Math Skills Analysis

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

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

In the figure above, [math]AB=25[/math] and [math]BC=15[/math]. What is [math]DB[/math]?

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

In function [math]f[/math] above, [math]a[/math] and [math]b[/math] are positive constants. If

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

where [math]n[/math] is a constant, what is the value of [math]b[/math]?

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

If [math]f(m)=f(4)[/math], what is [math]m[/math]?

There are three distinct numbers [math]a, b[/math], and [math]c[/math] on a number line. If [math]|a-b|=|b-c|[/math], 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 [math]k\sqrt{3}[/math] where [math]k[/math] is a rational number, what is the value of [math]k[/math]?

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

If [math]x-2[/math] and [math]x-3[/math] are the factors of the polynomial above, what is the value of [math]a[/math]?

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 [math]a\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the minimum of [math]b[/math]?

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

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

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