IIT JEE 1982 Maths - 'Adapted' - FIB to Single-Digit Integer Q1

Algebra Level 4

The coefficient of x 99 x^{99} in the polynomial ( x − 1 ) ( x − 2 ) ( x − 3 ) ⋯ ( x − 100 ) (x-1)(x-2)(x-3)\cdots(x-100) is q q . Then find the value of sgn ( q ) + ⌊ ∣ q ∣ 1010 ⌋ \text{sgn}(q) + \left\lfloor\frac{\lvert q\rvert}{1010}\right \rfloor .

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The answer is 4.

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Shubhamkar Ayare by Shubhamkar Ayare , 22, India

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1 solution

Md Zuhair
Feb 1, 2017

In the expression ( x − 1 ) ( x − 2 ) ( x − 3 ) . . . ( x − 100 ) (x-1)(x-2)(x-3)...(x-100) . Now coeffiecient of x 99 x^{99} = − ( 1 + 2 + 3 + . . . + 100 ) -(1+2+3+...+100) = − 5050 -5050 .

Now q = − 5050 q = -5050 So sgn ( q ) + ⌊ ∣ q ∣ 1010 ⌋ \text{sgn}(q) + \lfloor\frac{|q|}{1010}\rfloor [Subsituting q] we get s g n ( − 5050 ) sgn(-5050) + ⌊ ∣ − 5050 ∣ 1010 ⌋ \lfloor\frac{|-5050|}{1010}\rfloor = 4 \boxed{4}

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