Advanced Quizzes — Quiz 16

Questions in this quiz (10)

1. For all positive real numbers $$a$$ and $$b$$, the expression $$\frac{a^2-b^2}{a^{4/3}+a^{2/3}b^{2/3}+b^{4/3}}$$ is equivalent to which of the following?

   • $$a^{2/3}-b^{2/3}$$
   • $$a^{1/3}+b^{1/3}$$
   • $$a^{2/3}+b^{2/3}$$
   • $$a^{1/3}-b^{1/3}$$

2. A right circular cone has a height of $$15$$ units and a base radius of $$10$$ units. A plane parallel to the base cuts the cone at a point $$6$$ units above the base, forming a smaller cone and a frustum. What is the volume, in cubic units, of the frustum?

   • $$30\pi$$
   • $$392\pi$$
   • $$150\pi$$
   • $$300\pi$$

3. The system of equations $$y=x^2-8x+20$$ $$y=4x+k$$ has exactly one real solution, where $$k$$ is a constant. What is the value of $$k$$?

   • -16

4. A university surveyed its students about whether they participate in varsity sports and whether they have part-time jobs. The results are summarized in the table below.If a student is chosen at random from those who do not have a part-time job, what is the probability that the student is not a varsity athlete?

   • $$\frac{11}{19}$$
   • $$\frac{19}{40}$$
   • $$\frac{33}{57}$$
   • $$\frac{66}{240}$$

5. The graph of the line $$ax+by=30$$ in the $$(x,y)$$-plane passes through the point $$(4,3)$$. If the $$x$$-intercept of the line is $$3$$ times its $$y$$-intercept, where $$a\ne0$$ and $$b\ne0$$, what is the value of $$b$$?

   • $$3$$
   • $$\frac{15}{4}$$
   • $$5$$
   • $$\frac{90}{13}$$

6. The function $$f$$ is defined by $$f(x)=px^2+qx+r$$, where $$p$$, $$q$$, and $$r$$ are constants and $$0$$ < $$q$$ < $$12$$. The graph of $$y=f(x)$$ is a parabola such that $$f(-4)=f(8)$$. If $$q$$ is an integer, what is the greatest possible value of $$q-p$$?

   • $$\frac{11}{4}$$
   • $$\frac{1}{2}$$
   • $$\frac{55}{4}$$
   • $$\frac{1}{4}$$

7. In $$\triangle XYZ$$, point $$M$$ lies on $$\overline{YZ}$$ such that $$XM$$ bisects $$\angle YXZ$$. If $$MN\parallel XY$$ and $$MP\parallel XZ$$, where $$N$$ is on $$\overline{XZ}$$ and $$P$$ is on $$\overline{XY}$$, and $$\angle YXZ=80^\circ$$ and $$\angle XYZ=50^\circ$$, what is the measure of $$\angle NMP$$?

   • $$40^\circ$$
   • $$50^\circ$$
   • $$80^\circ$$
   • $$100^\circ$$

8. A circle lies entirely in the first quadrant of the $$(x,y)$$-plane and is tangent to both axes. The circle is also tangent to the line $$5x+12y=180$$ What is the sum of all possible values for the radius of the circle?

   • $$12$$
   • $$24$$
   • $$36$$
   • $$51$$

9. The numbers $$a$$, $$b$$, $$c$$, $$m$$, and $$n$$ are positive constants, where $$b$$ > $$c$$. The number $$a$$ is $$500\%$$ greater than the positive difference between $$b$$ and $$c$$, and $$c$$ is $$80\%$$ of $$b$$. If $$a$$ is $$m\%$$ greater than $$c$$, and $$c$$ is $$n\%$$ of $$a$$, what is the product of $$m$$ and $$n$$? Round your answer to the nearest integer.

   • 3333

10. In $$\triangle ABC$$, $$BD$$ is an angle bisector of $$\angle ABC$$, with point $$D$$ on side $$AC$$. A line is drawn through $$D$$ parallel to $$BC$$, intersecting side $$AB$$ at point $$E$$. If $$\angle ABD=35^\circ$$ and $$\angle BDC=115^\circ$$, what is the measure of $$\angle AED$$?

    • $$65^\circ$$
    • $$70^\circ$$
    • $$80^\circ$$
    • $$115^\circ$$