Advanced Quizzes — Quiz 18

Questions in this quiz (10)

1. In right triangle $$\triangle ABC$$, the right angle is at vertex $$B$$. Point $$D$$ is on the hypotenuse $$\overline{AC}$$ such that $$BD\perp AC$$. The length of the altitude $$BD$$ is $$12$$, and the length of segment $$\overline{DC}$$ is $$7$$ units greater than the length of segment $$\overline{AD}$$. What is the perimeter of $$\triangle ABC$$?

   • 60

2. A sphere has a great circle represented by the equation $$2x^2-8x+2y^2+20y-50=0$$ A regular tetrahedron is inscribed inside the sphere. What is the volume of the tetrahedron?

   • $$72\sqrt{2}$$
   • $$144\sqrt{2}$$
   • $$216\sqrt{2}$$
   • $$288\sqrt{2}$$

3. If $$(x+y)^2=5xy$$, which of the following is equivalent to the expression $$\frac{x^4+y^4}{xy(x+y)^2}$$?Assume $$x\ne0$$, $$y\ne0$$, and $$x+y\ne0$$.

   • $$1$$
   • $$2$$
   • $$3$$
   • $$\frac{7}{5}$$

4. A circle with center $$O$$ has two tangent lines $$PA$$ and $$PB$$ drawn from an external point $$P$$ to the circle at points $$A$$ and $$B$$, respectively. The measure of angle $$APB$$ is $$50^\circ$$. The length of line segment $$PA$$ is $$12$$ units. What is the area of the region that is both inside quadrilateral $$OAPB$$ and outside the circle, in square units? Round your answer to the nearest hundredth.

   • 31.61

5. The expression $$\sqrt[4]{432}\cdot\sqrt[6]{18}$$ can be rewritten in the form $$a\cdot\sqrt[b]{c}$$ where $$a$$, $$b$$, and $$c$$ are positive integer constants. What is the smallest possible value of $$a+b+c$$?

   • $$13$$
   • $$16$$
   • $$19$$
   • $$30$$

6. A linear function $$f(x)$$ has a slope of $$(3p-2)$$ and a $$y$$-intercept of $$(2p+5)$$, where $$p$$ is a constant. If the graph of $$f(x)$$ passes through the point $$(p,23)$$, what is the positive value of $$p$$?

   • $$6$$
   • $$\frac{7}{3}$$
   • $$4$$
   • $$\sqrt{6}$$

7. The equation $$(mx-2)^2=(kx+1)(3x-p)$$ has infinitely many solutions, where $$m$$, $$k$$, and $$p$$ are constants. What is the value of $$m$$?

   • $$-\frac{3}{2}$$
   • $$-\frac{2}{3}$$
   • $$\frac{3}{2}$$
   • $$\frac{2}{3}$$

8. Data set $$R$$ consists of $$17$$ distinct positive integers. The mean of $$R$$ is $$24$$, and the median of $$R$$ is $$21$$. A new data set $$U$$ is formed by removing the largest integer from $$R$$. Which of the following statements must be true?

   • The mean of $$U$$ is less than the mean of $$R$$.
   • The median of $$U$$ is less than the median of $$R$$.
   • The range of $$U$$ is equal to the range of $$R$$.
   • The standard deviation of $$U$$ is greater than the standard deviation of $$R$$.

9. The shaded region is bounded by the lines shown in the graph. If $$(x,y)$$ is an integer point in this region, what is the maximum possible value of $$x+y$$?

   • $$4$$
   • $$5$$
   • $$6$$
   • $$7$$

10. Which of the following expressions is equivalent to $$\frac{x^4-16}{x^2+8x+16}\cdot\frac{x^2+6x+8}{x^2-4x}$$ for all values of $$x$$ for which the expression is defined?

    • $$\frac{x+2}{x}$$
    • $$\frac{x-2}{x}$$
    • $$\frac{x(x+2)}{x-4}$$
    • $$\frac{(x-2)(x+2)^2(x^2+4)}{x(x-4)(x+4)}$$