Binomial Theorem - Algebra -Multiple Correct Type Questions(JEE ADVANCED)

 Q.1

Correct Answer is:C

Solution Explain:

37n+2 = (5 × 7 + 2)n+2 = a multiple of 7 + 2n+2
16n+1 = (2 × 7 + 2)n+1 = a multiple of 7 + 2n+1 and
30n = (4 × 7 + 2)n =  a multiple of 7 + 2n
Thus 37n+2 + 16n+1 + 30n
= 7k + 2n (22 + 2 + 1) for same k ∈ N
= 7(k + 2n )
Therefore,
37n+2 + 16n+1 + 30n is divisible by 7.

Q.2

Correct Answer is:A,B,C

Solution Explain:

Putting x = w in the equation,

0 = a0 + a1ω + a2ω2  + a3 + ......                                  .... (i)

Putting x = ω2 in the equation,

0 = a0 + a1ω2 + a2ω + a3 +.....                                     .... (ii)

Putting x = 1 in the equation,

3n = a0 + a1 + a2 + a3 + ....                                          .....(iii)

adding (i), (ii) and (iii),

3n =   3(a0 + a3 + a6 + …..)         .....(a)

⇒   a0 + a3 + a6 + ....... = 3n–1    (option C)

subtracting (ii) from (i),

0 =  (ω – ω2) (a1 – a2 + a4 – a5 +.....)

Since ω – ω2 ≠ 0, a1 + a4 + a7 +... = a2 + a5 + a8 +  ...                  .....(iv)

Also from (3) – (a), a1 + a2 + a4 + a5 +.... =  3n – 3n–1 = 2.3n–1      ..... (v)

From (iv) and (v),  a1 + a4 + a7 +...  = a2 + a5 + a8 +  ... = 3n–1 = a0 + a3 + a6 + ....

Q.3

Correct Answer is:B,C

Solution Explain:

Q.4

Correct Answer is:A,B

Solution Explain:

Q.5

Correct Answer is:A,B,C,D

Solution Explain:

Sr = coefficient xr in the extension of (1+x)n (1- (1+x) )n 

⇒ Sr = (-1)n if r = n

⇒ Sr = 0  if r < n

Q.6

Correct Answer is:A,D

Solution Explain: