csat-suneung 2016 Q10

csat-suneung · South-Korea · csat__math-A 3 marks Discriminant and conditions for roots Parameter range for specific root conditions (location/count)

For a sequence $\left\{ a _ { n } \right\}$, the curve $y = x ^ { 2 } - ( n + 1 ) x + a _ { n }$ intersects the $x$-axis, and the curve $y = x ^ { 2 } - n x + a _ { n }$ does not intersect the $x$-axis. What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n ^ { 2 } }$? [3 points]
(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$

For a sequence $\left\{ a _ { n } \right\}$, the curve $y = x ^ { 2 } - ( n + 1 ) x + a _ { n }$ intersects the $x$-axis, and the curve $y = x ^ { 2 } - n x + a _ { n }$ does not intersect the $x$-axis.\\
What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n ^ { 2 } }$? [3 points]\\
(1) $\frac { 1 } { 20 }$\\
(2) $\frac { 1 } { 10 }$\\
(3) $\frac { 3 } { 20 }$\\
(4) $\frac { 1 } { 5 }$\\
(5) $\frac { 1 } { 4 }$

Paper Questions

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