Math Skills Analysis

$$2x^2+3x-6=0$$

If $p$ and $q$ are two solutions of the equation above, what is $\dfrac{1}{p}+\dfrac{1}{q}$?

In the figure above, $AB=25$ and $BC=15$. What is $DB$?

$$f(x)=a\cdot b^x$$

In function $f$ above, $a$ and $b$ are positive constants. If

$$f(n+2)=f(n)+\dfrac{1}{3}f(n+2)$$

where $n$ is a constant, what is the value of $b$?

$$f(x)=-3(x-5)(x+7)$$

If $f(m)=f(4)$, what is $m$?

There are three distinct numbers $a, b$, and $c$ on a number line. If $|a-b|=|b-c|$, which of the following statements must be true?

An equilateral triangle inscribed in a circle. If the length of the triangle is 12 and the radius of the circle is $k\sqrt{3}$ where $k$ is a rational number, what is the value of $k$?

$$x^3+ax^2+x+b$$

If $x-2$ and $x-3$ are the factors of the polynomial above, what is the value of $a$?

The figure above shows a sector with radius 15 and a rectangle. The area of the rectangle is 45. If the perimeter of the rectangle is $a\sqrt{b}$ where $a$ and $b$ are integers, what is the minimum of $b$?

$$3px^2-27qx+3p=0$$

In the equation above, $p>0$ and $q<0$ are constants. If the equation has two real solutions, how many possible integer values does $\dfrac{p}{q}$ have?

The figure above shows sector OAB with radius 2. If $\angle AOB=1$, what is the area of the shaded region?