Advanced Quizzes — Quiz 1

Questions in this quiz (10)

1. The graph of the quadratic function $$f(x)=ax^2+bx+c$$ has vertex at $$(4,-18)$$ and passes through the point $$(7,9)$$. What is the value of $$a+b+c$$?

   • $$-15$$
   • $$-9$$
   • $$3$$
   • $$9$$

2. The function $$f$$ is defined by $$f(x)=a(x-h)^2+k$$ where $$a$$, $$h$$, and $$k$$ are constants. The graph of $$y=f(x)$$ has vertex at $$(-5,2)$$. Additionally, it is known that $$f(-2)$$ > $$f(-8)$$. Which of the following must be true?

   • $$f(1)$$ > $$f(3)$$
   • $$f(4)$$ > $$f(6)$$
   • $$f(-1)$$ > $$f(-3)$$
   • $$f(-2)=f(-8)$$

3. For the quadratic equation $$kx^2-(14w)x+49k=0$$ suppose it has exactly one real solution and $$kw=400$$. What is the value of $$|k+w|$$?

   • 40

4. The system of equations below has two distinct solutions $$(x_1,y_1)$$ and $$(x_2,y_2)$$.$$y=x^2-6x+13$$$$y=kx+9$$If $$x_1+x_2=14$$, what is the value of $$k$$?

   • $$6$$
   • $$8$$
   • $$10$$
   • $$12$$

5. In the system:$$\frac{a}{6}x+\frac{7}{12}=\frac{b}{6}y-\frac{5}{12}$$$$-30x+8y=9-4y$$where $$a$$ and $$b$$ are constants, the system has no solution. The ratio $$a:b$$ is equivalent to $$18:k$$. What is the value of $$k$$?

   • $$\frac{36}{5}$$
   • $$\frac{7}{5}$$
   • $$\frac{12}{7}$$
   • $$7$$

6. Consider the system of equations:$$(x-4)^2+(y+2)^2=49$$$$y=3x-k$$where $$k$$ is a constant. If the system has exactly one real solution $$(x,y)$$, what is the sum of all possible values of $$k$$?

   • 28

7. A project requires between 540 and 900 labor-hours, inclusive. A team consists of 8 senior workers and 5 junior workers. Each senior worker contributes between 32 and 48 hours per week. Each junior worker contributes between 20 and 30 hours per week. If the project is completed in exactly $$w$$ weeks, which inequality represents the possible values of $$w$$?

   • $$\frac{90}{89}\le w\le\frac{225}{89}$$
   • $$\frac{45}{59}\le w\le\frac{45}{29}$$
   • $$\frac{15}{29}\le w\le\frac{90}{59}$$
   • $$\frac{90}{59}\le w\le\frac{15}{29}$$

8. From an external point $$P$$, two tangent segments, $$PA$$ and $$PB$$, are drawn to a circle with center $$O$$. The radius of the circle is $$18$$. The length of chord $$AB$$ is $$24$$. What is the distance between point $$O$$ and point $$P$$?

   • $$18$$
   • $$24$$
   • $$\frac{54\sqrt{5}}{5}$$
   • $$36$$

9. A circle in the $$xy$$-plane has the equation $$x^2+y^2-4x+8y-20=0$$ What is the length of a tangent segment drawn from the point $$(11,-4)$$ to the circle?

   • $$5$$
   • $$\sqrt{41}$$
   • $$4\sqrt{2}$$
   • $$3\sqrt{5}$$

10. In the $$xy$$-plane, a circle has its center at the origin. An external point $$P$$ is located at $$(16,0)$$. Two lines tangent to the circle are drawn from point $$P$$, touching the circle at points $$A$$ and $$B$$. The measure of the angle formed by these two tangent lines, $$\angle APB$$, is $$120^\circ$$. What is the radius of the circle?

    • $$4$$
    • $$8$$
    • $$8\sqrt{3}$$
    • $$16$$