+y)^n is 1024 , then the value of the greatest coefficient in the expansion is

Accepted Answer

Given, sum of the coefficient =1024 array ll i.e. & 2^n=1024=2^ 10 \ & n=10 array Since, n is even, so greatest coefficient = ^n C_ n / 2 = ^ 10 C_5=252

Question 9

The coefficient of the term independent of x in the expansion of ( x 3 + 3 2 x^ 2 )^ 10 is

Accepted Answer

The (r+1) th term in the expansion of ( x 3 + 3 2 x^ 2 )^ 10 is given by array l T _ r+1 = ^ 10 C _ r ( x 3 )^ 10-r ( 3 2 x^ 2 )^ r \ = ^ 10 C _ r x^ 5- ( r 2 ) 3^ 5- ( r 2 ) 3^ r 2^ r x^ 2 r \ = ^ 10 C _ r 3^ ( 3 r 2 )-5 -5- ( 5 r 2 ) 2^ r x array T _ r+1 = ^ 10 C _ r ( x 3 )^ 10-r ( 3 2 x^ 2 )^ r = ^ 10 C _ r x^ 5- ( r 2 ) 3 3^ r 2^ r x^ 2 r = ^ 10 C _ r 3^ ( 3 r 2 )-5 - ( 5 2 )- ( 5 r 2 ) 2^ r x For T _ r+1 to be independent of x, we must have 5- ( 5 r 2 )=0 or r=2. Thus, the 3 rd term is i

Question 10

Coefficient of x^ 13 in the expansion of (1-x)^ 5 (1+x+x^ 2 +x^ 3 )^ 4 is

Accepted Answer

Expression =(1-x)^ 5 (1+x)^ 4 (1+x^ 2 )^ 4 array l =(1-x) (1-x^ 2 )^ 4 (1+x^ 2 )^ 4 \ =(1-x) (1-x^ 4 )^ 4 array Coefficient of x^ 13 =- ^ 4 C _ 3 (-1)^ 3 =4

Question 11

If a, b, c are three natural numbers in AP and a + b + c =21 then the possible number of values of the ordered triplet ( a , b , c ) is

Accepted Answer

Let a=b-d and c=b+d, then a+b+c=21 b =7 So, the equation is a + c =14 No. of solution = coeff. of x^ 14 in (x+x^ 2 + . ) = ^ 13 C _ 12 =13

Binomial Theorem — Previous Year Questions & Solutions