HPC Q2-Q3 quiz

Solve the equation [math]\sin2x=\sqrt{3}\cos2x[/math].

Solve the equation [math]3\cos^2x-\cos2x=1[/math]

(Calculator) Consider a triangle [math]ABC[/math] with [math]AB=10, AC=8, \angle B=45^\circ[/math]. Find [math]BC[/math]. If you have two answers, write the biggest one.

There is a fort that is surrounded by mines in an 80 meters radius. A Soviet tank KV-2 measures the fort on a bearing of [math]060^\circ[/math]. After moving 50 meters to the east, it takes another measurement, and finds the fort on a bearing of [math]045^\circ[/math]. Suppose KV-2 keeps heading east. Can KV-2 avoid the mines? Just write “Yes” or “No”.

(Calculator) An aged dinosaur watches a meteor falling vertically from the sky. Initially, the dinosaur’s angle of elevation is [math]55^\circ[/math]. After 10 minutes, the angle of elevation changes to [math]40^\circ[/math]. Assuming the horizontal distance between the meteor and the dinosaur is 1500 meters, after how many more minutes will the meteor hit the ground, causing dinosaur extinction? Round to integer.

(Calculator) Three forces [math]F_1, F_2[/math], [math]F_3[/math] are applied on an object. The figure shows [math]F_1[/math] and [math]F_2[/math] and the object. The magnitudes of [math]F_1[/math] and [math]F_2[/math] are 3 and 5 respectively. If the resultant force of the object is zero, find the magnitude and direction angle of [math]F_3[/math].

Let [math]A(1,3), B(-2,-1)[/math], and [math]C(0,5)[/math] be three points.

(a) Find [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math].

(b) If [math]\cos\theta=\dfrac{a}{\sqrt{b}}[/math] where [math]a[/math] and [math]b[/math] are positive integers and [math]\theta[/math] is the angle between [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math], what is the value of [math]\dfrac{a}{b}[/math]?

(c) Find the area of triangle [math]ABC[/math].

(d) Find the unit vectors [math]\overrightarrow{u}[/math] parallel to [math]\overrightarrow{AB}[/math].

(e) If [math]\dfrac{1}{\sqrt{p}}(q\textbf{i}-r\textbf{j})[/math] is a unit vector perpendicular to [math]\overrightarrow{AC}[/math] where [math]p, q[/math], and [math]r[/math] are positive integers, what is the value of [math]p+q+r[/math]?

Use the synthetic division to find the quotient of [math]\dfrac{f(x)}{g(x)}[/math] where [math]f(x)=4x^5+x^3-7x^2+2x-3[/math] and [math]g(x)=2x+1[/math].

When polynomial [math]f(x)=x^3+2x^2+ax+b[/math] is divided by [math]x-1[/math], the remainder is 4; when [math]f(x)[/math] is divided by [math]x+2[/math], the remainder is 16. Find [math]a[/math] and [math]b[/math].

Consider function [math]f(x)=12x^4+ax^3+bx^2+cx+10[/math] where [math]a, b[/math], and [math]c[/math] are integers. Which of the following cannot be the factor of [math]f(x)[/math]?

How many integer solutions are there to the equation [math]14x^4-8x^3+28x=3[/math]?

Let [math]f(a)[/math] be a real polynomial with zeros [math]3+\sqrt{2}i, -4[/math], and [math]0[/math]. If the leading coefficient of [math]f(x)[/math] is 1, what is [math]f(1)[/math]?

The figure shows the graph of polynomial function [math]f(x)[/math]. Which of the following could be [math]f(x)[/math]?

How many negative integer solutions does the inequality [math]\displaystyle{\frac{(x^3+1)(x+3)x}{x^4-1}}\geq0[/math] have?

Let [math]f(x)=\dfrac{(x-1)^2(x+2)(2x+3)}{(x-1)(x+2)^2(x-3)}[/math]. Which of the following statements is true?