HPC SEM2 practice 2

(Calculator) Given the coordinates [math]A(3,-2), B(4,5)[/math], and [math]C(1,-3)[/math], find the following:

(a) The angle [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math]. Round to the nearest integer.

(b) The area of triangle [math]ABC[/math]

(c) The orthogonal projection of [math]\overrightarrow{AB}[/math] onto [math]\overrightarrow{AC}[/math]

(Calculator) Determine how many triangles can be formed with the given information. If your answer is “infinite”, just write “inf”

(a) [math]m\angle A=36^\circ, m\angle B=71^\circ, b=3[/math]

(b) [math]m\angle A=51^\circ, c=4, b=11[/math]

(c) [math]m\angle B=40^\circ, a=12, b=9[/math]

(d) [math]a=3, b=12, c=6[/math]

(Calculator) Find the direction angle of the vector [math]\overrightarrow{u}=-12\textbf{i}+7\textbf{j}[/math]

The polar function [math]r=4\sin\theta+6\cos\theta[/math] shows a circle. If the center and radius are [math](a, b)[/math] and [math]r[/math] respectively, what is the value of [math]a+b+r^2[/math]?

The polar function [math]r=\dfrac{2}{4\sin\theta-3\cos\theta}[/math] shows a line, what is the slope of the line?

When polynomial [math]f(x)[/math] is divided by [math]x^2-3x+2[/math], the remainder is [math]4x-7[/math]. What is the remainder when [math]P(x)[/math] is divided by [math]x-1[/math]?

Find [math]f(4)[/math] where [math]f(x)=x^5-3x^4-2x^3-x^2-10x+27[/math].

Given that [math]x+2-i[/math] is a factor of [math]f(x)=x^3+x^2-7x-15[/math], what is the [math]x[/math]-intercept of [math]f(x)[/math].

Let [math]f(x)=2x^n+ax^{n-1}+bx^{n-2}+\cdots+7[/math]. If the solutions to the equation [math]f(x)=0[/math] are all rational with mutiplicity 1, what is the maximum of [math]n[/math]?

$$\dfrac{(x-1)(x^2+2)(x^2-4)^2}{(x-1)(x-3)}\geq0$$What is the minimum integer solution to the inequality? If the minimum integer solution does not exist, just write “None”.

Let [math]f(x)[/math] be a quadratic function with [math]y[/math]-intercept [math]-6[/math] and [math]x[/math]-intercepts [math]1[/math] and [math]3[/math]. Find the minimum [math]k[/math] such that [math]g(x)=f(x)-k[/math] does not pass through the first quadrant.

If the range of quadratic function [math]f(x)=mx^2+mx+5[/math] is [math][0,\infty)[/math] where [math]m[/math] is a constant, what is the value of [math]m[/math]? If you have two or more answers, write the smallest one.

Find the real part of the number [math]\dfrac{z^2-6z+11}{z^*}[/math] where [math]z=3-i[/math]

Mr. Fang wrote two functions [math]f(x)=\dfrac{(x^2-1)(x-3)}{x(x-3)^2}[/math] and [math]g(x)=1+\dfrac{3x-1}{x^2-3x}[/math] on a board and askes his students to find their difference. Whose answer(s) is/are correct?

Mr. Fang wrote two functions [math]f(x)=(1+x)(2-x)^2(3-x)^3(4+x)^4[/math] and [math]g(x)=xf(x)[/math] on a board and askes his students to find their difference. Whose answer(s) is/are correct?

Find the remainder of [math]x^{1000}+x^{500}+x^{125}+x^5+1[/math] is divided by [math]g(x)=x^2-1[/math]

Mr. Fang is a farmer. He feeds many wolves and sheep. All the wolves and sheep stay in the same area, but to prevent the wolves from eating the sheep, Mr. Fang constructs a fence to separate the two, as shown by the figure on the right. Suppose the fence is 120 meters long. Find the maximum area, in square meters, of his animal’s activity range.

The curve to the right shows the graph of [math]f(x)=ax^2+bx+c[/math]. Which of the following numbers is/are positive?