csat-suneung 2019 Q11

csat-suneung · South-Korea · csat__math-science 3 marks Quadratic trigonometric equations
When $0 \leq \theta < 2 \pi$, the range of all values of $\theta$ such that the quadratic equation in $x$ $$6 x ^ { 2 } + ( 4 \cos \theta ) x + \sin \theta = 0$$ has no real roots is $\alpha < \theta < \beta$. What is the value of $3 \alpha + \beta$? [3 points]
(1) $\frac { 5 } { 6 } \pi$
(2) $\pi$
(3) $\frac { 7 } { 6 } \pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$
When $0 \leq \theta < 2 \pi$, the range of all values of $\theta$ such that the quadratic equation in $x$
$$6 x ^ { 2 } + ( 4 \cos \theta ) x + \sin \theta = 0$$
has no real roots is $\alpha < \theta < \beta$. What is the value of $3 \alpha + \beta$? [3 points]\\
(1) $\frac { 5 } { 6 } \pi$\\
(2) $\pi$\\
(3) $\frac { 7 } { 6 } \pi$\\
(4) $\frac { 4 } { 3 } \pi$\\
(5) $\frac { 3 } { 2 } \pi$