HPC SEM2 practice 1

Let [math]f(x)[/math] be a quadratic function with vertex [math](k,0)[/math] where [math]k[/math] is a positive number. The [math]y[/math]-intercept of [math]f(x)[/math] is [math]k[/math]. If [math]f(ak-b\sqrt{k})=1[/math] where [math]a[/math] and [math]b[/math] are positive constants, what is the value of [math]a+b[/math]?

Let [math]f(x)=x^7-4x^6+3x^4-7x^3+8x^2+10[/math]. Evaluate [math]f(2)[/math].

If a polynomial [math]x^4+ax^3+bx^2+cx+6[/math] has 4 distinct rational zeros, what is the value of [math]a+b+c[/math]?

Consider a polynomial [math]f(x)=x^4+4x^3+2x^2-12x-15[/math]. If [math]f(-2+i)=f(-\sqrt{a})=0[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

Which of the following numbers is/are a solution to the rational inequality [math]\dfrac{(x-2)(x+2)(1-x)^3}{x(2x+1)(3-x)^2}<0[/math]?

Which of the following functions best represents the graph above?

Find the limits: [math]\displaystyle{\lim_{x\to0}\dfrac{1-\sec x}{x}}[/math]

Consider a function [math]f(x)=e^{\cos x}[/math] with domain [math]\left[0,\dfrac{\pi}{2}\right][/math]. How many possible integers [math]a[/math] such that [math]f(x)=a[/math] must have a solution?

Which of the following is the derivative of [math]f(x)=\tan x[/math]?

(Calculator) In triangle [math]ABC[/math], [math]b=6, c=9[/math], and [math]\angle B=30^\circ[/math]. Find [math]\angle C[/math]. If you have two answers, write the bigger one.

Consider a triangle [math]ABC[/math] that satisfies [math]a=12, b=10[/math], and [math]c=8[/math]. If the area of [math]ABC[/math] can be written as [math]a\sqrt{b}[/math] where [math]b[/math] is a prime number, what are [math]a [/math] and [math]b[/math]?

(Calculator) Find the direction angle and magnitude of [math]\overrightarrow{a}=2\textbf{i}-7\textbf{j}[/math]

If [math]\overrightarrow{u}=3\textbf{i}-7\textbf{j}[/math] and [math]\overrightarrow{v}=k\textbf{i}-12\textbf{j}[/math] are parallel, what is the value of [math]k[/math]?

(Calculator) What force is required to keep a 1000 kg vehicle from rolling down a ramp inclined at [math]10^\circ[/math] from the horizontal? Acceleration due to gracity is 9.8 [math]m/s^2[/math], so the car has a force due to gravity of approximately 10,000 N ([math]kgm/s^2[/math])

Consider a vector [math]\overrightarrow{a}=-6\textbf{i}+\textbf{j}[/math]. If [math]\overrightarrow{b}[/math] is a unit vector with direction angle [math]\theta[/math] such that [math]\overrightarrow{a}[/math] is perpendicular to [math]\overrightarrow{b}[/math] where [math]\pi<\theta<2\pi[/math], what is the [math]x[/math]-component of [math]\overrightarrow{b}[/math]?

(Calculator) Let [math]\overrightarrow{u}=2\textbf{i}+3\textbf{j}[/math] and [math]\overrightarrow{v}=-\textbf{i}+4\textbf{j}[/math] be two vectors. Find [math]\|\text{proj}_{\overrightarrow{u}}\overrightarrow{v}\|[/math]

Let [math]A(0,0), B(1,3)[/math], and [math]C(7,-1)[/math] be three points, what is the area of triangle [math]ABC[/math]?

Consider a polar curve [math]r=\dfrac{6}{2\cos\theta+3\sin\theta}[/math]

(a) The curve shows a
(A) circle
(B) line
(C) petal
(D) heart
(E) None of the above

(b) Find the [math]y[/math]-intercept(s) of the curve. If you have two or more answers, write the smallest one.

Consider a polar function [math]r(\theta)=4\cos(2\theta)+1[/math]. If the distance between two points [math]P=\left[r\left(\dfrac{3\pi}{2}\right),\dfrac{3\pi}{2}\right][/math] and [math]Q=[r(\pi), \pi][/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?