Advanced Quizzes — Quiz 6

Questions in this quiz (10)

1. If $$\frac{\sqrt[3]{x^2}}{\sqrt[8]{x^5}}=x^{\frac{a}{b}}$$ for all positive values of $$x$$, what is the value of $$\frac{a}{b}$$?

   • $$\frac{1}{24}$$
   • $$\frac{1}{4}$$
   • $$\frac{3}{25}$$
   • $$\frac{2}{3}$$

2. The function $$p$$ is defined by $$p(x)=a((x+6)^2-b)((x+6)^2-c)$$ where $$a$$, $$b$$, and $$c$$ are constants. In the $$xy$$-plane, the graph of $$y=p(x)$$ passes through the points $$(-7,49)$$ and $$(0,784)$$. What is the value of $$p(-12)+p(-5)$$?

   • 833

3. $$2\cos(90-a)^\circ\cdot\cos b^\circ+\sin(a+y)^\circ\cdot\sin(90-b)^\circ$$In the given expression, $$a$$ and $$b$$ are constants such that $$\sin a^\circ=0.50$$, $$\cos b^\circ=0.99$$, and $$0$$ < $$b$$ < $$90$$. What is the value of the expression when $$y=0$$?

   • $$1.758$$
   • $$1.485$$
   • $$2.325$$
   • $$2.675$$

4. Function $$f$$ is defined by $$f(x)=a^x+b$$, where $$a$$ and $$b$$ are constants. In the $$xy$$-plane, the graph of $$y=f(x)-11$$ has a $$y$$-intercept at $$(0,-\frac{29}{4})$$. The product of $$a$$ and $$b$$ is $$\frac{66}{4}$$. What is the value of $$a$$?

   • 6

5. Consider the system$$2x+3y=7$$$$10x+15y=35$$For each real number $$r$$, which of the following points lies on the graph of each equation in the $$xy$$-plane for the given system?

   • $$\left(\frac{r}{5}+7,-\frac{r}{5}+35\right)$$
   • $$\left(-\frac{3r}{2}+\frac{7}{2},r\right)$$
   • $$\left(r,\frac{2r}{3}+\frac{7}{3}\right)$$
   • $$\left(r,-\frac{3r}{2}+\frac{7}{2}\right)$$

6. In the equation $$(3x+p)(5x^2-45)(2x^2-16x+6p)=0$$ $$p$$ is a positive constant. The sum of the solutions to the equation is $$\frac{20}{3}$$. What is the value of $$p$$?

   • 4

7. The given polynomial $$54x^4+216x^2+210$$ has factors in the form $$k(ax^2+b)(cx^2+d)$$ If $$a$$, $$b$$, $$c$$, $$d$$, and $$k$$ are integers, what is the smallest possible value of $$ab$$?

   • 15

8. For the function $$f$$, for each increase in the value of $$x$$ by $$c$$, where $$c$$ is a positive constant, the value of $$f(x)$$ increases by a factor of $$27$$. Which of the following equivalent forms of the function $$f$$ displays $$\frac{1}{c}$$ as a coefficient of $$x$$?

   • $$f(x)=48(3)^{\frac{1}{2}x}$$
   • $$f(x)=48(3^3)^{\frac{1}{c}x}$$
   • $$f(x)=48(9)^{\frac{1}{4}x}$$
   • $$f(x)=48\left(27^{\frac{1}{3}x}\right)^{\frac{1}{2}}$$

9. In the given equation, $$57x^2+(57b+a)x+ab=0$$ where $$a$$ and $$b$$ are positive constants. The product of the solutions to the given equation is $$kab$$, where $$k$$ is a constant. What is the value of $$k$$?

   • $$\frac{1}{57}$$
   • $$\frac{1}{19}$$
   • $$1$$
   • $$57$$

10. In the given figure, $$\theta$$ is an angle. If $$\sin\theta=\frac{\sqrt{3}}{2}$$, what is $$\cos\theta$$?

    • $$\frac{\sqrt{3}}{2}$$
    • $$\frac{1}{2}$$
    • $$-\frac{1}{2}$$
    • $$-\frac{\sqrt{3}}{2}$$