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TS EAMCET 11-Sep-2020 Shift 1 Solved Paper

Section: Mathematics

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Question : 65 of 160

Marks: +1, -0

If p(x) be a polynomial satisfying

p(2x)=p'(x)\cdot p''(x)

\sum\limits_{x=1}^{5} p(x)=

200
100
50
450

Solution:

It is given a polynomial satisfying

p(2x)=p'(x)p''(x)

Let the degree of polynomial is

n

, then degree of

p'(x)

and

p''(x)

are

n-1

and

n-2

respectively, so

n=n-1+n-2 \Rightarrow n=3

Now, let the polynomial

p(x)=a x^{3}

\therefore \;\; p(2x)=p'(x) \cdot p''(x) \Rightarrow 8 a x^{3}=(3 a x^{2}) \cdot (6 a x)

\Rightarrow \;\; a=\frac{4}{9}

\therefore

The required polynomial is

p(x) \;\frac{4}{9} x^{3}

Therefore

\sum\limits_{x=1}^{5} p(x)=\;\frac{4}{9}[1^{3}+2^{3}+3^{3}+4^{3}+5^{3}]

=\;\frac{4}{9}\left(\;\frac{5 \times 6}{2}\right)^{2}=100

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