### Question 1

$$(x-6)^{2}+(y+5)^{2}=16$$

In the $$xy-$$plane, the graph of the equation above is a circle. Point $$P$$ is on the circle and has coordinates $$(10,-5)$$. If $$\overline{P Q}$$ is a diameter of the circle, what are the coordinates of point $$Q$$?

A) $$(2,-5)$$

B) $$(6,-1)$$

C) $$(6,-5)$$

D) $$(6,-9)$$

Choice A is correct. The standard form for the equation of a circle is $$(x-h^{2})+(y-k)^{2}$$, where $$(h,k$$) are the coordinates of the centre and $$r$$ is the length of the radius. According to the given equation, the centre of the circle is $$(6,-5)$$. Let $$(x_{1},y_{1})$$ represents the coordinates of point $$Q$$. Since point $$P(10,-5)$$ and point $$Q(x_{1},y_{1})$$ are the endpoints of the diameter of the circle, the centre $$(6,-5)$$ lies on the diameter halfway between $$P$$ and $$Q$$. Therefore, the following relationships hold:

$$\frac{x_{1}+10}{2}=6$$ and $$\frac{y_{1}+(-5)}{2}=-5$$

Solving the equations for $$x_{1}$$ and $$y_{1}$$, respectively, yields $$x_{1}=2$$ and $$y_{1}=-5$$. Therefore, the coordinates of point $$Q$$ are $$(2,-5)$$.

Alternative approach: Since point $$P(10,-5)$$ on the circle and the centre of the circle $$(6,-5)$$ have the same $$y$$-coordinates, it follows that the diameter $$\overline{P Q}$$ must have the same $$y$$-coordinates as $$P$$ and be $$4$$ units away from the centre. Hence, the coordinates of point $$Q$$ must be $$(2,-5)$$.

Choice B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter $$\overline{P Q}$$. If either of these points were point $$Q$$, then $$\overline{P Q}$$ would not be the diameter of the circle.

Choice C is incorrect because$$(6,-5)$$ is the centre of the circle and does not lie on the circle.

### Question 2

A group of $$202$$ people went on an overnight camping trip, taking $$60$$ tents with them. Some tents held $$2$$ people each, and the rest held $$4$$ people each. Assuming all the tents were filled to capacity, and every person got to sleep in a tent, exactly how many of the tents were $$2$$-person tents?

A) $$30$$

B) $$20$$

C) $$19$$

D) $$18$$

Choice C is correct. Let $$x$$ represent the number of $$2$$-person tents and let $$y$$ represent the number $$4$$-person tents. It is given that the total number of tents was $$60$$, and the total number of people in the group was $$202$$. This situation can be expressed as a system of two equations, $$x+y=60$$ and $$2x+4y=202$$.

The first equation can be rewritten as $$y=-x+60$$. Subtracting $$-x++60$$ for $$y$$ in the equation $$2x+4y=202$$ yields $$2x+4(-x+60)=202$$.

Distributing and combining like terms gives $$-2x+240=202$$ and then dividing both sides by $$-2$$ gives $$x=19$$. Therefore, the number of $$2$$-person tents is $$19$$.

Alternate approach: If each of the 60 tents held $$4$$ people, the total number of people that could be accommodated in tents would be $$240$$. However, the actual number of people who slept in tents was $$202$$. The difference of $$38$$ accounts for the $$2$$-person tents. Since each of these tents hold $$2$$ people fewer than a $$4$$-person tent, $$\frac{38}{2}=19$$ gives the number of $$2$$-person tents.

### Question 3 Note: Figure is not drawn to scale In this circle above, point $$A$$ is the centre and the length of $$\overset{\huge\frown}{BC}$$ is $$\frac{2}{5}$$ of the circumference of the circle. What is the value of $$x$$?

The correct answer is 144. In a circle, the ratio of the length of a given arc to the circle’s circumference is equal to the ratio of the measure of the arc, in degrees, to $$360^{\large\circ}$$. The ratio between the arc length and the circle’s circumference is given as $$\frac{2}{5}$$. It follows that $$\frac{2}{5}=\frac{x}{360}$$. Solving this proportion for $$x$$ gives $$x=144$$.

Source of these questions can be found here.