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Question

Let (1x+x4)10=a0+a1x+a2x2++a40x40. Then the correct option(s) is (are)
  1. a0+a2+a4+.....+a40=310+12
  2. a0=a40=1
  3. a1=a39
  4. a39=0


Solution

The correct options are
A a0+a2+a4+.....+a40=310+12
B a0=a40=1
D a39=0
(1x+x4)10=a0+a1x+a2x2++a40x40     (1)
Put x=1, we get
a0+a1+a2++a40=1
Put x=1, we get
a0a1+a2+a40=310
Adding both, 
a0+a2+a4++a40=310+12

Put x=0 in (1), we get a0=1

Differentiating (1) w.r.t. x and then putting x=0,
a1=10

Replace x by 1x in (1),
(11x+1x4)10=a0+a1x+a2x2++a40x40(x4x3+1)4=a0x40+a1x39+a2x38++a40     (2)
Putting x=0 in (2), we get a40=1

Differentiating (2) w.r.t. x and then putting x=0,
a39=0

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