Question

Sum of coefficient in the expansion of $(1+ax-2x^2{)}^n$ is

• A. Positive when $a<1$ and $n=2k,k\in N$
• B. Negative when $a<1$ and $n=2k+1,k\in N$
• C. Positive when $a>1$ ad $n\in N$
• D. zero, when $a=1$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Positive when $a<1$ and $n=2k,k\in N$
B Negative when $a<1$ and $n=2k+1,k\in N$
C Positive when $a>1$ ad $n\in N$
D zero, when $a=1$

To find sum of co-efficent of

$(1+ax-2x^2{)}^n$

We put $x=1$

$\therefore$ Sum $=(1+a-2{)}^n$

$=(a-1{)}^n$

a) $a<1,n=2k,k\in N$

$\therefore a-1<0$

$\therefore (a-1{)}^{2k}$ is positive.

$\therefore$ option (a) is true.

b) $a<1,n=2k+1,k\in N$

$\therefore a-1<0$

$\therefore (a-1{)}^{2k+1}$ is negative.

option (b) is true.

c) $a<1,n\in N$

$\therefore a-1<0$ when $n$ is odd

then $(a-1{)}^n$ is negative.

option (c) is wrong.

d) $a=1$

$\therefore a-1=0$

$\therefore (a-1{)}^n=0$

(d) is true.

$\therefore$ option (a), (b), (d) is true.