IIT JEE 1982 Maths - 'Adapted' - FIB to Single-Digit Integer Q4

Algebra Level 3

If the sum of the coefficients of the polynomial $(1 + x - 3x^2)^{2163}$ is $-2^{9s}$ , then find $s-\text{sgn}(s)$ .

Notation: $\text{sgn } (\cdot)$ denotes the sign or signum function .

In case you are preparing for IIT JEE, you may want to try IIT JEE 1982 Mathematics Archives

1 solution

Md Zuhair
Feb 1, 2017

The expression $(1 + x - 3x^2)^{2163}$ can be written as $a_0 + a_1 x + a_2 x^2 + ....... + a_{2163} x^{2163}$ .

Now Putting x=1 we get ,

$(1+1-3)^{2163}$ = $a_0 + a_1 + a_2 + ... a_{2163}$

Now $-1 =$ $a_0 + a_1 + a_2 + ... a_{2163}$

Hence $-2^{9s} = -1$

Or $s= 0$ .

Now We know $sgn$ means Signum function.

And for $0, sgn(0) = 0$

So $s - sgn(0) = \boxed{0}$ (ANSWER)