KVPY SX SB Exam 19-Nov-2017 Question Paper

Question : 6 of 120

Marks: +1, -0

Let n > 2 be an integer and define a polynomial

p(x) = x^n + a_{n–1} \ x ^{n–1} + ...... + a_1x + a_0

where

a_0, a_1, ........ a_{n–1}

are integers. Suppose we know that np(x) = (1 + x)p'(x). If b = p(1), then

Solution:

n[x^n + a_{n-1} \ x^{ n-1} + a_{n-2}\ x ^{n-2} -a_1x + a_0]

=(1 + x) (nx^{n-1} + a _{n-1}{(n -1)}x ^{n-2}

+ a_{n-2}\ x ^{n-3 }(n -2)

+ a_{n-3}(n-3)x^{n-4} + .......)

compare coefficient of

x^{n-1}

na_{n-1} = (n-1)a_{n-1} + n

Solve

a_{n-1} = n \or ^nC_1

compare coefficient of

x^{n–2}

na_{n-2} = (n-2)a_{n-2} + (n-1)a_{n-1}

a_{n-2}={n(n-1)}/2=^nC_2

similarly

a_{n-3} = ^nC_2

& So .....on

b = P(1) = 1 + a_{n-1 }+ a_{n-2} + ..... a_1

+ a_0

=^ nC_0 +^ nC_1 + ^nC_2 +....... ^nC_n = 2 n

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