isi-entrance 2021 Q5

isi-entrance · India · UGB Roots of polynomials Polynomial evaluation, interpolation, and remainder

Let $a _ { 0 } , a _ { 1 } , \cdots , a _ { 19 } \in \mathbb { R }$ and
$$P ( x ) = x ^ { 20 } + \sum _ { i = 0 } ^ { 19 } a _ { i } x ^ { i } , \quad x \in \mathbb { R }$$
If $P ( x ) = P ( - x )$ for all $x \in \mathbb { R }$, and
$$P ( k ) = k ^ { 2 } , \text{ for } k = 0,1,2 \cdots , 9$$
then find
$$\lim _ { x \rightarrow 0 } \frac { P ( x ) } { \sin ^ { 2 } x }$$

Let $a _ { 0 } , a _ { 1 } , \cdots , a _ { 19 } \in \mathbb { R }$ and

$$P ( x ) = x ^ { 20 } + \sum _ { i = 0 } ^ { 19 } a _ { i } x ^ { i } , \quad x \in \mathbb { R }$$

If $P ( x ) = P ( - x )$ for all $x \in \mathbb { R }$, and

$$P ( k ) = k ^ { 2 } , \text{ for } k = 0,1,2 \cdots , 9$$

then find

$$\lim _ { x \rightarrow 0 } \frac { P ( x ) } { \sin ^ { 2 } x }$$

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8