taiwan-gsat 2022 Q12

taiwan-gsat · Other · gsat__math-a 5 marks Discriminant and conditions for roots Proving no real roots exist for a given expression

Let $f ( x ) , g ( x )$ both be real-coefficient polynomials, where $g ( x )$ is a quadratic with positive leading coefficient. It is known that the remainder when $( g ( x ) ) ^ { 2 }$ is divided by $f ( x )$ is $g ( x )$ , and the graph of $y = f ( x )$ has no intersection with the $x$-axis. Select the option that cannot be the $y$-coordinate of the vertex of the graph of $y = g ( x )$.
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2
(5) $\pi$

Let $f ( x ) , g ( x )$ both be real-coefficient polynomials, where $g ( x )$ is a quadratic with positive leading coefficient. It is known that the remainder when $( g ( x ) ) ^ { 2 }$ is divided by $f ( x )$ is $g ( x )$ , and the graph of $y = f ( x )$ has no intersection with the $x$-axis. Select the option that cannot be the $y$-coordinate of the vertex of the graph of $y = g ( x )$.\\
(1) $\frac { \sqrt { 2 } } { 2 }$\\
(2) 1\\
(3) $\sqrt { 2 }$\\
(4) 2\\
(5) $\pi$

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20