Advanced Quizzes — Quiz 22

Questions in this quiz (10)

1. If $$(x+y)^3=7xy(x+y)$$, which of the following is equivalent to $$\frac{x^4+y^4}{xy(x+y)^2}$$?Assume $$x\ne0$$, $$y\ne0$$, and $$x+y\ne0$$.

   • $$3$$
   • $$7$$
   • $$14$$
   • $$\frac{23}{7}$$

2. In $$\triangle ABC$$, $$\angle ABC$$ is a right angle. Point $$D$$ lies on hypotenuse $$AC$$ such that $$BD$$ is an altitude to the hypotenuse. The altitude $$BD$$ has length $$15$$, and the difference between $$DC$$ and $$AD$$ is $$16$$. What is the perimeter of $$\triangle ABC$$?

   • $$34\sqrt{34}$$
   • $$34+8\sqrt{34}$$
   • $$34$$
   • $$8\sqrt{2}$$

3. $$\frac{ax^4+bx^3+cx^2+dx+e}{(x^2-9)(x-2)}=\frac{5}{x-3}+\frac{2x-1}{x+3}$$The equation above is true for all values of $$x$$ for which the expressions are defined. What is the value of $$a+b+d+e$$?

   • -12

4. In the $$xy$$-plane, a line with equation $$y=3x+k$$ is tangent to the circle $$x^2+y^2=45$$. The system has exactly one solution at a point $$(a,b)$$, where $$a$$ > $$0$$ and $$b$$ < $$0$$. What is the value of $$k$$?

   • $$-15\sqrt{2}$$
   • $$-\sqrt{10}$$
   • $$\sqrt{10}$$
   • $$15\sqrt{10}$$

5. In the $$xy$$-plane, a circle passes through the points $$(2,1)$$ and $$(8,5)$$. The center of the circle lies on the line $$y=2x-7$$. What is the radius of the circle?

   • $$2\sqrt{5}$$
   • $$4\sqrt{2}$$
   • $$5$$
   • $$\sqrt{13}$$

6. If $$x\ne0$$ and $$a=x+\frac{2}{x}$$, which of the following expressions is equivalent to $$x^2+\frac{4}{x^2}$$?

   • $$a^2-2a$$
   • $$a^2-4$$
   • $$a^2-8$$
   • $$a^2+4$$

7. The polynomial $$P(x)=x^3+Ax^2+Bx-40$$ has a factor of $$(x-5)$$, and $$\frac{P(x)}{x-5}=x^2+4x+C$$ for all values of $$x$$ for which the expression is defined. What is the value of $$A+B+C$$?

   • $$-19$$
   • $$-11$$
   • $$-5$$
   • $$9$$

8. The magnitude of the gravitational force between two objects is given by $$F=G\frac{m_1m_2}{d^2}$$, where $$G$$ is a constant, $$m_1$$ and $$m_2$$ are the masses of the objects, and $$d$$ is the distance between their centers of mass. In a first system, two objects have masses $$2M$$ and $$5M$$ and are separated by distance $$D$$. In a second system, the masses become $$6M$$ and $$10M$$, and the distance becomes $$\frac{1}{3}D$$. What is the ratio of the gravitational force in the second system to the gravitational force in the first system?

   • $$9:1$$
   • $$18:1$$
   • $$27:1$$
   • $$54:1$$

9. A circle in the $$xy$$-plane has equation $$(x-4)^2+(y+1)^2=50$$. From an external point $$P(13,5)$$, a tangent segment $$PT$$ is drawn to the circle at point $$T$$. What is the length of $$PT$$?

   • $$5\sqrt{2}$$
   • $$6\sqrt{2}$$
   • $$8\sqrt{2}$$
   • $$\sqrt{67}$$

10. The box plots shown summarize two different data sets, Data Set A and Data Set B. Which of the following statements correctly compares the standard deviations of Data Set A and Data Set B?

    • The standard deviation of Data Set A is greater than the standard deviation of Data Set B.
    • The standard deviation of Data Set B is greater than the standard deviation of Data Set A.
    • The standard deviations of Data Set A and Data Set B are equal.
    • It is impossible to compare the standard deviations of Data Set A and Data Set B from the given box plots.