Let E denote the set of all integers a such t | vedclass

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(C) The points of intersection of the parabola $y = x^2 + 2ax + 2021$ and the $x$-axis $(y = 0)$ are given by the roots of the quadratic equation $x^2

Vedclass · Accepted Answer

$1011$

Vedclass · Answer

(C) The points of intersection of the parabola $y = x^2 + 2ax + 2021$ and the $x$-axis $(y = 0)$ are given by the roots of the quadratic equation $x^2 + 2ax + 2021 = 0$.For the roots to be rational,the discriminant $D$ must be a perfect square.$D = (2a)^2 - 4(1)(2021) = 4a^2 - 8084 = 4(a^2 - 2021)$.Thus,$a^2 - 2021$ must be a perfect square,say $\lambda^2$ for some non-negative integer $\lambda$.$a^2 - \lambda^2 = 2021 \Rightarrow (a - \lambda)(a + \lambda) = 2021$.Since $2021 = 43 	imes 47$,we consider the factors of $2021$ as $(1, 2021)$ and $(43, 47)$.For the largest value of $a$,we set $a + \lambda = 2021$ and $a - \lambda = 1$.Adding these equations: $2a = 2022 \Rightarrow a = 1011$.Checking the other case: $a + \lambda = 47$ and $a - \lambda = 43$ $\Rightarrow 2a = 90$ $\Rightarrow a = 45$.The largest element of $E$ is $1011$.

Let $E$ denote the set of all integers $a$ such that the point of intersection of the parabola $y = x^2 + 2ax + 2021$ with the $x$-axis has rational coordinates. The largest element of $E$ is

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